Theory and Applications of Categories, Vol. 3, No. 4, 1997, pp. 50–84.

LAX OPERAD ACTIONS AND COHERENCE FOR MONOIDAL N-CATEGORIES, A_{∞} RINGS AND MODULES

GERALD DUNN

Transmitted by James Stasheff

ABSTRACT. We establish a general coherence theorem for lax operad actions on an n-category which implies that an n-category with such an action is lax equivalent to one with a strict action. This includes familiar coherence results (e.g. for symmetric monoidal categories) and many new ones. In particular, any braided monoidal n-category is lax equivalent to a strict braided monoidal n-category. We also obtain coherence theorems for A_{∞} and E_{∞} rings and for lax modules over such rings. Using these results we give an extension of Morita equivalence to A_{∞} rings and some applications to infinite loop spaces and algebraic K-theory.

Introduction

The purpose of this paper is to introduce the idea of a lax operad action on an n-category and to show how it can be used to simplify the coherence conditions for an n-category with an algebraic structure. This approach leads to coherence theorems for lax monoidal n-categories (Theorems 1.7, 3.13 and 3.14). Coherence for braidings and other forms of higher commutativity for lax monoidal structures are also covered by these theorems.

It is well known that the categories Mon , SymMon and BrMon of monoidal, symmetric monoidal and braided monoidal categories each has the structure of a 2-category. We show in Theorem 1.6 that the 2-category structure can be described more simply in terms of lax braided cat-operad actions. For example, there is a braided cat-operad B̃ (see 1.2) such that giving a lax action by B̃ on a category A is equivalent to giving a braided monoidal structure on A. Similarly, a strict B̃ action on A corresponds to a strict braided monoidal structure on A. Thus a coherence theorem for braided monoidal categories can be approached via B̃ actions. The monoidal and symmetric monoidal cases arise from different choices of operad. In Theorem 1.7 we establish a coherence result for lax C-categories for any braided cat-operad C. In particular this proves coherence for monoidal, symmetric monoidal and braided monoidal categories.

In section 2 we extend these results to categories with a coherent ring structure which we call lax G-ring categories. The main coherence result here, Theorem 2.5, implies that a braided (resp. symmetric) bimonoidal category is lax equivalent to a strict braided bimonoidal (resp. bipermutative) category. We also give some applications of Theorem 2.5. For example we show that the K-theory of a lax G-ring category is a ring spectrum (Theorem 2.8).

Received by the editors 1995 September 22 and, in revised form, 1996 September 10.

Published on 1997 February 28

1991 Mathematics Subject Classification : 18C15, 18D05, 18D10, 19D23, 55P47, 55U40.

Key words and phrases: braided monoidal n-category, operad, ring spectrum, A_{∞} ring, Morita equivalence.

\begin{matrix}c \\ ◯ \end{matrix} Gerald Dunn 1997. Permission to copy for private use granted.

In [8] Kapranov and Voevodsky define the notions of lax monoidal 2-category and of braiding of such an object. They show that this is a natural setting for defining Yang-Baxter and tetrahedra equations and that lax structures provide solutions for these equations. The idea of lax operad action allows these structures to be defined in a much simpler and more conceptual way (see Remarks 3.10). In section 3 we define lax monoidal n-category in this way and establish some coherence results, Theorems 3.13 and 3.14. Not only does this point of view simplify some of the basic notions of [8], but it also seems to be the proper setting for higher dimensional Yang-Baxter equations.

The proofs of the coherence theorems in sections 1 through 3 follow the same pattern since in each case the coherence conditions are encoded by a lax operad action. For example, the standard coherence result for monoidal categories due to Isbell (see [12;4.2]) is the special case of Theorem 1.7 for the trivial A_{∞} operad M, [11]. Essentially, the proof of Theorem 1.7 (in the monoidal case) is the proof of [12;4.2] “paramaterized” by the operad M. Theorem 1.7 follows from the observation that this works more generally for any braided cat-operad and some additional considerations for the 2-category structure. Similarly, Theorem 2.5 is a parameterized version of [13;VI,3.5]. This idea also extends to n-categories with a coherent algebraic structure (Theorems 3.13 and 3.14). All ncategories considered in this paper are assumed to be topological (see 3.1) because of the following considerations. If C is a braided cat-operad and the n-category A has a lax C action, then our coherence theorems give an n-category PA having a strict C action which is lax C-equivalent to A. The n-category PA will not in general be topologically discrete even if A is. Moreover, the applications in section 2 and elsewhere require the topological structure. However, the reader not interested in the topological case need only assume that the categories C_{j} comprising C are topologically discrete, for then PA will be discrete whenever A is. In particular there are nontopological versions of the coherence theorems for monoidal, braided monoidal and symmetric monoidal n-categories.

In section 4 we define (lax) modules over A_{∞} and E_{∞} rings R and develop some of their basic properties. The coherence conditions for a lax R-module are not encoded by an operad action, but can be expressed very simply using k-symmetric monoidal functors and natural transformations. These are functors and natural transformations of kvariables that are symmetric monoidal in each variable and satisfy some additional compatibility conditions. We rely heavily on k-symmetric monoidal structures throughout this section.

The main goal is to set up a useful 2-category of modules having tensor product and hom 2-functors with reasonable properties. The first step is to prove a coherence result (Theorem 4.7). This and the use of k-symmetric monoidal structures makes the construction of tensor product and hom 2-functors more manageable. Next, one would like to establish some basic properties, such as adjointness, but virtually all of the expected results are unavailable using just the 2-category structure. For example, the tensor and hom 2-functors are lax (but not strictly) 2-adjoint, so we must also consider the 3-category structure. The relevant terminology is given in Definition 4.9. It should be pointed out that all of the coherence results of this paper, except Theorem 3.13, also refer to a 3category structure, though this has been suppressed (through Theorem 4.7) for the sake of concreteness. See the discussion after Definition 4.9 for the proper statements.

In section 5 we use the constructions of section 4 to prove a version of Morita equivalence for A_{∞} rings, Theorems 5.8 and 5.9. These results also depend on a 3-category structure, but unlike the coherence theorems they are essentially impossible to state without the terminology of Definition 4.9.

The material on modules was developed with a view towards applications in algebraic K-theory and stable homotopy, some of which will appear in [5] and are indicated in brief remarks at the ends of sections 4 and 5.

Throughout this paper Cat will denote the 2-category of (small) based topological categories; functors and natural transformations are required to be based and continuous. The symbols ˙ ≅ and ≅ will denote equivalence and isomorphism of 2-categories respectively. The 2-categories with 0-cells the strict monoidal, permutative and strict braided monoidal categories are denoted stMon, Perm and Braid respectively. The 1cells are the strict morphisms. The 1-cells F of Mon , SymMon and BrMon satisfy F (a ⊕ b) ≅ F a ⊕ F b and (since F is based) F 0 = 0. We assume the reader is familiar with the basic ideas concerning 2-categories, [10], and operads, [11], although very little of either is actually needed here. In the latter case all of the relevant definitions are given in section 1.

1. Lax braided cat-operad actions

Let p : B_{j} ⟶ Σ_{j} be the usual homomorphism from the jth braid group to the jth symmetric group.

1.1. Definition. A braided cat-operad C consists of unbased categories C_{j}, j ≥ 0 with C_{0} the trivial category, functors γ : C_{k} × C_{j_{1}} × … × C_{j_{k}} ⟶ C_{Σj_{i}} and functorial actions C_{j} × B_{j} ⟶ C_{j} satisfying

(i) γ(u; γ(u_{1}; u_{11},…, u_{1 j_{1}}),…, γ(u_{k}; u_{k1},…, u_{k j_{k}})) = γ(γ(u; u_{1},…, u_{k}); u_{11},…, u_{k j_{k}})

(ii) there is an object 1 in C_{1} such that γ(1_{1}; u) = u and γ(u; 1_{1},… 1_{1}) = u

(iii) γ(u σ; u_{1},…, u_{k}) = γ(u; u_{pσ^{−1}(1)},…, u_{pσ^{−1}(k)}) σ(j_{1},…, j_{k}) γ(u; u_{1} τ_{1},…, u_{k} τ_{k}) = γ(u; u_{1},…, u_{k}) (τ_{1} ⊕ ⋯ ⊕ τ_{k})

for morphisms u in C_{k}, u_{i} in C_{j_{i}} and u_{pq} in C_{r_{pq}}, and σ ∈ B_{k}, τ_{i} ∈ B_{j_{i}}. Here σ(j_{1},…, j_{k}) ∈ B_{Σ j_{i}} is obtained by replacing the ith strand of σ with j_{i} parallel strands and τ_{1} ⊕⋯⊕τ_{k} ∈ B_{Σ j_{i}} is the usual sum of braids.

C is a braided operad, [6], if all morphisms are identities. C is a cat-operad if the braid group actions factor through p : B_{j} ⟶ Σ_{j} and is an operad if in addition all morphisms are identities.

A morphism F : C ⟶ C′ of braided cat-operads consists of B_{j}-equivariant functors F_{j} : C_{j} ⟶ C′_{j} such that F_{1}(1) = 1′ and F (γ(u; u_{1},…, u_{k})) = γ′(F (u); F (u_{1}),…, F (u_{k})). Morphisms for the other types of operads are defined similarly.

1.2. Examples. (i) Let S_{j} = Σ̃_{j}, the translation category of Σ_{j} ; the objects are the elements of Σ_{j} and there is a unique morphism σ → τ for σ, τ ∈ Σ_{j}. Thus it suffices to define the cat-operad structure just on objects. The Σ_{j}-action is given by right multiplication. If e_{j} ∈ Σ_{j} is the identity element, let γ(e_{k}; e_{j_{1}},…, e_{j_{k}}) = e_{Σj_{i}}. This determines γ on all objects by the equivariance conditions 1.1 (iii).

(ii) There is a braided cat-operad B̃ with B̃_{j} = B̃_{j}, the translation category of B_{j}. The composition γ for B̃ is as for the cat-operad S.

(iii) If C is any (braided) cat-operad, then taking objects in each degree determines a (braided) operad. For the cat-operad S this gives the A_{∞} operad M with M(j) = Σ_{j}.

Further examples and relations among the four categories of operads can be found in [3].

For any category A let t : C_{k} × C_{j_{1}} × ⋯ × C_{j_{k}} × A^{Σj_{i}} ⟶ C_{k} × C_{j_{1}} × A^{j_{1}} × ⋯ × C_{j_{k}} × A^{j_{k}} be the obvious isomorphism.

1.3. Definition. Let C be a braided cat-operad. A lax C-object in Cat is a category A with functors θ_{j} : C_{j} × A^{j} ⟶ A and natural isomorphisms

σ : θ_{1}(1; −) ⟶ 1_{A} α : θ_{k} ∘ (1 × θ_{j_{1}} × ⋯ × θ_{j_{k}}) ∘ t ⟶ θ_{Σj_{i}} ∘ (γ × 1)

such that

(i) θ_{k}(uτ; f) = θ_{k}(u; τ · f), for τ ∈ B_{k}, and morphisms u in C_{k} and f in A^{k}

(ii) θ_{0}(∗; 0) = 0

(iii) α((c; 1,…, 1); a_{1},…, a_{j}) = θ_{j}(1_{c}; σ(a_{1}),…, σ(a_{j})) α((1; c); a_{1},…, a_{j}) = σ(θ_{j}(c; a_{1},…, a_{j}))

(iv) α((γ(c; c_{1},…, c_{k}); c_{11},…, c_{k j_{k}}); a_{1}^{11},…, a_{r_{k j_{k}}}^{k j_{k}}) ∘ α((c; c_{1},…, c_{k}); x_{11},…, x_{1 j_{1}},…, x_{k1},…, x_{k j_{k}}) = α((c; γ(c_{1}; c_{11},…, c_{1 j_{1}}),…, γ(c_{k}; c_{k1},…, c_{k j_{k}})); a_{1}^{11},…, a_{r_{k j_{k}}}^{k j_{k}}) ∘ θ_{k}(1_{c}; α_{1},…, α_{k})

where x_{pq} = θ_{r_{pq}}(c_{pq}; a_{1}^{pq},…, a_{r_{pq}}^{pq}) and α_{i} = α((c_{i}; c_{i1},…, c_{i j_{i}}); a_{1}^{i1},…, a_{r_{i j_{i}}}^{i j_{i}}). A is a (strict) C-object if the components of σ and α are identities.

1.4. Definition. Let A and A′ be lax C-categories. A functor F : A ⟶ A′ is a lax C-functor if there are natural isomorphisms

h : F ∘ θ_{j} ⟶ θ′_{j} ∘ (1 × F^{j})

(i) h(∗; 0) = 1_{0′}

(ii) σ′(F a) ∘ h(1; a) = F σ(a)

(iii) h(γ(c; c_{1},…, c_{k}); a_{11},…, a_{1 j_{1}},…, a_{k1},…, a_{k j_{k}}) ∘ F α((c; c_{1},…, c_{k}); a_{11},…, a_{1 j_{1}},…, a_{k1},…, a_{k j_{k}}) = α′((c; c_{1},…, c_{k}); F a_{11},…, F a_{1 j_{1}},…, F a_{k1},…, F a_{k j_{k}}) ∘ θ′_{k}(1_{c}; h(c_{1}; a_{11},…, a_{1 j_{1}}),…, h(c_{k}; a_{k1},…, a_{k j_{k}})) ∘ h(c; θ_{j_{1}}(c_{1}; a_{11},…, a_{1j_{1}}),…, θ_{j_{k}}(c_{k}; a_{k1},…, a_{k j_{k}}))

F is a (strict) C-functor if the components of h are identities.

1.5. Definition. Let F, F′ : A ⟶ A′ be lax C-functors. A lax C-natural transformation is a natural transformation τ : F ⟶ F′ such that

h′(c; a_{1},…, a_{j}) ∘ τ(θ_{j}(c; a_{1},…, a_{j})) = θ′_{j}(1_{c}; τ(a_{1}),…, τ(a_{j})) ∘ h(c; a_{1},…, a_{j})

τ is a (strict) C-natural transformation if F and F′ are C-functors.

The lax C-categories, lax C-functors and lax C-natural transformations are the 0, 1 and 2-cells of a 2-category denoted C(Cat). The strict C-categories, C-functors and C-natural transformations form a sub 2-category denoted C[Cat].

1.6. Theorem. There are equivalences and isomorphisms of 2-categories

(i) M(Cat) ˙ ≅ Mon and M[Cat] ≅ stMon

(ii) S(Cat) ˙ ≅ SymMon and S[Cat] ≅ Perm

(iii) B̃(Cat) ˙ ≅ BrMon and B̃[Cat] ≅ Braid

each the identity on underlying 0, 1 and 2-cells.

Proof. We show B̃(Cat) ˙≅BrMon. If (A, θ) is a lax B̃-category define a monoidal structure □ on A by a□b = θ_{2}(e_{2}; a, b) on objects and f□g = θ_{2}(1_{e_{2}}; f, g) on morphisms. The associativity isomorphism a□(b□c) ≅ (a□b)□c is given by the isomorphisms

θ_{2}(e_{2}; a, θ_{2}(e_{2}; b, c)) ⟶^{α_{1}} θ_{3}(e_{3}; a, b, c) ⟵^{α_{2}} θ_{2}(e_{2}; θ_{2}(e_{2}; a, b), c)

where α_{1} = α((e_{2}; 1, e_{2}); a, b, c) and α_{2} = α((e_{2}; e_{2}, 1); a, b, c).

The unit isomorphism 0□a ≅ a is σ(a) ∘ α((e_{2}; ∗, 1); 0, a) ∘ θ_{2}(1_{e_{2}}; 1_{0}, σ(a))^{−1} and similarly for a□0 ≅ a. The braiding is the isomorphism

θ_{2}(e_{2} → τ; 1_{a}, 1_{b}) : θ_{2}(e_{2}; a, b) ⟶ θ_{2}(τ; a, b) = θ_{2}(e_{2}; b, a)

where τ is the generator of B_{2}. This defines a braided monoidal structure on A.

Now suppose (A, □) is braided monoidal and define a lax B̃-action θ on A as follows. Let θ_{1}(1; a)=a, θ_{2}(e_{2}; a_{1}, a_{2})=a□b and θ_{j}(e_{j}; a_{1},…, a_{j}) = θ_{2}(e_{2}; a_{1}, θ_{j−1}(e_{j−1}; a_{2},…, a_{j})) for j ≥ 3.

In view of the equivariance conditions 1.1(iii), in order to define the natural isomorphisms α we need only specify the components

α((e_{k}; e_{j_{1}},…, e_{j_{k}}); a_{11},…, a_{1 j_{1}},…, a_{k1},…, a_{k j_{k}}).

This is taken to be the unique isomorphism

θ_{k}(e_{k}; θ_{j_{1}}(e_{j_{1}}; a_{11},…, a_{1 j_{1}}),…, θ_{j_{k}}(e_{j_{k}}; a_{k1},…, a_{k j_{k}})) ⟶ θ_{j}(e_{j}; a_{11},…, a_{k j_{k}})

provided by coherence, where j = Σj_{i}. The natural isomorphisms σ are taken to be the identity.

This defines 2-functors B̃(Cat) → BrMon → B̃(Cat) on objects. The correspondence on functors and natural transformations is similar and is left to the reader.

1.7. Theorem. Let U : C[Cat] ⟶ C(Cat) be the inclusion. There is a 2-functor P : C(Cat) ⟶ C[Cat] such that (P, U) is a 2-adjoint pair. Moreover the unit and counit η(A) : A ⟶ UP(A) and ε(X) : PU(X) ⟶ X are equivalences in C(Cat) (more precisely U(ε) is an equivalence).

Proof. For A ∈ C(Cat) the category PA has objects the objects of ∐_{j≥0} C_{j}×_{B_{j}} A^{j} and morphisms the triples ( (c; a_{1},…, a_{j}), f, (d; b_{1},…, b_{l}) ) with f : θ_{j}(c; a_{1},…, a_{j}) ⟶ θ_{l}(d; b_{1},…, b_{l}) in A. Composition and identities in PA are induced by that in A.

Define a C-action \overline{θ} on PA by

\overline{θ}_{k}(c; (c_{1}; a_{11},…, a_{1 j_{1}}),…, (c_{k}; a_{k1},…, a_{k j_{k}})) = (γ(c; c_{1},…, c_{k}); a_{11},…, a_{k j_{k}})

and for \overline{f}_{i} = ( (c_{i}; a_{i1},…, a_{i j_{i}}), f_{i}, (d_{i}; b_{i1},…, b_{i l_{i}}) ), i = 1,…, k, and u : c → d in C_{k}, let

\overline{θ}_{k}(u; \overline{f}_{1},…, \overline{f}_{k}) = ((γ(c; c_{1},…, c_{k}); a_{11},…, a_{k j_{k}}), \overline{f}, (γ(d; d_{1},…, d_{k}); b_{11},…, b_{k l_{k}})),

where

\overline{f} = α((d; d_{1},…, d_{k}); b_{11},…, b_{k l_{k}}) ∘ θ_{k}(u; f_{1},…, f_{k}) ∘ α((c; c_{1},…, c_{k}); a_{11},…, a_{k j_{k}})^{−1}.

If F : A ⟶ A′ is a lax C-functor, define a strict C-functor PF : PA ⟶ PA′ by PF ((c; a_{1},…, a_{j}), f, (d; b_{1},…, b_{l})) = ((c; F a_{1},…, F a_{j}), f′, (d; F b_{1},…, F b_{l})) where f′ = h(d; b_{1},…, b_{l}) ∘ F f ∘ h(c; a_{1},…, a_{j})^{−1}.

If τ:F ⟶ F′ is a lax C-natural transformation, define a strict C-natural transformation Pτ : PF ⟶ PF′ by

Pτ(c; a_{1},…, a_{j}) = ((c; F a_{1},…, F a_{j}), θ′_{j}(1_{c}; τ(a_{1}),…, τ(a_{j})), (c; F′a_{1},…, F′a_{j})).

Define 2-natural transformations η : Id ⟶ UP and ε : PU ⟶ Id as follows. For a lax C-category A, let η_{A}(f : a → b) = ((1; a), θ_{1}(1_{1}; f), (1; b)) , and for a C-category X, let ε_{X}((c; a_{1},…, a_{j}), g, (d; b_{1},…, b_{l})) = g.

It is straightforward to verify that (P, U) is a 2-adjoint pair with unit η and counit ε having the stated properties.

A braided cat-operad determines a monad in Cat as follows. This will be needed in section 4.

1.8. Definition. If C is a braided cat-operad, the associated monad C : Cat ⟶ Cat is defined on objects by

CA = ∐_{j≥0} C_{j} ×_{B_{j}} A^{j}/ ∼

where (c; a_{1},…, a_{j}) ∼ (γ(c; 1^{i−1} × ∗ × 1^{j−i}); a_{1},…, â_{i},… a_{j}) if a_{i} = 0.

The definitions of the unit η : Id ⟶ C and multiplication μ : CC ⟶ C are similar to the operad case, [11]. We also note that CA is the free C-object on the category A.

2. Lax G-ring categories

2.1. Definition. Let G be a braided cat-operad. A lax G-ring category (A, ⊕, θ) consists of a symmetric monoidal category (A, ⊕, 0, α′, γ′) and a lax G-category (A, θ, 1, σ, α) satisfying

(i) θ_{j}(g; f_{1},…, f_{j}) = 1_{0} if f_{i} = 1_{0} for some i. α((y; x_{1},…, x_{k}); a_{11},…, a_{k j_{k}}) = 1_{0} if a_{rs} = 0 for some r, s.

(ii) There are natural distributivity isomorphisms δ_{j}^{i}(x; a_{1},…, a_{i}, a′_{i},…, a_{j}) ,1 ≤ i ≤ j,

θ_{j}(x; a_{1},…, a_{i} ⊕ a′_{i},…, a_{j}) ⟶^{δ_{j}^{i}} θ_{j}(x; a_{1},…, a_{i},…, a_{j}) ⊕ θ_{j}(x; a_{1},…, a′_{i},…, a_{j})

such that

(a) [α((y; x_{1},…, x_{k}); a_{11},…, a_{k j_{k}}) ⊕ α((y; x_{1},…, x_{k}); a_{11},…, a′_{il},…, a_{k j_{k}})] ∘ δ_{k}^{i}(y; a_{1},…, a_{i}, θ_{j_{i}}(x_{i}; a_{i1},…, a′_{il},…, a_{ij_{i}}),…, a_{k}) ∘

θ_{k}(1_{y}; 1_{a_{1}},…, δ_{j_{i}}^{l}(x_{i}; a_{i1},…, a_{il}, a′_{il},…, a_{ij_{i}}),…, 1_{a_{k}}) =
δ_{j}^{t}(γ_{G}(y; x_{1},…, x_{k}); a_{11},…, a_{il}, a′_{il},…, a_{kj_{k}}) ∘
α((y; x_{1},…, x_{k}); a_{11},…, a_{il} ⊕ a′_{il},…, a_{k j_{k}})

where a_{r} = θ_{j_{r}}(x_{r}; a_{r1},…, a_{rj_{r}}), t = Σ_{r=1}^{i−1} j_{r} + l and γ_{G} is the operad composition in G.

(b) (σa ⊕ σb) ∘ δ_{1}^{1}(1; a, b) = σ(a ⊕ b)

(c) δ_{j}^{i}(xτ; a_{1},…, a_{i}, a′_{i},…, a_{j}) = δ_{j}^{pτ(i)}(x; a_{pτ^{−1}(1)},…, a_{pτ^{−1}(k)}, a′_{pτ^{−1}(k)},…, a_{pτ^{−1}(j)})

where τ ∈ B_{j}, pτ is the associated permutation and pτ(i) = k.

(d) ρ′(θ_{j}(x; a_{1},…, a_{j})) ∘ δ_{j}^{i}(x; a_{1},…, a_{i}, 0,…, a_{j}) = θ_{j}(1_{x}; 1_{a_{1}},…, ρ′(a_{i}),…, 1_{a_{j}}) λ′(θ_{j}(x; a_{1},…, a_{j})) ∘ δ_{j}^{i}(x; a_{1},…, 0, a_{i},…, a_{j}) = θ_{j}(1_{x}; 1_{a_{1}},…, λ′(a_{i}),…, 1_{a_{j}}) δ_{j}^{i}(x; a_{1},…, a_{i}, a′_{i},…, a_{j}) = λ′(0)^{−1} = ρ′(0)^{−1} if a_{r} = 0, for some r ≠ i

where λ′(a) : 0 ⊕ a → a and ρ′(a) : a ⊕ 0 → a are the unit isomorphisms.

Theory and Applications of Categories, Vol. 3, No. 4 57

(e) δ_{j}^{i}(x; a_{1},…, a′_{i}, a_{i},…, a_{j}) ∘ θ_{j}(1_{x}; 1_{a_{1}},…, γ′(a_{i}, a′_{i}),…, 1_{a_{j}}) =

γ′(a, a′) ∘ δ_{j}^{i}(x; a_{1},…, a_{i}, a′_{i},…, a_{j})

where a = θ_{j}(x; a_{1},…, a_{j}) and a′ = θ_{j}(x; a_{1},…, a′_{i},…, a_{j}).

(f) (δ_{j}^{i}(x; a_{1},…, a_{i}, a′_{i},…, a_{j}) ⊕ 1_{a′′}) ∘ δ_{j}^{i}(x; a_{1},…, a_{i} ⊕ a′_{i}, a′′_{i},…, a_{j}) ∘ θ_{j}(1_{x}; 1_{a_{1}},…, α′(a_{i}, a′_{i}, a′′_{i}),…, 1_{a_{j}}) = α′(a, a′, a′′) ∘ (1_{a} ⊕ δ_{j}^{i}(x; a_{1},…, a′_{i}, a′′_{i},…, a_{j})) ∘ δ_{j}^{i}(x; a_{1},…, a_{i}, a′_{i} ⊕ a′′_{i},…, a_{j}) with a, a′ as in (e) and a′′ = θ_{j}(x; a_{1},…, a′′_{i},…, a_{j}).

(g) α′(a ⊕ c, b, d) ∘ (δ_{j}^{l}(x; a_{1},…, a_{l}, a′_{l},…, a_{j}) ⊕ δ_{j}^{l}(x; a_{1},…, a′_{k},…, a_{l}, a′_{l},…, a_{j})) ∘

δ_{j}^{k}(x; a_{1},…, a_{k}, a′_{k},…, a_{l} ⊕ a′_{l},…, a_{j}) = (α′(a, c, b) ⊕ 1_{d}) ∘ (1_{a} ⊕ γ′(b, c) ⊕ 1_{d}) ∘ (α′(a, b, c)^{−1} ⊕ 1_{d}) ∘ α′(a ⊕ b, c, d) ∘ (δ_{j}^{k}(x; a_{1},…, a_{k}, a′_{k},…, a_{j}) ⊕ δ_{j}^{k}(x; a_{1},…, a_{k}, a′_{k},…, a′_{l},…, a_{j})) ∘

δ_{j}^{l}(x; a_{1},…, a_{k} ⊕ a′_{k},…, a_{l}, a′_{l},…, a_{j}) for k < l,where a = θ_{j}(x; a_{1},…, a_{j}) ,b = θ_{j}(x; a_{1},…, a′_{k},…, a_{j}), c = θ_{j}(x; a_{1},…, a′_{l},…, a_{j}) and d = θ_{j}(x; a_{1},…, a′_{k},…, a′_{l},…, a_{j}).

A is a (strict) G-ring category if (A, ⊕) is permutative and (A, θ) is a G-category. 2.2. Definition. A lax G-ring functor (F, w, h) : A ⟶ A′ is a symmetric monoidal functor (F, w) and a lax G-functor (F, h) satisfying:

(i) h(x; a_{1},…, a_{j}) = 1_{0}, if a_{i} = 0 for some i (ii) [h(x; a_{1},…, a_{j}) ⊕ h(x; a_{1},…, a′_{i},…, a_{j})] ∘

w(θ_{j}(x; a_{1},…, a_{j}) , θ_{j}(x; a_{1},…, a′_{i},…, a_{j})) ∘ F δ_{j}^{i}(x; a_{1},…, a_{i}, a′_{i},…, a_{j}) = (δ′)_{j}^{i}(x; F a_{1},…, F a_{i}, F a′_{i},…, F a_{j}) ∘ θ′_{j}(1_{x}; 1_{F a_{1}},…, w(a_{i}, a′_{i}),…, 1_{F a_{j}}) ∘ h(x; a_{1},…, a_{i} ⊕ a′_{i},…, a_{j})

F is a (strict) G-ring functorof G-ring categories if F is permutative and a G-functor. 2.3. Definition. A lax G-ring natural transformation of lax G-ring functors is a natural transformation τ : F ⟶ F′ such that τ : (F, w) ⟶ (F′, w′) is a symmetric monoidal natural transformation and τ : (F, h) ⟶ (F′, h′) is a lax G-natural transformation.

τ is strict if F and F′ are strict G-ring functors. The 2-category of lax G-ring categories is denoted G −Rng(Cat) and the 2-category

of (strict) G-ring categories is denoted G−RngCat.

2.4. Examples. (i) Let SymBiMon be the 2-category with 0-cells the symmetric bimonoidal categories (see LaPlaza [9], except that here we assume distributivity isomorphisms). We also have the 2-category BrBiMon having 0-cells the braided bimonoidal categories. It contains SymBiMon as a full sub 2-category. Similarly the 2-category BiPerm with 0-cells the bipermutative categories is a full sub 2-category of BrPerm with 0-cells the braided permutative categories (such a category is additively permutative and multiplicatively strict braided monoidal). For the braided cat-operads S and B̃ we have equivalences and isomorphisms of 2-categories

S −Rng(Cat) ˙ ≅ SymBiMon S −RngCat ≅ BiPerm

B̃−Rng(Cat) ˙ ≅ BrBiMon  B̃−RngCat ≅ BrPerm

each the identity on underlying 0, 1 and 2-cells. Coherence for these 2-categories is a special case of Theorem 2.5 below.

(ii) If R is a bialgebra over a commutative ring k , then the category R-Mod of R-modules is a monoidal category under the tensor product ⊗ = ⊗_{k} . Drinfel’d, [2], has shown that R-Mod is braided monoidal if there is an invertible element β in R ⊗ R satisfying certain conditions. In fact there is a one-to-one correspondence between braidings on R-Mod and such elements β, [7]. We say (R, β) is a braided bialgebra. In this case the category R-Mod is braided bimonoidal with additive operation the direct sum of R-modules and hence is a (large) lax B̃-ring category. If we drop the braiding on R, then R-Mod is a lax M-ring category.

2.5. Theorem. Let U : G−RngCat ⟶ G−Rng(Cat) be the inclusion. There is a 2functor P : G−Rng(Cat) ⟶ G−RngCat such that (P, U) is a 2-adjoint pair and the unit η and counit ε are equivalences in G−Rng(Cat) (i.e. U(ε) is an equivalence).

Proof. Denote the functors of Theorem 1.7 by P′ and U′. For a lax G-ring category A define the category PA as follows. Let A_{0} be the space obj (P′A)/ ∼ where ∼ is the relation (c; a_{1},…, a_{j}) ∼ 0 ≡ (∗; 0) if a_{i} = 0 for some i. Let obj (PA) be the free based monoid on A_{0} and define π : obj (PA) ⟶ obj A by

π((c_{1}; a_{11},…, a_{1 j_{1}})□ ⋯ □(c_{k}; a_{k1},…, a_{k j_{k}})) = a_{1} ⊕ (a_{2} ⊕ ⋯ (a_{k−1} ⊕ a_{k}) ⋯),

where a_{r} = θ_{j_{r}}(c_{r}; a_{r1},…, a_{r j_{r}}) for r = 1,…, k and the sum is taken over the a_{r} ≠ 0.

For objects a, b in PA let PA(a, b) = {a}×A(π(a), π(b))×{b}. Composition is induced by the composition in A. Note that coherence determines isomorphisms π(a□b) → π(a)⊕ π(b).

For morphisms (a, f, a′),(b, g, b′) in PA let (a, f, a′)□(b, g, b′) = (a□b, f□g, a′□b′), where f□g is the composite

π(a□b) ⟶ π(a) ⊕ π(b) ⟶^{f⊕g} π(a′) ⊕ π(b′) ⟶ π(a′□b′)

PA is a permutative category with commutativity isomorphisms \overline{γ}(a, b) the composites

π(a□b) ⟶ π(a) ⊕ π(b) ⟶^{γ′} π(b) ⊕ π(a) ⟶ π(b□a)

where γ′ = γ′(π(a), π(b)) is the symmetry of A.

Define a strict G action \overline{θ} on PA as follows. The G action on P′A determined by θ as in the proof of Theorem 1.7 passes to an action θ′ on A_{0}. Let a_{1} …, a_{k} be objects in PA, say a_{i} = □_{j=1}^{r_{i}} b_{ij} with b_{ij} an object of A_{0}. Now for c ∈ G_{k} let

\overline{θ}_{k}(c; a_{1},…, a_{k}) = □_{(i_{1},...,i_{k})} θ′_{k}(c; b_{1i_{1}},…, b_{ki_{k}})

where 1 ≤ i_{j} ≤ r_{j} and the □ sum has the lexicographic order. Then coherence provides isomorphisms π(\overline{θ}_{k}(c; a_{1},…, a_{k})) → θ_{k}(c; π(a_{1}),…, π(a_{k})).

For morphisms \overline{f}_{i} = (a_{i}, f_{i}, a′_{i}) in PA, i = 1,…, k and u : c → d in G_{k} let

\overline{θ}_{k}(u; \overline{f}_{1},…, \overline{f}_{k}) = (a, f, a′)

where a = \overline{θ}_{k}(c; a_{1},…, a_{k}), a′ = \overline{θ}_{k}(d; a′_{1},…, a′_{k}) and f is the composite

π(a) ⟶ θ_{k}(c; π(a_{1}),…, π(a_{k})) ⟶^{f′} θ_{k}(d; π(a′_{1}),…, π(a′_{k})) ⟶ π(a′)

in which f′ = θ_{k}(u; f_{1},…, f_{k}).

We next define distributivity isomorphisms δ_{k}^{i} : U → V where U = \overline{θ}_{k}(x; a_{1},…, a_{i}□a′_{i}, …, a_{k}) and V = \overline{θ}_{k}(x; a_{1},…, a_{i},…, a_{k})□\overline{θ}_{k}(x; a_{1},…, a′_{i},…, a_{k}). First suppose that each a_{r} and a′_{i} are in A_{0}. Then

\overline{θ}_{k}(x; a_{1},…, a_{i}□a′_{i},…, a_{k}) = θ′_{k}(x; a_{1},…, a_{k})□θ′_{k}(x; a_{1},…, a′_{i},…, a_{k})
= \overline{θ}_{k}(x; a_{1},…, a_{k})□\overline{θ}_{k}(x; a_{1},…, a′_{i},…, a_{k})

and we let δ_{k}^{i}(x; a_{1},…, a_{i}, a′_{i},…, a_{k}) = id for objects in A_{0}.

In general expanding U and V using the definition of \overline{θ} we see that each is a □ sum of objects of P′A with exactly the same terms, but in different orders. It follows that coherence determines an isomorphism d : π(U) ⟶ π(V) in A and we let δ_{k}^{i} = (U, d, V).

Given a lax G-ring functor (F, w, h) : A ⟶ A′ define a G-ring functor PF : PA ⟶ PA′ as follows. On objects let PF (□_{i=1}^{k} (c_{i}; a_{i1},…, a_{i j_{i}})) = □_{i=1}^{k} (c_{i}; F a_{i1},…, F a_{i j_{i}}).

Note that the natural isomorphisms w and h determine an isomorphism π′(PF (a)) → F (π(a)). Now for a morphism (a, f, b) in PA let PF (a, f, b) = (PF (a), f′, PF (b)) where f′ is the composite

π′(PF (a)) ⟶ F (π(a)) ⟶^{F f} F (π(b)) ⟶ π′(PF (b))

If τ:F ⟶ F′ is a lax G-ring natural transformation, then for a=□_{i=1}^{k} (c_{i}; a_{i1},…, a_{i j_{i}}) let Pτ(a) = (PF (a), g, PF′(a)) where g is the composite

π′(PF (a)) ⟶ F (π(a)) ⟶^{τ(π(a))} F′(π(a)) ⟶ π′(PF′(a))

This defines a strict natural transformation Pτ : PF ⟶ PF′.

It is straightforward to check that P : G-Rng(Cat) ⟶ G-RngCat is a 2-functor. The unit η(A) : A ⟶ UP(A) and counit ε(X) : PU(X) ⟶ X are given by η(f : a → b) = ((1; a), θ_{1}(1_{1}; f), (1; b)) and ε(x, g, y) = g.

The previous theorem can be used to unify and extend the results of [3] and [4]. It is possible to prove significantly stronger results than those stated here. We only give the weaker versions below in order to avoid excessive technicalities. The reader should refer to [3] and [4] for additional details.

If G is a braided cat-operad the category of G-spectra is denoted GSp and the zeroth space of a G-spectrum E is denoted E_{0}. Let L be the linear isometries operad, [13].

2.6. Theorem. There is a functor W : G-Rng (Cat) ⟶ (G × L)Sp and an additive group completion BA ⟶ (W A)_{0}.

Proof. A functor W′ : G-RngCat ⟶ (G × L)Sp and an additive group completion η : BA ⟶ (W′A)_{0} were constructed in [3]. Let W = W′ ∘ P. The group completion is obtained by composing η with the unit of the (P, U) adjunction.

If A is a category with cofibrations and weak equivalences, write co A and w A for the subcategories of cofibrations and weak equivalences respectively. The K-theory of A is denoted K(w; A).

2.7. Definition. Let A be a category with cofibrations and weak equivalences, and let ∨ denote the symmetric monoidal structure induced by the cofibrations. A lax G-ring category with cofibrations and weak equivalences is an object (A, ⊕, θ) ∈ G-Rng(Cat) such that Id : (A, ⊕) ≅ (A, ∨) is a symmetric monoidal natural isomorphism and co A, w A are sub lax G-ring categories of A. The category having these objects is denoted G-Rng(Cat)_{cw}; the morphisms are the exact functors in G-Rng(Cat). The subcategory of objects and morphisms with strict G-structure is denoted G-RngCat_{cw}.

2.8. Theorem. There is a functor K_{G} : G-Rng(Cat)_{cw} ⟶ (G × L)Sp such that the additive spectrum of K_{G}(w; A) is equivalent to the spectrum of K(w; A) (as spectra).

Proof. This was shown for G-RngCat_{cw} in [4]. The present case is immediate from Theorem 2.5.

3. Lax C n-categories

The first three definitions are from [14] with slight changes for the topological case. This approach to n-categories is well-suited for our development of lax n-functors and lax n-natural transformations in this section.

3.1. Definition. A category is a space A with maps s, t : A ⟶ A and ∗ : {(a, b) ∈ A × A : t(a) = s(b)} ⟶ A satisfying:

(i) ss = ts = s, tt = st = t and the inclusion A_{0} ⊆ A is a cofibration, where A_{0} = s(A) = t(A)

(ii) s(b ∗ a) = s(a) and t(b ∗ a) = t(b)

(iii) s(a) = t(v) = v implies a ∗ v = a u = s(u) = t(a) implies u ∗ a = a (iv) s(a) = t(b), s(b) = t(c) imply a ∗ (b ∗ c) = (a ∗ b) ∗ c

where b ∗ a denotes the value of ∗ at (a, b).

We will write (A, μ) for the category (A, s, t, ∗) where μ = (s, t, ∗) is thought of as a partially defined operation on A.

3.2. Definition. A 2-category (A, μ_{0}, μ_{1}) with μ_{r} = (s_{r}, t_{r}, ∗_{r}) consists of categories (A, μ_{0}) and (A, μ_{1}) satisfying:

(i) s_{1}s_{0} = s_{0} = s_{0}s_{1} = s_{0}t_{1}, t_{0} = t_{0}s_{1} = t_{0}t_{1} and the inclusions A_{0} ⊆ A_{1} ⊆ A are cofibrations, where A_{r} = s_{r}(A) = t_{r}(A).

(ii) s_{0}(a) = t_{0}(a′) implies s_{1}(a ∗_{0} a′) = s_{1}(a) ∗_{0} s_{1}(a′) and t_{1}(a ∗_{0} a′) = t_{1}(a) ∗_{0} t_{1}(a′)

(iii) s_{1}(a) = t_{1}(b), s_{1}(a′) = t_{1}(b′), s_{0}(a) = t_{0}(a′) imply (a ∗_{1} b) ∗_{0} (a′ ∗_{1} b′) = (a ∗_{0} a′) ∗_{1} (b ∗_{0} b′)

3.3. Definition. An n-category (A, {μ_{r}}_{r=0}^{n−1}) consists of categories (A, μ_{r}) such that (A, μ_{r}, μ_{s}) is a 2-category for 0 ≤ r < s ≤ n − 1. A_{r} = s_{r}(A) = t_{r}(A) is called the space of r-cells.

An n-category (A, {μ_{r}}_{r=0}^{n−1}) determines an (n − 1)-category σA = (A, {μ_{r}}_{r=1}^{n−1}) with r-cells the (r +1)-cells of A. From this we get an (n−1)-category (σA×_{0}σA, {μ_{r}}_{r=1}^{n−1}) the pullback of σA ⟶^{t_{0}} A_{0} ⟵^{s_{0}} σA with ith composition coordinatewise (and still denoted μ_{i}). Note that μ_{0} : σA×_{0}σA ⟶ σA is an (n − 1)-functor. We also regard the space of 0-cells as an (n − 1)-category (A_{0}, {μ_{r}}_{r=1}^{n−1}) with all cells identities. The inclusion I_{0} : A_{0} ⟶ σA with I_{0}(a) = 1_{a} is an (n − 1)-functor. The full sub (n − 1)-category of A_{0} with one object (the basepoint) is denoted A^{0}. A map F : A ⟶ A′ of n-categories is a based map compatible with sources and targets. F a based map means F (1_{a}) = 1_{F a} for a in A^{0}. F determines maps σF : σA ⟶ σA′, σF ×_{0}σF : σA×_{0}σA ⟶ σA′×_{0}σA′ and F_{0} : A_{0} ⟶ A′_{0} with F_{0}(1_{a}) = 1_{F_{0}(a)}.

If α : A_{0} ⟶ σA′ is a map of (n − 1)-categories we can form composite maps

σ_{s}(α, F) : σA ⟶^{(s_{0},1_{σA})} A_{0}×_{0}σA ⟶^{α×_{0}σF} σA′×_{0}σA′ ⟶^{μ′_{0}} σA′

σ_{t}(F, α) : σA ⟶^{(1_{σA},t_{0})} σA×_{0}A_{0} ⟶^{σF ×_{0}α} σA′×_{0}σA′ ⟶^{μ′_{0}} σA′

In what follows F will be a lax n-functor and α will be the zeroth component of a lax n-natural transformation. The composite maps above will then be lax (n − 1)-functors.

To help motivate the definitions of lax n-functor and lax n-natural transformation we first reformulate them in the case n = 2. Vertical and horizontal composition in the 2-category Cat are denoted by • and ∘ respectively.

3.4. Definition. A lax 2-functor (F, φ, η) : (A, μ_{0}, μ_{1}) ⟶ (A′, μ′_{0}, μ′_{1}) consists of a map F : A ⟶ A′ of 2-categories such that σF : σA ⟶ σA′ is a functor, and natural isomorphisms φ : σF ∘ μ_{0} ⟶ μ′_{0} ∘ (σF ×_{0}σF) and η : σF ∘ I_{0} ⟶ I′_{0} ∘ F_{0} satisfying

(i) (μ′_{0} ∘ (φ×_{0}σF)) • (φ ∘ (μ_{0}×_{0}1_{σA})) = (μ′_{0} ∘ (σF ×_{0}φ)) • (φ ∘ (1_{σA}×_{0}μ_{0}))

(ii) σ_{t}(F, η) • (φ ∘ (1_{σA}, I_{0}t_{0})) = 1_{σF} = σ_{s}(η, F) • (φ ∘ (I_{0}s_{0}, 1_{σA}))

F is a strict 2-functor if φ = id and η = id.

3.5. Definition. Let (F, φ, η) , (F′, φ′, η′) : (A, μ) ⟶ (A′, μ′) be lax 2-functors. A lax 2-natural transformation α : (F, φ, η) ⟶ (F′, φ′, η′) consists of a functor α^{0} : A_{0} ⟶ σA′ with α^{0}(a) : F a ⟶ F′a and a natural isomorphism σα = α^{1} : σ_{s}(α^{0}, F′) ⟶ σ_{t}(F, α^{0}) satisfying

(i) (μ′_{0} ∘ (σF ×_{0}σα)) • (μ′_{0} ∘ (σα×_{0}σF′)) • (μ′_{0} ∘ (α^{0}s_{0}, φ′)) = (μ′_{0} ∘ (φ, α^{0}t_{0})) • (σα ∘ μ_{0})

(ii) (μ′_{0} ∘ (η, α^{0})) • (σα ∘ I_{0}) = μ′_{0} ∘ (α^{0}, η′)

α is a (strict) 2-natural transformation if the components of α^{1} are identities in (A′, μ′_{1}), i.e. σα : σ_{s}(α^{0}, F′) ⟶ σ_{t}(F, α^{0}) is the identity transformation. α is a (lax or strict) 2-natural isomorphism if the components of α^{0} are isomorphisms in (A′, μ′_{0}).

3.6. Definition. Let α, β : (F, φ, η) ⟶ (F′, φ′, η′) be lax 2-natural transformations. A modification ρ : α ⟶ β consists of a map ρ : A_{0} ⟶ σA′ with ρ(a) : α^{0}(a) ⟶ β^{0}(a) such that σβ • σ_{s}(ρ, F′) = σ_{t}(F, ρ) • σα.

The definition of modification is included here for convenience and will not be needed until section 4.

We next define lax n-functor, lax n-natural transformation and certain composites of the latter simultaneously by induction on n.

3.7. Definition. A lax n-functor (F, φ, η) : A ⟶ A′ consists of a map F : A ⟶ A′ of n-categories such that σF = (F, (φ_{i})_{i=1}^{n−2}, (η_{i})_{i=1}^{n−2}) : σA ⟶ σA′ is a lax (n−1)-functor, and lax (n−1)-natural isomorphisms φ_{0} : σF ∘μ_{0} ⟶ μ′_{0}∘(σF ×_{0}σF) and η_{0} : σF ∘I_{0} ⟶ I′_{0}∘F_{0} satisfying

(i) (μ′_{0} ∘ (φ_{0}×_{0}σF)) • (φ_{0} ∘ (μ_{0}×_{0}1_{σA})) = (μ′_{0} ∘ (σF ×_{0}φ_{0})) • (φ_{0} ∘ (1_{σA}×_{0}μ_{0}))

(ii) σ_{t}(F, η_{0}) • (φ_{0} ∘ (1_{σA}, I_{0}t_{0})) = 1_{σF} = σ_{s}(η_{0}, F) • (φ_{0} ∘ (I_{0}s_{0}, 1_{σA}))

F is a strict n-functor if φ = id and η = id.

3.8. Definition. Let (F, φ, η) , (F′, φ′, η′) : (A, μ) ⟶ (A′, μ′) be lax n-functors. A lax n-natural transformation α = (α^{i})_{i=0}^{n−1} : (F, φ, η) ⟶ (F′, φ′, η′) consists of an (n − 1)- functor α^{0} : A_{0} ⟶ σA′ with α^{0}(a) : F a ⟶ F′a and a lax (n − 1)-natural isomorphism σα = (α^{i})_{i=1}^{n−1} : σ_{s}(α^{0}, F′) ⟶ σ_{t}(F, α^{0}) satisfying

(i) (μ′_{0} ∘ (σF ×_{0}σα)) • (μ′_{0} ∘ (σα×_{0}σF′)) • (μ′_{0} ∘ (α^{0}s_{0}, φ′_{0})) = (μ′_{0} ∘ (φ_{0}, α^{0}t_{0})) • (σα ∘ μ_{0})

(ii) (μ′_{0} ∘ (η_{0}, α^{0})) • (σα ∘ I_{0}) = μ′_{0} ∘ (α^{0}, η′_{0})

α is a ( strict) n-natural transformation if the components of α^{i} are identities in (A′, μ′_{i}) for i ≥ 1, i.e. σα : σ_{s}(α^{0}, F′) ⟶ σ_{t}(F, α^{0}) is the identity transformation. α is a lax (or strict) n-natural isomorphism if the components of α^{0} are isomorphisms in (A′, μ′_{0}).

The compositions • and ∘ in (i) and (ii) are defined inductively as follows. Suppose lax n-functor, lax n-natural transformation and the compositions • and ∘ for lax (n − 1)- natural transformations are defined. If F, F′, F′′ : A ⟶ A′ are lax n-functors and α : F ⟶ F′, β : F′ ⟶ F′′ are lax n-natural transformations, then the vertical composite β • α : F ⟶ F′′ is the lax n-natural transformation defined by (β • α)^{0} = μ′_{0} ∘ (α^{0}, β^{0}) and σ(β •α) = σ_{t}(α, β^{0})•σ_{s}(α^{0}, β), where the latter is vertical composition of lax (n−1)- natural transformations each of which is the horizontal composite of lax (n − 1)-natural transformations. For example σ_{t}(α, β^{0}) = μ′_{0} ∘ (σα×_{0}β^{0}) ∘ (1_{σA}, t_{0}).

If G : A′′′ ⟶ A and H : A′ ⟶ A′′ are lax n-functors, then we can define horizontal composites α ∘ G and H ∘ α as follows. First α ∘ G : F ∘ G ⟶ F′ ∘ G is the lax n-natural transformation defined by (α ∘ G)^{0} = α^{0} ∘ G_{0} and

σ(α ∘ G) : σ_{s}(α^{0} ∘ G_{0}, F′ ∘ G) = σ_{s}(α^{0}, F′) ∘ σG ⟶^{σα∘σG} σ_{t}(F, α^{0}) ∘ σG = σ_{t}(F ∘ G, α^{0} ∘ G_{0})

where the identities for σ_{s} and σ_{t} are equalities of lax (n − 1)-functors.

H ∘ α : H ∘ F ⟶ H ∘ F′ is the lax n-natural transformation defined by (H ∘ α)^{0} = σH ∘ α^{0} and

σ(H∘α) : σ_{s}(σH∘α^{0}, H∘F′) ⟶^{g} σH∘σ_{s}(α^{0}, F′) ⟶^{σH∘σα} σH∘σ_{t}(F, α^{0}) ⟶^{h} σ_{t}(H∘F, σH∘α^{0})

where g = (φ_{0})^{−1} ∘ (α^{0}×_{0}σH) ∘ (s_{0}, 1_{σA′}), h = φ_{0} ∘ (σF ×_{0}α^{0}) ∘ (1_{σA}, t_{0}) and the composite is vertical composition of lax (n − 1)-natural transformations.

In defining these compositions for degree n we have implicitly used the fact that each is strictly associative in degree n − 1. This is obvious for n = 2 and can be verified by induction in general.

Let n − Cat denote the category with objects the n-categories and morphisms the lax n-functors. It is not a 2-category with the lax n-natural transformations as 2-cells. We do get a 2-category n−Cat_{st} by restricting to strict n-functors and strict n-natural transformations.

If C is a braided cat-operad let γ(j_{1},…, j_{k}) : C_{k} × C_{j_{1}} × ⋯ × C_{j_{k}} ⟶ C_{Σj_{i}} be the composition of C and t(j_{1},…, j_{k}) : C_{k}×C_{j_{1}}×⋯×C_{j_{k}}×A^{Σj_{i}} ⟶ C_{k}×C_{j_{1}}×A^{j_{1}}×⋯×C_{j_{k}}×A^{j_{k}} the obvious isomorphism. Also let I_{A} : A ⟶ C_{1} × A be the n-functor given by I_{A}(f) = (1_{1}, f).

3.9. Definition. Let C be a braided cat-operad. A lax C-object in n−Cat is an n-category (A, μ) with lax n-functors (θ_{j}, φ_{j}, η_{j}) : C_{j} × A^{j} ⟶ A and lax n-natural isomorphisms

σ : θ_{1} ∘ I_{A} ⟶ 1_{A}

α(j_{1},…, j_{k}) : θ_{k} ∘ (1_{C_{k}} × θ_{j_{1}} × ⋯ × θ_{j_{k}}) ∘ t(j_{1},…, j_{k}) ⟶ θ_{Σj_{i}} ∘ (γ(j_{1},…, j_{k}) × 1_{A^{Σj_{i}}})

such that

Theory and Applications of Categories, Vol. 3, No. 4 64

(i) θ_{k}(uσ; a) = θ_{k}(u; σa), for σ ∈ Σ_{k}, and morphisms u in C_{k} and a in A^{k} (ii) θ_{0}(0; ∗) = ∗ and θ_{0} is a strict n-functor (iii) σ ∘ θ_{j} = α(j) ∘ I where I : C_{j} × A^{j} ⟶ C_{1} × C_{j} × A^{j} with I(u, f) = (1_{1}, u, f) (iv) θ_{j} ∘ (σ × ⋯ × σ) = α(1,…, 1) ∘ J where J : C_{j} × A^{j} ⟶ C_{j} × (C_{1})^{j} × A^{j} with

J(u, f) = (u, (1_{1})^{j}, f)

(v) [α(r_{1},…, r_{k}) ∘ (1_{C_{k}} × γ(r_{11},…, r_{1 j_{1}}) × 1_{A^{r_{1}}} × ⋯ × γ(r_{k1},…, r_{k j_{k}}) × 1_{A^{r_{k}}})] • [θ_{k} ∘ (1_{C_{k}} × α(r_{11},…, r_{1 j_{1}}) × ⋯ × α(r_{k1},…, r_{k j_{k}}))] = [α(r_{11},…, r_{k j_{k}}) ∘ (γ(j_{1},…, j_{k}) × 1_{C_{r_{11}}×A^{r_{11}}×···×C_{r_{k j_{k}}} ×A^{r_{k j_{k}}}}) ∘ T^{−1} ∘ (1_{C_{k}} × t(r_{11},…, r_{1 j_{1}}) × ⋯ × t(r_{k1},…, r_{k j_{k}}))] • [α(r_{1},…, r_{k})∘(1_{C_{k}} ×θ_{r_{11}} ×⋯×θ_{r_{k j_{k}}})∘(1_{C_{k}} ×t(r_{11},…, r_{1 j_{1}})×⋯×t(r_{k1},…, r_{k j_{k}}))]

where T : C_{k} × C_{j_{1}} × ⋯ × C_{j_{k}} × C_{r_{11}} × A^{r_{11}} × ⋯ × C_{r_{k j_{k}}} × A^{r_{k j_{k}}} ⟶ C_{k} × C_{j_{1}} × C_{r_{11}} × A^{r_{11}} × ⋯ × C_{r_{1 j_{1}}} × A^{r_{1 j_{1}}} × ⋯ × C_{j_{k}} × C_{r_{k1}} × A^{r_{k1}} × ⋯ × C_{r_{k j_{k}}} × A^{r_{k j_{k}}} is the obvious isomorphism and r_{i} = Σ_{l=1}^{j_{i}} r_{il}.

We say A is a strict C n-category if each θ_{j} is a strict n-functor, σ = id and α = id. 3.10. Remarks. (i) A lax C-object in 1−Cat is the same thing as a lax C-object in Cat, so there is no conflict with the terminology of section 1.

(ii) The notion of lax (braided) monoidal 2-category as defined in [8] is a special case of Definition 3.9. A lax monoidal 2-category is just a lax M-object in 2−Cat and a lax braided monoidal 2-category is just a lax B̃-object in 2−Cat. 3.11. Definition. Let A and A′ be lax C n-categories. A lax n-functor F : A ⟶ A′ is a lax C lax n-functor if there are lax n-natural isomorphisms

h(j) : F ∘ θ_{j} ⟶ θ′_{j} ∘ (1_{C_{j}} × F^{j})

such that (i) h(0) = id (ii) (σ′ ∘ F) • (h(1) ∘ I) = F ∘ σ where I : A ⟶ C_{1} × A with I(f) = (1_{1}, f) (iii) [h(j) ∘ (γ(j_{1},…, j_{k}) × 1_{A^{j}})] • [F ∘ α(j_{1},…, j_{k})] =

[α′(j_{1},…, j_{k}) ∘ (1_{C_{j}} × F^{j})] • [θ′_{k} ∘ (1_{C_{k}} × h(j_{1}) × ⋯ × h(j_{k})) ∘ t(j_{1},…, j_{k})] • [h(k) ∘ (1_{C_{k}} × θ_{j_{1}} × ⋯ × θ_{j_{k}}) ∘ t(j_{1},…, j_{k})]

where j = Σj_{i}.

F is a (strict) C lax n-functor if h = id.

3.12. Definition. Let F, F′ : A ⟶ A′ be lax C lax n-functors. A lax C lax n-natural transformation is a lax n-natural transformation τ : F ⟶ F′ such that h′(j) • (τ ∘ θ_{j}) = (θ′_{j} ∘ (1_{C_{j}} × τ^{j})) • h(j).

If F and F′ are strict C lax n-functors (so that h and h′ are identities), then we call τ a (strict) C lax n-natural transformation.

Notation. A lax n-natural transformation α = (α^{i})_{i=0}^{n−1} : F ⟶ F′ determines shifts σ^{k}α = (α^{i})_{i=k}^{n−1} : s_{k}α ⟶ t_{k}α, a lax (n − k)-natural transformation, for k = 0,…, n − 1. Thus s_{0}α = F, s_{1}α = σ_{s}(α^{0}, F′), etc.

The coherence theorem 3.13 below is based on the following construction of an ncategory A′_{F} from a lax n-functor F : A ⟶ A′. First let (A′_{F})_{n−1} = A_{n−1} as (n − 1)- categories. The n-cells of A′_{F} are the triples (f, x, g) where f and g are (n − 1)-cells of A such that s_{r}(f) = s_{r}(g) and t_{r}(f) = t_{r}(g) for r = 0,…, n − 2 and x : F f ⟶ F g is an n-cell of A′.

If (f, x, g) and (f′, x′, g′) are n-cells such that s_{r}(f′, x′, g′) = t_{r}(f, x, g) the rth composition is defined as follows. If r = n − 1, then f′ = g and we let (f′, x′, g′)□_{r}(f, x, g) = (f, x′∗_{r}x, g′). For r < n−1 let (f′, x′, g′)□_{r}(f, x, g) = (f′∗_{r}f, y, g′∗_{r}g) where y is the n-cell described below.

Consider the lax (n−r−1)-natural transformation φ_{r} = (φ_{r}^{i})_{i=0}^{n−r−2} : σ^{r+1}F ∘μ_{r} ⟶ μ′_{r}∘ (σ^{r+1}F ×_{r}σ^{r+1}F) and its shifts σ^{k}φ_{r} = (φ_{r}^{i})_{i=k}^{n−r−2} : s_{k}φ_{r} ⟶ t_{k}φ_{r} for k = 0,…, n − r − 2. We can apply the map μ′_{r} ∘ (σ^{r+1}F ×_{r}σ^{r+1}F) to the pair (p, p′) = ((f, x, g), (f′, x′, g′)) and obtain μ′_{r}(x, x′) : μ′_{r}(F f, F f′) ⟶ μ′_{r}(F g, F g′) an n-cell of A′. In a similar way we can apply the shifts of this map to the pair (p, p′). These shifts are t_{0}φ_{r}, s_{1}φ_{r}, t_{2}φ_{r}, etc. Also, if y′ : F μ_{r}(f, f′) ⟶ F μ_{r}(g, g′) is an n-cell we can apply the shifts of σ^{r+1}F ∘ μ_{r} to y′.

Now take k = n − r − 2 above and consider the natural isomorphism σ^{n−r−2}φ_{r} = φ_{r}^{n−r−2} : s_{n−r−2}φ_{r} ⟶ t_{n−r−2}φ_{r}. If n − r is even, then s_{n−r−2}φ_{r} and t_{n−r−2}φ_{r} are shifts of σ^{r+1}F ∘ μ_{r} and μ′_{r} ∘ (σ^{r+1}F ×_{r}σ^{r+1}F) respectively. In this case there is a unique n-cell y : F μ_{r}(f, f′) ⟶ F μ_{r}(g, g′) satisfying

s_{n−r−2}φ_{r}(y) = φ_{r}^{n−r−2}(g, g′)^{−1}∗′_{n−1}t_{n−r−2}φ_{r}(p, p′)∗′_{n−1}φ_{r}^{n−r−2}(f, f′).

The case when n−r is odd is similar. In principle one could obtain an explicit description of y as a pasting composite by unravelling the iterated shift, but this is extremely tedious. The above descirption has the advantage that checking associativity and the interchange of compositions is relatively painless.

The n-cells that are identities for □_{r} are easy to describe. If f is an r-cell the identity n-cell is (1_{f}^{n−r−1}, F (1_{f}^{n−r−1}), 1_{f}^{n−r−1}) where 1_{f}^{n−r−1} is the identity (n − 1)-cell on f. To see that this is an identity for □_{r} requires unravelling the shifts for the lax (n − r − 1)-natural transformation η_{r}. This completes the construction of A′_{F}.

Let C(n − Cat) be the category with 0-cells the lax C n-categories (A, θ, α, σ) and 1-cells the lax C lax n-functors (F, h). The subcategory C<n−Cat> has 0-cells the strict C n-categories and 1-cells the strict C lax n-functors. Let U : C<n−Cat>⟶ C(n−Cat) be the inclusion.

3.13. Theorem. There is a functor P : C(n−Cat) ⟶ C<n−Cat> such that (P, U) is an adjoint pair. Moreover the unit and counit η(A) : A ⟶ UP(A) and ε(X) : PU(X) ⟶ X are equivalences in C(n−Cat) (i.e. U(ε) is an equivalence).

Proof. If A is a 0-cell in C(n−Cat) let PA = A_{θ}, the n-category associated to the lax n-functor θ̂: ∐_{j≥0} C_{j} ×_{B_{j}} A^{j} ⟶ A with θ̂(u; f_{1},…, f_{j}) = θ_{j}(u; f_{1},…, f_{j}).

Define a C-action \overline{θ} on (PA)_{n−1} as follows. On r-cells, r < n, let

\overline{θ}_{k}(u; (u_{1}; f_{11},…, f_{1 j_{1}}),…, (u_{k}; f_{k1},…, f_{k j_{k}})) = (γ(u; u_{1},…, u_{k}); f_{11},…, f_{k j_{k}})

For n-cells x̂_{i} = ((u_{i}; f_{i1},…, f_{i j_{i}}), x_{i}, (v_{i}; g_{i1},…, g_{i j_{i}})), i = 1,…, k with s_{n−2}(u_{i}) = c_{i}, t_{n−2}(u_{i}) = d_{i}, s_{n−2}(f_{il}) = s_{n−2}(g_{il}) = a_{il} and t_{n−2}(f_{il}) = t_{n−2}(g_{il}) = b_{il}, and u : c ⟶ d in C_{k}, let \overline{θ}_{k}(u; x̂_{1},…, x̂_{k}) = (f, x, g) where f = (γ(u; u_{1},…, u_{k}); f_{11},…, f_{k j_{k}}), g = (γ(u; v_{1},…, v_{k}); g_{11},…, g_{k j_{k}}) and x is the unique n-cell of A such that

[α^{n−2}((d; d_{1},…, d_{k}); b_{11},…, b_{k j_{k}})∗_{n−2} θ_{k}(u; x_{1},…, x_{k})] ∗_{n−1} α^{n−1}((u; u_{1},…, u_{k}); f_{11},…, f_{k j_{k}}) = α^{n−1}((u; v_{1},…, v_{k}); g_{11},…, g_{k j_{k}}) ∗_{n−1}[x ∗_{n−2} α^{n−2}((c; c_{1},…, c_{k}); a_{11},…, a_{k j_{k}})]

If (F, h) : A ⟶ A′ is a lax C lax n-functor, define a strict C lax n-functor PF : PA ⟶ PA′ as follows. On r-cells, r < n, let PF ((u; f_{1},…, f_{j})) = (u; F f_{1},…, F f_{j}). If (f, x, g) = ((u; f_{1},…, f_{j}), x, (v; g_{1},…, g_{j})) is an n-cell, then we let PF (f, x, g) = ((u; F f_{1},…, F f_{j}), y, (v; F g_{1},…, F g_{j})) where y is defined as follows. Consider the shift σ^{n−1}h(j) = h(j)^{n−1} : s_{n−1}h(j) ⟶ t_{n−1}h(j). If n is even, then s_{n−1}h(j) is a shift of F ∘ θ_{j} and as in the construction prior to the theorem the n-cell s_{n−1}h(j)(x) is defined. Then y is the unique n-cell such that y ∗_{n−1} h(j)^{n−1}(u; f_{1},…, f_{j}) = h(j)^{n−1}(v; g_{1},…, g_{j}) ∗_{n−1} s_{n−1}h(j)(x). y is defined in a similar way if n is odd. The remaining verifications are straightforward.

Theorem 3.13 has the following variant obtained by replacing the underlying category n−Cat with the 2-category n−Cat_{st}. Thus let C(n−Cat_{st}) be the 2-category with 0,1 and 2-cells the lax C-objects in n−Cat, lax C strict n-functors and lax C strict n-natural transformations, and let C[n−Cat_{st}] be the sub 2-category in which the C-structures on 0,1 and 2-cells are strict as well. Let V : C[n−Cat_{st}] ⟶ C(n−Cat_{st}) be the inclusion.

3.14. Theorem. There is a 2-functor P : C(n−Cat_{st}) ⟶ C[n−Cat_{st}] such that (P, V) is a 2-adjoint pair. Moreover the unit and counit η(A) : A ⟶ VP(A) and ε(X) : PV(X) ⟶ X are equivalences in C(n−Cat_{st}) (i.e. V(ε) is an equivalence).

Proof. The functor P of Theorem 3.13 is defined on the 0 and 1-cells of C(n−Cat_{st}), and it extends to the 2-cells in an obvious way.

4. Tensor products for A_{∞} rings and modules

4.1. Definition. Let A,A_{1} and A_{2} be symmetric monoidal categories. A functor F : A_{1} × A_{2} ⟶ A is 2-symmetric monoidal if there are natural isomorphisms

w^{1}(a_{1}, a_{2}, b) : F (a_{1} ⊕ a_{2}, b) ⟶ F (a_{1}, b) ⊕ F (a_{2}, b)
w^{2}(a, b_{1}, b_{2}) : F (a, b_{1} ⊕ b_{2}) ⟶ F (a, b_{1}) ⊕ F (a, b_{2})

such that

(i) (F (−, b), w^{1}(−, b)) and (F (a, −), w^{2}(a, −)) are symmetric monoidal functors

(ii) w^{1}(a_{1}, a_{2}, −) : F (a_{1} ⊕ a_{2}, −) ⟶ F (a_{1}, −) ⊕ F (a_{2}, −) is a symmetric monoidal natural transformation (equivalently w^{2}(−, b_{1}, b_{2}) : F (−, b_{1} ⊕ b_{2}) ⟶ F (−, b_{1}) ⊕ F (−, b_{2}) is a symmetric monoidal natural transformation)

for all objects a, a_{1}, a_{2} ∈ A_{1} and b, b_{1}, b_{2} ∈ A_{2}.

4.2. Definition. Let (F, w^{1}, w^{2}) , (G, v^{1}, v^{2}) : A_{1} × A_{2} ⟶ A be 2-symmetric monoidal functors. A 2-symmetric monoidal natural transformation is a natural transformation h : F ⟶ G such that h(−, b) : (F (−, b), w^{1}(−, b)) ⟶ (G(−, b), v^{1}(−, b)) and h(a, −) : (F (a, −), w^{2}(a, −)) ⟶ (G(a, −), v^{2}(a, −)) are symmetric monoidal natural transformations for all objects a ∈ A_{1} and b ∈ A_{2}.

h is a 2-symmetric monoidal natural isomorphism if h is also a natural isomorphism.

k-symmetric monoidal functor (F, w^{1},…, w^{k}) : A_{1} × ⋯ × A_{k} ⟶ A and k-symmetric monoidal natural transformation are defined similarly.

If F_{i} : A_{i1}, × ⋯ × A_{i r_{i}} ⟶ B_{i} is an r_{i}-symmetric monoidal functor for i = 1,…, k and H : B_{1} × ⋯ × B_{k} ⟶ A is a k-symmetric monoidal functor, then H ∘ (F_{1} × ⋯ × F_{k}) is a Σ r_{i}-symmetric monoidal functor. The verification is straightforward.

The use of k-symmetric monoidal structures simplifies the coherence conditions for a lax G-ring category (Definition 2.1).

4.3. Definition. Let G be a braided cat-operad. A lax G-ring category (A, ⊕, θ) consists of a symmetric monoidal category (A, ⊕, 0, α′, γ′) and a lax G-category (A, θ, 1, σ, α) satisfying

(i) θ_{j}(g; f_{1},…, f_{j}) = 1_{0} if f_{i} = 1_{0} for some i.

α((y; x_{1},…, x_{k}); a_{11},…, a_{k j_{k}}) = 1_{0} if a_{rs} = 0 for some r, s.

(ii) There are natural distributivity isomorphisms δ_{j}^{i}(x; a_{1},…, a_{i}, a′_{i},…, a_{j}) ,1 ≤ i ≤ j,

θ_{j}(x; a_{1},…, a_{i} ⊕ a′_{i},…, a_{j}) ⟶^{δ_{j}^{i}} θ_{j}(x; a_{1},…, a_{i},…, a_{j}) ⊕ θ_{j}(x; a_{1},…, a′_{i},…, a_{j})

such that

(a) (θ_{j}(x; −), δ_{j}^{1}(x; −),…, δ_{j}^{j}(x; −)) is a j-symmetric monoidal functor.

(b) α : θ_{k} ∘ (1 × θ_{j_{1}} × ⋯ × θ_{j_{k}}) ∘ t ⟶ θ_{Σj_{i}} ∘ (γ × 1) is a Σj_{i}-symmetric monoidal natural transformation.

(c) σ : (θ_{1}(1; −), δ_{1}^{1}(1; −)) ⟶ 1_{A} is a symmetric monoidal natural transformation.

(d) (θ_{j}(xτ; −), δ_{j}^{1}(xτ; −),…, δ_{j}^{j}(xτ; −)) =

(θ_{j}(x; pτ(−)), δ_{j}^{pτ(1)}(x; pτ(−)),…, δ_{j}^{pτ(j)}(x; pτ(−))) as j-symmetric monoidal functors, where τ ∈ B_{j}.

A is a (strict) G-ring category if (A, ⊕) is permutative and (A, θ) is a G-category, i.e. σ = id and α = id.

From now on we consider only strict G-ring categories which can always be arranged by Theorem 2.5. We also assume G is one of the braided cat-operads M , B̃ or S of section one. In each case there is a morphism of braided cat-operads M ⟶ G, so any G-ring category has an underlying M-ring category. This assumption is not overly restrictive by a standard change of operad argument, [3;§4]. In fact the most important examples of A_{∞} and E_{∞} rings are already covered by the above list (cf. Examples 2.4). We say a G-ring category satisfies strict left distributivity if δ_{j}^{i}(e_{j}; a_{1},…, a_{i}, a′_{i},…, a_{j}) = id when a_{k} = 1 for k > i (notation as in 4.3). Imposing this condition results in no loss of generality since it holds for the G-ring category PA of Theorem 2.5.

Let R be a G-ring category. The multiplicative operation θ_{2}(e_{2}; −) : R × R → R will be denoted μ(r, s) or r ⊗ s. Note that this is a 2-symmetric monoidal functor.

4.4. Definition. A lax (left) R- module is a symmetric monoidal category M with a 2-symmetric monoidal functor (α, w^{1}, w^{2}) : R × M ⟶ M, a symmetric monoidal natural isomorphism σ : (α(1, −), w^{2}(1, −)) ⟶ Id and a3-symmetric monoidal natural isomorphism v : α ∘ (1_{R} × α) ⟶ α ∘ (μ × 1_{M}) such that:

(i) v(1, s, a) = σ(α(s, a)) and v(r, 1, a) = α(1_{r}, σ(a))

(ii) v(r ⊗ s, t, a) ∘ v(r, s, α(t, a)) = v(r, s ⊗ t, a) ∘ α(1_{r}, v(s, t, a))

M is an R-module if M is a permutative category and σ = id; M is a strict R-module if in addition v = id and w^{2} = id.

4.5. Definition. A lax R-module morphism is a symmetric monoidal functor F : M ⟶ M′ with a 2-symmetric monoidal natural isomorphism h : F ∘ α ⟶ α′ ∘ (1_{R} × F) such that

(i) σ′(F a) ∘ h(1, a) = F σ(a)

(ii) h(r ⊗ s, a) ∘ F v(r, s, a) = v′(r, s, F a) ∘ α′(1_{r}, h(s, a)) ∘ h(r, α(s, a))

F is an R-module morphism if M and M′ are R-modules. F is a strict R-module morphism if M and M′ are strict R-modules, F is permutative and h = id.

4.6. Definition. Let F, G : M ⟶ M′ be lax R-module morphisms. A lax R-natural transformation is a symmetric monoidal natural transformation τ : F ⟶ G such that h_{G}(r, a) ∘ τ(α(r, a)) = α′(1_{r}, τ(a)) ∘ h_{F} (r, a). τ is a lax R-natural isomorphism if its components are isomorphisms in M′. τ is an R-natural transformation if F and G are R-module morphisms, and τ is strict if F and G are strict.

The 2-categories of lax R-modules, R-modules and strict R-modules are denoted by R−(Mod), R−Mod and R−[Mod] respectively. Similarly there are 2-categories of right modules (Mod)−R and Mod−R. Regarding R as a left R-module, the condition of strict left distributivity on R is equivalent to saying R is a strict left R-module, and similarly for right modules. Since a G-ring category will rarely satisfy strict right distributivity at the same time, it is not useful to consider strict right modules.

Let U : R−[Mod] ⟶ R−(Mod) and V : R−[Mod] ⟶ R−Mod be the inclusions.

4.7. Theorem. There is a 2-functor P : R−(Mod) ⟶ R−[Mod] such that (P, U) is a 2-adjoint pair and the unit η and counit ε are equivalences in R−(Mod) (i.e. U(ε) is an equivalence). Moreover, (P, U) restricts to a 2-adjoint pair (P, V) between R−Mod and R−[Mod] with unit and counit equivalences in R−Mod.

Proof. For a lax R-module (M, α, σ, v) let M_{0} be the smash product (obj R) ∧ (obj M) formed using the basepoints 0_{R} and 0_{M}. The space of objects of PM is the free based monoid on M_{0} and we define a map π : obj (PM) ⟶ obj M by π((r_{1}, a_{1})□ ⋯ □(r_{k}, a_{k})) = α(r_{1}, a_{1}) ⊕ (α(r_{2}, a_{2}) ⊕ ⋯ (α(r_{k−1}, a_{k−1}) ⊕ α(r_{k}, a_{k})) ⋯). mor (PM) is the space of triples (a, f, b) with a, b ∈ obj (PM) and f : π(a) ⟶ π(b) a morphism of M. For objects a, b in PM coherence determines an isomorphism π(a□b) ⟶ π(a) ⊕ π(b). PM is a permutative category via (a, f, b) ⊕ (a′, f′, b′) = (a□a′, g, b□b′) where g is the composite

π(a□a′) ⟶ π(a) ⊕ π(a′) ⟶^{f⊕f′} π(b) ⊕ π(b′) ⟶ π(b□b′)

The commutativity isomorphism \overline{γ} has components (a□b, \overline{γ}(a, b), b□a) where \overline{γ}(a, b) is the composite

π(a□b) ⟶ π(a) ⊕ π(b) ⟶^{g} π(b) ⊕ π(a) ⟶ π(b□a)

with g = γ(π(a), π(b)), the commutativity isomorphism of M.

R acts on the objects of PM by \overline{α}(r, (r_{1}, a_{1})□ ⋯ □(r_{k}, a_{k})) = (r ⊗ r_{1}, a_{1})□ ⋯ □(r ⊗ r_{k}, a_{k}), and coherence gives isomorphisms π(\overline{α}(r, a)) ⟶ α(r, π(a)). On morphisms we let \overline{α}(u, (a, f, b)) = (\overline{α}(r, a), \overline{f}, \overline{α}(s, b)) where u : r → s in R and \overline{f} is the composite

π(\overline{α}(r, a)) ⟶ α(r, π(a)) ⟶^{α(u,f)} α(s, π(b)) ⟶ π(\overline{α}(s, b))

We have a 2-symmetric monoidal functor (\overline{α}, \overline{w}^{1}, \overline{w}^{2}) with \overline{w}^{2} = id and \overline{w}^{1} having components \overline{w}^{1}(r, s, a) = (\overline{α}(r ⊕ s, a), g, \overline{α}(r, a)□\overline{α}(s, a)) where g is the composite

π(\overline{α}(r ⊕ s, a)) ⟶ α(r ⊕ s, π(a)) ⟶^{w^{1}(r,s,π(a))} α(r, π(a)) ⊕ α(s, π(a))

⟶ π(\overline{α}(r, a)) ⊕ π(\overline{α}(s, a)) ⟶ π(\overline{α}(r, a)□(\overline{α}(s, a))

This defines a strict R-module structure on PM.

If F : M ⟶ M′ is a lax R-module morphism define PF : PM ⟶ PM′ on objects by PF ((r_{1}, a_{1})□… □(r_{k}, a_{k})) = (r_{1}, F a_{1})□… □(r_{k}, F a_{k}) and note that coherence provides isomorphisms π′(PF (a)) ⟶ F (π(a)). For (a, f, b) in PM let PF (a, f, b) = (PF (a), \overline{f}, PF (b)) where \overline{f} is the composite

π′(PF (a)) ⟶ F (π(a)) ⟶^{F (f)} F (π(b)) ⟶ π′(PF (b))

PF is a strict R-module morphism.

If τ : F ⟶ G is a lax R-natural transformation let Pτ : PF ⟶ PG be the strict R-natural transformation with components Pτ(a) = (PF (a), g, PG(a)) where g is the composite

π′(PF (a)) ⟶ F (π(a)) ⟶^{τ(π(a))} G(π(a)) ⟶ π′(PG(a))

The unit and counit are defined as in the proof Theorem 2.5 or 1.7. The remaining verifications are routine.

4.8. Example. The commutative monoid S^{0} = {0, 1} is an S-category with zero and hence S(S^{0}) is an S-ring category (see [3;§3]) where S is the monad associated to S. It satisfies strict left distributivity. S(S^{0}) is the analogue for (lax) modules of the ring of integers. If R is a G-ring category (satisfying strict left distributivity), then the inclusion S^{0} ↪ R extends to a permutative functor S(S^{0}) ⟶ R which is in fact a morphism of G-ring categories. Thus an R-module has an underlying S(S^{0})-module. Also, there are equivalences of 2-categories

S(S^{0})−(Mod) ˙ ≅ SymMon S(S^{0})−Mod ˙ ≅ Perm

For the latter let J : Mod−S(S^{0}) ⟶ Perm be the 2-functor that sends a right S(S^{0})- module to its underlying permutative category and define I : Perm ⟶ Mod−S(S^{0}) by I(M) = (M, α) where α(a, [e_{n}; 1,…, 1]) = a ⊕ ⋯ ⊕ a (n terms). I(M) is an S(S^{0})- module with v = id (but is not strict). Then J ∘ I = id and I ∘ J ≅ id. Now combine this equivalence with the isomorphism S(S^{0})−Mod ≅ Mod−S(S^{0}) (see the discussion prior to 4.16).

4.9. Definition. (i) Let F, F′ : A ⟶ A′ be lax 2-functors and σ : F ⟶ F′ a lax 2-natural transformation. σ is a lax 2-natural equivalence if there is a lax 2-natural transformation τ : F′ ⟶ F and modifications ρ : τ ∘ σ ⟶ 1_{F} and λ : σ ∘ τ ⟶ 1_{F′} whose component 2-cells ρ(a), λ(a′) are isomorphisms in (A′, μ′_{1}).

(ii) Let F : A ⟶ A′ and G : A′ ⟶ A be lax 2-functors. We say (F, G) is a lax 2adjoint pair if there are lax 2-natural transformations η : Id ⟶ G ∘ F and ε : F ∘ G ⟶ Id, and modifications ρ : εF ∘ F η ⟶ 1_{F} and λ : Gε ∘ ηG ⟶ 1_{G} whose components ρ(a), λ(a′) are isomorphisms in (A′, μ′_{1}) and (A, μ_{1}) respectively. Note that for F, G strict 2-functors, (F, G) is a 2-adjoint pair when η and ε are strict and the components ρ(a), λ(a′) are identities.

If also η and ε are lax 2-natural equivalences we say (F, G) is a lax 2-adjoint, lax 2-equivalence. If we only have F η and Gε lax 2-natural equivalences we say (F, G) is a lax 2-adjoint, weak lax 2-equivalence. Finally, if we drop the adjointness condition, but still assume η and ε are lax 2-natural equivalences, we say F (or G) is a lax 2-equivalence of the 2-categories A and A′. If the functors or transformations are strict we drop the adjective “lax”.

The coherence theorems in sections 1 through 4 (except for Theorem 3.13) say that (P, U) is a 2-adjoint, weak 2-equivalence. In Theorem 4.7, for example, the unit η is a 2-equivalence while Uε is a 2-equivalence but ε is not. Note also that equivalence of 2-categories is the special case of 2-equivalence when all four modifications are identities.

We will shortly construct tensor products for R-modules. This is much simpler than for lax R-modules and is no real limitation by the previous theorem. On the other hand, some basic properties of the tensor product fail in R−[Mod], but hold in R−Mod (e.g. 4.17 (i)). The reverse is true in other situations, free modules for example, so it is necessary to work in both categories.

4.10. Definition. Let R, T ∈ G −RngCat with both multiplicative operations denoted by ⊗. An R−T -bimodule is a permutative category M that is a left R-module (M, α_{R}, v_{R}) and a right T -module (M, α_{T} , v_{T}) with a 3-symmetric monoidal natural isomorphism v_{R,T} : α_{T} ∘ (α_{R} × 1_{T}) ⟶ α_{R} ∘ (1_{R} × α_{T}) such that:

(i) v_{R,T} (r, a, 1) = 1_{α_{R}(r,a)} and v_{R,T} (1, a, t) = 1_{α_{T} (a,t)}

(ii) v_{R,T} (r_{1} ⊗ r_{2}, a, t_{2} ⊗ t_{1}) ∘ v_{T} (α_{R}(r_{1} ⊗ r_{2}, a), t_{2}, t_{1}) ∘ α(α_{T} (v_{R}(r_{1}, r_{2}, a), 1_{t_{2}}, 1_{t_{1}}) = v_{R}(r_{1}, r_{2}, α_{T} (a, t_{2} ⊗ t_{1})) ∘ α_{R}(1_{r_{1}}, α_{R}(1_{r_{2}}, v_{T} (a, t_{2}, t_{1}))) ∘ α_{R}(1_{r_{1}}, v_{R,T} (r_{2}, α_{T} (a, t_{2}), t_{1})) ∘ α_{R}(1_{r_{1}}, α_{T} (v_{R,T} (r_{2}, a, t_{2}), 1_{t_{1}})) ∘ v_{R,T} (r_{1}, α_{T} (α_{R}(r_{2}, a), t_{2}), t_{1}) ∘ α_{T} (v_{R,T} (r_{1}, α_{R}(r_{2}, a), t_{2}), 1_{t_{1}})

F : M ⟶ M′ is an R−T -bimodule morphism if it is a left R-module morphism (F, h_{R}) and a right T -module morphism (F, h_{T}) such that

v′_{R,T} (r, F a, t) ∘ α′_{T} (h_{R}(r, a), 1_{t}) ∘ h_{T} (α_{R}(r, a), t) = α′_{R}(1_{r}, h_{T} (a, t)) ∘ h_{R}(r, α_{T} (a, t)) ∘ F v_{R,T} (r, a, t)

An R−T -natural transformation is a symmetric monoidal natural transformation τ : F ⟶ G of R−T -bimodule morphisms that is both R-natural and T -natural.

The 2-category of R−T -bimodules is denoted R−Mod−T.

4.11. Definition. Let M be a right R-module, N a left R-module and P a permutative category. An R-biadditive functor is a 2-symmetric monoidal functor (F, w^{1}, w^{2}) : M × N ⟶ P with a 3-symmetric monoidal natural isomorphism

h : F ∘ (α_{M} × 1_{N}) ⟶ F ∘ (1_{M} × α_{N})

(i) h(−, 1, −) = 1_{F}

(ii) h(a, r ⊗ s, b) ∘ F (v_{M}(a, r, s), 1_{b}) = F (1_{a}, v_{N}(r, s, b)) ∘ h(a, r, α_{N}(s, b)) ∘ h(α_{M}(a, r), s, b)

Suppose R is an S-ring category, P is a left R-module and let t : M × R × N ⟶ R×M ×N be the obvious isomorphism. An R-bilinear functor is a 2-symmetric monoidal functor (F, w^{1}, w^{2}) : M × N ⟶ P with 3-symmetric monoidal natural isomorphisms

h^{1} : F ∘ (α_{M} × 1_{N}) ⟶ α_{P} ∘ (1_{R} × F) ∘ t
h^{2} : F ∘ (1_{M} × α_{N}) ⟶ α_{P} ∘ (1_{R} × F) ∘ t

such that F is R-biadditive via h = (h^{2})^{−1} ∘ h^{1} and

(i) h^{1}(−, 1, −) = 1_{F} = h^{2}(−, 1, −)

(ii) v_{P} (s, r, F (a, b)) ∘ α_{P} (1_{s}, h^{1}(a, r, b)) ∘ h^{2}(α_{M}(a, r), s, b) = α_{P} (γ_{R}(r, s), 1_{F (a,b)}) ∘ v_{P} (r, s, F (a, b)) ∘ α_{P} (1_{r}, h^{2}(a, s, b)) ∘ h^{1}(a, r, α_{N}(s, b))

where γ_{R} is the multiplicative commutativity isomorphism of R.

4.12. Example. A G-ring category R becomes an R−R-bimodule via the multiplicative operation μ : R × R ⟶ R. In this case v_{R,R} = id. Moreover μ is R-biadditive and if R is an S-ring category, then μ is R-bilinear.

If we take R = S(S^{0}), then R-biadditive is equivalent to R-bilinear and they imply 2-symmetric monoidal, but not conversely.

We now construct a tensor product functor ⊗_{R} : Mod−R × R−Mod ⟶ Perm. Let M be a right R-module, N a left R-module and A = S(M × N), the free permutative category on M × N, where S is the monad associated to the cat-operad S. Now let T be a set of formal morphisms between objects of A of the following types:

w_{i}^{1}((a_{1}, b_{1}),…, (a_{i}, a′_{i}, b_{i}),…, (a_{j}, b_{j})) : [e_{j}; (a_{1}, b_{1}),…, (a_{i} ⊕ a′_{i}, b_{i}),…, (a_{j}, b_{j})] ⟶

[e_{j+1}; (a_{1}, b_{1}),…, (a_{i}, b_{i}), (a′_{i}, b_{i}),…, (a_{j}, b_{j})]

w_{i}^{2}((a_{1}, b_{1}),…, (a_{i}, b_{i}, b′_{i}),…, (a_{j}, b_{j})) : [e_{j}; (a_{1}, b_{1}),…, (a_{i}, b_{i} ⊕ b′_{i}),…, (a_{j}, b_{j})] ⟶

[e_{j+1}; (a_{1}, b_{1}),…, (a_{i}, b_{i}), (a_{i}, b′_{i}),…, (a_{j}, b_{j})]

h_{i}((a_{1}, b_{1}),…, (a_{i}, r, b_{i}),…, (a_{j}, b_{j})) : [e_{j}; (a_{1}, b_{1}),…, (α_{M}(a_{i}, r), b_{i}),…, (a_{j}, b_{j})] ⟶

[e_{j}; (a_{1}, b_{1}),…, (a_{i}, α_{N}(r, b_{i})),…, (a_{j}, b_{j})]

Let A_{T} be the graph with the same objects as A and mor A_{T} = mor A ∐ T ∐ T^{op}. Now form the “localization” A[T^{−1}] of A with respect to T as the free category on the graph A_{T} modulo the identifications

(i) ( x_{0} ⟵^{u_{1}} x_{1} ⟵^{u_{2}} ⋯ ⟵^{u_{n}} x_{n}) = ( x_{0} ⟵^{u_{1}} ⋯ ⟵^{u_{i−1}} x_{i−1} ⟵^{u_{i}∘u_{i+1}} x_{i+1} ⟵^{u_{i+2}} ⋯ ⟵^{u_{n}} x_{n})

if u_{i} and u_{i+1} are morphisms in A

(ii) ( x ←^{t} y ←^{t^{op}} x) = 1_{x} and ( y ←^{t^{op}} x ←^{t} y) = 1_{y}, for t ∈ T

(iii) ( x ←^{u} y ←^{1_{y}} y) = ( x ←^{u} y) and ( x ←^{1_{x}} x ←^{u} y) = ( x ←^{u} y), for u ∈ mor A_{T}.

A is a subcategory of A[T^{−1}] and we partially extend the permutative operation ⊕ to the morphisms in T ∐ T^{op}. Let a = [e_{j}; (a_{1}, b_{1}),…, (a_{j}, b_{j})] and define

1_{a} ⊕ w_{i}^{l}((c_{1}, d_{1}),…, (c_{k}, d_{k})) = w_{i+j}^{l}((a_{1}, b_{1}),…, (a_{j}, b_{j}), (c_{1}, d_{1}),…, (c_{k}, d_{k}))

w_{i}^{l}((c_{1}, d_{1}),…, (c_{k}, d_{k})) ⊕ 1_{a} = w_{i}^{l}((c_{1}, d_{1}),…, (c_{k}, d_{k}), (a_{1}, b_{1}),…, (a_{j}, b_{j}))

1_{a} ⊕ h_{i}((c_{1}, d_{1}),…, (c_{k}, d_{k})) = h_{i+j}((a_{1}, b_{1}),…, (a_{j}, b_{j}), (c_{1}, d_{1}),…, (c_{k}, d_{k}))

h_{i}((c_{1}, d_{1}),…, (c_{k}, d_{k})) ⊕ 1_{a} = h_{i}((c_{1}, d_{1}),…, (c_{k}, d_{k}), (a_{1}, b_{1}),…, (a_{j}, b_{j}))

and similarly for morphisms t^{op} = t^{−1} in T^{op}. For example if 1_{a} ⊕ t = u, then u is an isomorphism and we let 1_{a} ⊕ t^{−1} = u^{−1}.

Let η_{j} : (M × N)^{j} ⟶^{η^{j}} A^{j} ⟶^{⊕} A ↪ A[T^{−1}] where η : Id ⟶ S is the unit of the monad S. Also let Δ denote a diagonal functor and P : M^{2} × N^{2} ⟶ (M × N)^{2} the obvious isomorphism. Define the tensor product M ⊗_{R} N to be the quotient of A[T^{−1}] by the (implied) identifications

(i) w_{i}^{1} : η_{j} ∘ [1_{(M×N)^{i−1}} × (⊕_{M} × 1_{N}) × 1_{(M×N)^{j−i}}] ⟶

η_{j+1}∘[1_{(M×N)^{i−1}}×(P ∘(1_{M^{2}}×Δ_{N}))×1_{(M×N)^{j−i}}]

w_{i}^{2} : η_{j} ∘ [1_{(M×N)^{i−1}} × (1_{M} × ⊕_{N}) × 1_{(M×N)^{j−i}}] ⟶

η_{j+1}∘[1_{(M×N)^{i−1}}×(P ∘(Δ_{M} ×1_{N^{2}}))×1_{(M×N)^{j−i}}]

h_{i} : η_{j} ∘ [1_{(M×N)^{i−1}} × (α_{M} × 1_{N}) × 1_{(M×N)^{j−i}}] ⟶

η_{j} ∘ [1_{(M×N)^{i−1}} × (1_{M} × α_{N}) × 1_{(M×N)^{j−i}}]

are natural transformations

(ii) (η, w_{1}^{1}, w_{1}^{2}) is a 2-symmetric monoidal functor and h_{1} : η ∘ (α_{M} × 1_{N}) ⟶ η ∘ (1_{M} × α_{N}) is a 3-symmetric monoidal natural transformation such that η is R-biadditive

(iii) (u_{1} ⊕ 1_{y_{2}}) ∘ (1_{x_{1}} ⊕ u_{2}) = (1_{y_{1}} ⊕ u_{2}) ∘ (u_{1} ⊕ 1_{x_{2}})

for morphisms u_{i} : x_{i} ⟶ y_{i} in A_{T}.

Now for morphisms u = ( x_{0} ⟵^{u_{1}} x_{1} ⟵^{u_{2}} ⋯ ⟵^{u_{n}} x_{n}) and v = ( y_{0} ⟵^{v_{1}} y_{1} ⟵^{v_{2}} ⋯ ⟵^{v_{m}}

y_{m}) in M⊗_{R}N define u ⊕ v as follows. The identifications above imply that each diagram

[figure: x_{i} ⊕ y_{j}; x_{i+1} ⊕ y_{j}; u_{i+1} ⊕ 1_{y_{j}}; x_{i} ⊕ y_{j+1}; x_{i+1} ⊕ y_{j+1}; u_{i+1} ⊕ 1_{y_{j+1}}; 1_{x_{i}} ⊕ v_{j+1}; 1_{x_{i+1}} ⊕ v_{j+1}]

commutes. Thus the m × n diagram formed from all these squares commutes and so there is a uniquely determined morphism u ⊕ v : x_{n} ⊕ y_{m} ⟶ x_{0} ⊕ y_{0}. This defines the operation ⊕ and ensures it is a functor. It is clear from the construction that M⊗_{R}N is a permutative category and that the composite

M × N ⟶^{η} A ⟶ A[T^{−1}] ⟶ M⊗_{R}N

also denoted by η, is R-biadditive.

4.13. Theorem. Let M be a right R-module, N a left R-module and P a permutative category. If (F, w_{F}^{1} , w_{F}^{2} , h_{F}) : M × N ⟶ P is R-biadditive, then there is a unique permutative functor \overline{F} : M⊗_{R}N ⟶ P such that \overline{F} ∘η = F. Moreover, if G : M⊗_{R}N ⟶ P is symmetric monoidal with G ∘ η = F, then there is a symmetric monoidal natural isomorphism G ⟶ \overline{F}.

Proof. Let A = S(M × N) and T be as above. F induces a permutative functor F′ : A ⟶ P which extends to a morphism of graphs F′′ : A_{T} ⟶ P as follows. For morphisms of T ∐ T^{op} we let

F′′(w_{i}^{1}((a_{1}, b_{1}),…, (a_{i}, a′_{i}, b_{i}),…, (a_{j}, b_{j}))) =

1_{F (a_{1},b_{1})⊕···⊕F (a_{i−1},b_{i−1})} ⊕ w_{F}^{1} (a_{i}, a′_{i}, b_{i}) ⊕ 1_{F (a_{i+1},b_{i+1})⊕···⊕F (a_{j},b_{j})}

F′′(w_{i}^{2}((a_{1}, b_{1}),…, (a_{i}, b_{i}, b′_{i}),…, (a_{j}, b_{j}))) =

1_{F (a_{1},b_{1})⊕···⊕F (a_{i−1},b_{i−1})} ⊕ w_{F}^{2} (a_{i}, b_{i}, b′_{i}) ⊕ 1_{F (a_{i+1},b_{i+1})⊕···⊕F (a_{j},b_{j})}

F′′(h_{i}((a_{1}, b_{1}),…, (a_{i}, r, b_{i}),…, (a_{j}, b_{j}))) =

1_{F (a_{1},b_{1})⊕···⊕F (a_{i−1},b_{i−1})} ⊕ h_{F} (a_{i}, r, b_{i}) ⊕ 1_{F (a_{i+1},b_{i+1})⊕···⊕F (a_{j},b_{j})}

F′′(w_{i}^{1}((a_{1}, b_{1}),…, (a_{i}, a′_{i}, b_{i}),…, (a_{j}, b_{j}))^{op}) =

1_{F (a_{1},b_{1})⊕···⊕F (a_{i−1},b_{i−1})}⊕w_{F}^{1} (a_{i}, a′_{i}, b_{i})^{−1}⊕1_{F (a_{i+1},b_{i+1})⊕···⊕F (a_{j},b_{j})}

F′′(w_{i}^{2}((a_{1}, b_{1}),…, (a_{i}, b_{i}, b′_{i}),…, (a_{j}, b_{j}))^{op}) =

1_{F (a_{1},b_{1})⊕···⊕F (a_{i−1},b_{i−1})}⊕w_{F}^{2} (a_{i}, b_{i}, b′_{i})^{−1}⊕1_{F (a_{i+1},b_{i+1})⊕···⊕F (a_{j},b_{j})}

F′′(h_{i}((a_{1}, b_{1}),…, (a_{i}, r, b_{i}),…, (a_{j}, b_{j}))^{op}) =

1_{F (a_{1},b_{1})⊕···⊕F (a_{i−1},b_{i−1})} ⊕ h_{F} (a_{i}, r, b_{i})^{−1} ⊕ 1_{F (a_{i+1},b_{i+1})⊕···⊕F (a_{j},b_{j})}

Now F′′ induces a functor A[T^{−1}] ⟶ P which in turn induces a functor \overline{F} : M⊗_{R}N → P such that \overline{F} ∘ η = F. It is straightforward to check that \overline{F} is permutative and has the stated uniqueness property.

Given (G, w_{G}) with G ∘ η = F, w_{G} determines a symmetric monoidal natural isomorphism G ⟶ \overline{F}.

4.14. Corollary. ⊗_{R} is a 2-functor Mod−R × R−Mod ⟶ Perm.

Proof. If F : M ⟶ M′ and G : N ⟶ N′ are morphisms of right and left R-modules respectively, then

M × N ⟶^{F ×G} M′ × N′ ⟶^{η′} M′⊗_{R}N′

is R-biadditive, hence induces a permutative functor F ⊗_{R}G : M⊗_{R}N ⟶ M′⊗_{R}N′.

If τ:F ⟶ F′ and σ:G ⟶ G′ are R-natural transformations, let (τ⊗_{R}σ)([e_{j}; (a_{1}, b_{1}), …, (a_{j}, b_{j})]) = [e_{j} → e_{j}; (τa_{1}, σb_{1}),…, (τa_{j}, σb_{j})]. This defines a permutative natural transformation F ⊗_{R}G ⟶ F′⊗_{R}G′.

The proof of the expected result for bimodules is straightforward and is left to the reader.

4.15. Corollary. The tensor product ⊗_{R} induces a 2-functor

T −Mod−R × R−Mod−T′ ⟶ T −Mod−T′

If R is a G-ring category, then its (ring) opposite R^{o} has objects and morphisms a^{o} for a in R. R^{o} is a G-ring category with a^{o} ⊕b^{o} = (a⊕b)^{o}, θ_{j}^{o}(x; a_{1}^{o},…, a_{j}^{o}) = θ_{j}(xτ_{j}; a_{1},…, a_{j})^{o} and (δ^{o})_{j}^{i}(x; a_{1}^{o},…, a_{i}^{o}, (a′_{i})^{o},…, a_{j}^{o}) = δ_{j}^{i}(xτ_{j}; a_{1},…, a_{i}, a′_{i},…, a_{j})^{o}, where τ_{j}(i) = j+1−i.

If T is an S-ring category, then T and T^{o} are isomorphic in S−Rng(Cat). F : T ⟶ T^{o} defined by F (a) = a^{o} is permutative and a lax S-morphism where h : F ∘θ_{j} ⟶ θ_{j}^{o}∘(1×F^{j}) has components h(σ; a_{1},…, a_{j}) = θ_{j}(σ → στ_{j}; 1_{a_{1}},…, 1_{a_{j}})^{o}.

For a G-ring category R and an S-ring category T, there are isomorphisms of 2categories R−Mod ≅ Mod−R^{o}, Mod−T^{o} ≅ Mod−T (by Proposition 4.16) and T−Mod ≅ Mod−T. Also note that a (left or right) T -module is a T −T -bimodule in the obvious way, hence for T -modules M and N Corollary 4.15 implies M ⊗_{T} N is a T −T -bimodule.

4.16. Proposition. Let R and T be G-ring categories. If R and T are equivalent (respectively isomorphic) in G−Rng(Cat), then R−Mod and T−Mod are equivalent (respectively isomorphic) as 2-categories.

Proof. Let (φ, w_{φ}, h_{φ}) : R ⟶ T and (ψ, w_{ψ}, h_{ψ}) : T ⟶ R be morphisms in G − Rng(Cat) with lax G-ring natural isomorphisms τ : φ∘ψ ≅ Id and σ : ψ ∘φ ≅ Id. Define 2-functors I : R−Mod ⟶ T −Mod and J : T −Mod ⟶ R−Mod as follows.

If (M, α_{M}, v_{M}) is an R-module let I(M, α_{M}, v_{M}) = (M, α̂_{M}, v̂_{M}) where α̂_{M} = α_{M} ∘ (ψ × 1_{M}) and v̂_{M}(t_{1}, t_{2}, a) = α_{M}(h_{ψ}(t_{1}, t_{2})^{−1}, 1_{a}) ∘ v_{M}(ψ(t_{1}), ψ(t_{2}), a). If (F, w_{F} , h_{F}) : (M, α_{M}, v_{M}) ⟶ (M′, α_{M′}, v_{M′}) is an R-module morphism letI(F, w_{F} , h_{F}) = (F, w_{F} , ĥ_{F}) where ĥ_{F} (t, a) = h_{F} (ψ(t), a). If ω : F ⟶ G is an R-natural transformation let (Iω)(a) = ω(a). J is defined similarly.

Define a 2-natural isomorphism \overline{σ} : J ∘ I ⟶ Id with components \overline{σ}(M) = (ι, h_{ι}) : (J ∘ I)(M) ⟶ M where ι = 1_{M} and h_{ι}(r, a) = α_{M}(σ(r), 1_{a}). A 2-natural isomorphism \overline{τ} : I ∘ J ⟶ Id is defined similarly. Thus R − Mod and T − Mod are equivalent as 2-categories.

If σ and τ are identities, then I = J^{−1} so R−Mod and T −Mod are isomorphic.

4.17. Proposition. Let M be a right R-module and N a left R-module.

(i) M ⊗_{R} R ˙ ≅ M and R ⊗_{R} N ˙ ≅ N in R−Mod.

(ii) M ⊗_{R} N ≅ N ⊗_{R} M as R−R-bimodules, if R is an S-ring category.

(iii) (M⊗_{R}N)⊗_{T} P ≅ M⊗_{R}(N⊗_{T} P) if N is an R−T -bimodule and P is a T -module.

Proof. α_{M} : M × R ⟶ M is R-biadditive, so we get a permutative functor \overline{α}_{M} : M⊗_{R}R ⟶ M. Define I : M ⟶ M⊗_{R}R by I(a) = η(a, 1) = [e_{1}; (a, 1)]. Then \overline{α}_{M} and I are right R-module morphisms such that \overline{α}_{M} ∘ I = Id and I ∘ \overline{α}_{M} ≅ Id. We note that I is a 2-natural transformation, but \overline{α} is only lax 2-natural.

The R-bilinear morphism M × N ⟶ N × M ⟶ N⊗_{R}M induces a permutative functor M⊗_{R}N ⟶ N⊗_{R}M which is easily seen to be an isomorphism of R−R-bimodules.

The proof of (iii) is straightforward (but lengthy) and is left to the reader.

If M and N are R-modules let Hom_{R}(M, N) be the category with objects the R-module morphisms from M to N and with morphisms the R-natural transformations. The set O of objects is topologized as a subspace of Map(obj M, obj N) × Map(mor M, mor N) and the set of morphisms as a subspace of O ×Map(obj M, mor N)×O. This is a permutative category under pointwise operations. Moreover if M is an R − T -bimodule and N is an R − T′-bimodule, then Hom_{R}(M, N) is a T − T′-bimodule.

An elaboration of the usual argument gives the following adjointness property.

4.18. Proposition. If M is an R − T -bimodule, then (M⊗_{T} (−), Hom_{R}(M, −)) is a lax 2-adjoint pair. In particular, if N is a T -module and P is an R-module, then there is an equivalence of permutative categories (hence of S(S^{0})-modules)

Hom_{R}(M⊗_{T} N, P) ˙ ≅ Hom_{T} (N, Hom_{R}(M, P))

If M is an R-module, then Hom_{R}(M, M) is an M-ring category with multiplicative operation being composition of functors and natural transformations. Distributivity isomorphisms

F_{1} ∘ ⋯ ∘ (F_{i} ⊕ F′_{i}) ∘ ⋯ ∘ F_{j} ⟶^{δ_{j}^{i}} (F_{1} ∘ ⋯ ∘ F_{i} ∘ ⋯ ∘ F_{j}) ⊕ (F_{1} ∘ ⋯ ∘ F′_{i} ∘ ⋯ ∘ F_{j})

are defined inductively. For j = 2 let δ_{2}^{1} = id : (F_{1} ⊕ F′_{1}) ∘ F_{2} ⟶ (F_{1} ∘ F_{2}) ⊕ (F′_{1} ∘ F_{2}) and δ_{2}^{2} = w_{F_{1}}(F_{2}(−), F′_{2}(−)) : F_{1} ∘ (F_{2} ⊕ F′_{2}) ⟶ (F_{1} ∘ F_{2}) ⊕ (F_{1} ∘ F′_{2}). Higher distributivity isomorphisms are composites of these. Note that Hom_{R}(M, M) satisfies strict right distributivity, i.e. δ_{j}^{i}(e_{j}; F_{1},…, F_{i}, F′_{i},…, F_{j}) = id when F_{k} = 1_{M} for k < i.

The endomorphism ring End_{R}M is defined to be the M-ring category Hom_{R}(M, M)^{o} which satisfies strict left distributivity.

If R is an M-ring category, then R and End_{R}R are equivalent in M−Rng(Cat). Define φ : R ⟶ End_{R}R by φ(r) = ρ_{r}^{o} where ρ_{r} is right translation, and φ(f : r → s) = τ_{f}^{o} where τ_{f} has components τ_{f}(a) = 1_{a} ⊗ f. Also define ψ : End_{R}R ⟶ R by ψ(F^{o}) = F (1) and ψ(τ^{o}) = τ(1). Then ψ ∘ φ = Id and φ ∘ ψ ≅ Id. The result is also true for a G-ring category R when G is B̃ or S.

We next construct a tensor product for algebras over an S-ring category. Suppose A is a G-ring category with ψ : End_{A}A ⟶ A as above and let End_{A}A_{A} be the sub G-ring category consisting of the A−A-bimodule morphisms and A−A-natural transformations. It is in fact an S-ring category.

4.19. Definition. Let A be a G-ring category. The center of A is the sub G-ring category Z(A) = ψ(End_{A}A_{A}), a full subcategory of A.

If G is B̃ or S, then Z(A) is the full subcategory with objects a such that γ(a, x) = γ(x, a)^{−1} for all objects x in A, where γ is the commutativity isomorphism. Thus Z(A) = A if A is an S-ring category.

4.20. Definition. Let R be an S-ring category and A a G-ring category. A is an Ralgebra in G −RngCat if there is a morphism φ : R ⟶ A in G −Rng(Cat) such that φ(R) is contained in Z(A) and φ : R ⟶ Z(A) is a morphism in S −Rng(Cat).

A morphism of R-algebras is a morphism F : A ⟶ A′ in G −RngCat such that F ∘ φ = φ′. An R-algebra natural transformation is a G-ring natural transformation τ : F ⟶ F′ of R-algebra morphisms such that τ(φ(r)) = 1_{φ′(r)}. The 2-category of R-algebras is denoted R−Alg_{G}.

4.21. Examples. (i) Any object A in G−RngCat is a Z(A)-algebra in G−RngCat.

(ii) There is an isomorphism of 2-caregories G−RngCat ≅ S(S^{0})−Alg_{G} analogous to the case of ordinary rings (ring = Z-algebra).

(iii) An R-algebra A is an A−A-bimodule and hence an A−R, R−A and R−R-bimodule by restriction.

4.22. Proposition. ⊗_{R} is a 2-functor R−Alg_{G} × R−Alg_{G} ⟶ R−Alg_{G}.

Proof. Let (A, φ) and (A′, φ′) be R-algebras in G−RngCat. The construction follows that for the tensor product of modules with some additional requirements.

First A × A′ is a G-category with zero, where G acts diagonally, hence S(A × A′) is a G-ring category satisfying strict left distributivity, [3;§3]. Next, partially extend the multiplicative operation ⊗ (i.e. the M-action underlying the G-action) to the morphisms in T ∐ T^{op}. It suffices to define just the following products by strict left distributivity. Let a = [e_{j}; (a_{1}, b_{1}),…, (a_{j}, b_{j})].

1_{a} ⊗ w_{1}^{1}((c, c′, d)) = [e_{2j} → σ(2, j); (1_{a_{1}⊗c}, 1_{b_{1}⊗d}), (1_{a_{1}⊗c′}, 1_{b_{1}⊗d}),…, (1_{a_{j}⊗c}, 1_{b_{j}⊗d}), (1_{a_{j}⊗c′}, 1_{b_{j}⊗d})] ∘

w_{1}^{1}((a_{1} ⊗ c, a_{1} ⊗ c′, b_{1} ⊗ d)) ⊕ ⋯ ⊕ w_{1}^{1}((a_{j} ⊗ c, a_{j} ⊗ c′, b_{j} ⊗ d)) ∘

[e_{j}; (δ_{2}^{2}(e_{2}; a_{1}, c, c′), 1_{b_{1}⊗d}),…, (δ_{2}^{2}(e_{2}; a_{j}, c, c′), 1_{b_{j}⊗d})]

where σ(2, j) ∈ Σ_{2j} is the obvious shuffle permutation.

1_{a} ⊗ w_{1}^{2}((c, d, d′)) = [e_{2j} → σ(2, j); (1_{a_{1}⊗c}, 1_{b_{1}⊗d}), (1_{a_{1}⊗c}, 1_{b_{1}⊗d′}),…, (1_{a_{j}⊗c}, 1_{b_{j}⊗d}), (1_{a_{j}⊗c}, 1_{b_{j}⊗d′})] ∘

w_{1}^{2}((a_{1} ⊗ c, b_{1} ⊗ d, b_{1} ⊗ d′)) ⊕ ⋯ ⊕ w_{1}^{2}((a_{j} ⊗ c, b_{j} ⊗ d, b_{j} ⊗ d′)) ∘
[e_{j}; (1_{a_{1}⊗c}, δ_{2}^{2}(e_{2}; b_{1}, d, d′)),…, (1_{a_{j}⊗c}, δ_{2}^{2}(e_{2}; b_{j}, d, d′))]

w_{i}^{1}((c_{1}, d_{1}),…, (c_{k}, d_{k})) ⊗ 1_{[e_{1};(a,b)]} = w_{i}^{1}((c_{1} ⊗ a, d_{1} ⊗ b),…, (c_{i} ⊗ a, c′_{i} ⊗ a, d_{i} ⊗ b),…, (c_{k} ⊗ a, d_{k} ⊗ b)) ∘

[e_{k} → e_{k}; 1_{c_{1}⊗a}, 1_{d_{1}⊗b}),…, (δ_{2}^{1}(e_{2}; c_{i}, c′_{i}, a), 1_{d_{i}⊗b}),…, 1_{c_{k}⊗a}, 1_{d_{k}⊗b})]

w_{i}^{2}((c_{1}, d_{1}),…, (c_{k}, d_{k})) ⊗ 1_{[e_{1};(a,b)]} = w_{i}^{1}((c_{1} ⊗ a, d_{1} ⊗ b),…, (c_{i} ⊗ a, d_{i} ⊗ b, d′_{i} ⊗ b),…, (c_{k} ⊗ a, d_{k} ⊗ b)) ∘

[e_{k} → e_{k}; 1_{c_{1}⊗a}, 1_{d_{1}⊗b}),…, (1_{c_{i}⊗a}, δ_{2}^{1}(e_{2}; d_{i}, d′_{i}, b)),…, 1_{c_{k}⊗a}, 1_{d_{k}⊗b})]

1_{a} ⊗ h_{1}((c, r, d)) = [e_{j} → e_{j}; (1_{a_{1}⊗c}, γ_{A′}(φ′(r), b_{1}) ⊗ 1_{d}),…, (1_{a_{j}⊗c}, γ_{A′}(φ′(r), b_{j}) ⊗ 1_{d})] ∘

h_{1}((a_{1} ⊗ c, r, b_{1} ⊗ d)) ⊕ ⋯ ⊕ h_{1}((a_{j} ⊗ c, r, b_{j} ⊗ d))

h_{i}((c_{1}, d_{1}),…, (c_{i}, r, d_{i}),…, (c_{k}, d_{k})) ⊗ 1_{[e_{1};(a,b)]} = h_{i}((c_{1} ⊗ a, d_{1} ⊗ b),…, (c_{i} ⊗ a, r, d_{i} ⊗ b),…, (c_{k} ⊗ a, d_{k} ⊗ b)) ∘

[e_{k} → e_{k}; (1_{c_{1}⊗a}, 1_{d_{1}⊗b}),…, (1_{c_{i}} ⊗ γ_{A}(φ(r), a), 1_{d_{i}⊗b}),…, (1_{c_{k}⊗a}, 1_{d_{k}⊗b})]

and similarly for morphisms t^{op} = t^{−1} in T^{op}.

We add to requirement (iii) above that (u_{1} ⊗ 1_{y_{2}}) ∘ (1_{x_{1}} ⊗ u_{2}) = (1_{y_{1}} ⊗ u_{2}) ∘ (u_{1} ⊗ 1_{x_{2}}) for morphisms u_{i} : x_{i} ⟶ y_{i} in S(A × A′)_{T}. The remainder of the construction is as for the module case. One can check that A⊗_{R}A′ is a G-ring category and that defining φ′′ : R ⟶ A⊗_{R}A′ by φ′′(r) = [e_{1}; (φ(r), 1)] makes A⊗_{R}A′ into an R-algebra in G−RngCat. We note that A⊗_{R}A′ satisfies strict left distributivity (even if A and A′ do not), since this is true for S(A × A′)_{T}. For 1 and 2-cells, ⊗_{R} is defined as for modules.

Let R be a G-ring category and let A be an H-category with zero, [3;§3], where H is also one of the operads M, B̃ or S. Then R ⊗ S(A) is an S(S^{0})-algebra in M−RngCat, where ⊗ = ⊗_{S(S^{0})}. In section five we show R ⊗ S(A) is the free strict R-module on the category A. If H = G, then R ⊗ S(A) is an S(S^{0})-algebra in G−RngCat. If G = S, any H, then R ⊗ S(A) is an R-algebra in H−RngCat.

For a category A let A_{+} denote A with an object 0 and a morphism 1_{0} adjoined. If A is an H-category, then A_{+} is an H-category with zero and we define the A_{∞} monoid ring of A to be R[A] = R ⊗ S(A_{+}). If R is an S-ring category, then R[A] is an object of R−Alg_{H}; R[A] is called the H-monoid algebra, or more loosely A_{∞} monoid algebra of A.

In [5] we will show how the integral A_{∞} monoid algebra S(S^{0})[A] can be used to construct algebraic K-theory of spaces and stable homotopy.

5. Morita equivalence for A_{∞} rings

We begin with the construction of free modules in R−[Mod]. Throughout this section ⊗ denotes the tensor product over S(S^{0}). Let U : R−[Mod] ⟶ Cat be the underlying category functor and define F = R⊗S(−) : Cat ⟶ R−[Mod]. The following lemma implies that (F, U) is a 2-adjoint pair.

5.1. Lemma. R ⊗ S(A) is the free strict R-module on the category A. Moreover, any two R-module morphisms R ⊗ S(A) ⟶ N(N an R-module) that agree on A are R-naturally isomorphic.

Proof. Let τ:A ⟶ R⊗S(A) be the functor defined on objects by τ(a)=[e_{1}; (1, [e_{1}; a])]. For M a strict R-module, a functor F : A ⟶ M induces a permutative functor F̃ : S(A) ⟶ M and regarding this as an S(S^{0})-module morphism we have a strict R-module morphism \overline{F} = \overline{α}_{M} ∘ (1_{R} ⊗ F̃) such that \overline{F} ∘ τ = F. Here \overline{α}_{M} : R ⊗ M ⟶ M is induced by the R-action α_{M} which is S(S^{0})-biadditive. It is a strict R-module morphism since M is strict.

Let η_{A} : R × S(A) ⟶ R ⊗ S(A) be the universal S(S^{0})-biadditve functor. If G : R ⊗ S(A) ⟶ M is a strict R-module morphism such that G ∘ τ = F, then using the universal property of S(−), we get G ∘ η_{A} = \overline{F} ∘ η_{A} and therefore G = \overline{F}. Thus R ⊗ S(A) is the free strict R-module on A. The second statement is a consequence of Theorem 4.13.

The adjoint pair (F, U, η, ε) determines a comonad (i.e. cotriple) (G, δ, ε) in R−[Mod] where G = F ∘ U and δ = FηU. Moreover, it extends to a lax comonad in R−Mod which we also denote by (G, δ, ε). It fails to be a strict comonad only because ε is a lax 2-natural transformation.

Define an R-module M to be G-projective , [1], if there is an R-module morphism s : M ⟶ GM such that ε_{M} ∘ s = 1_{M}. If M is projective in the usual sense, then it is G-projective, but not conversely.

5.2. Example. The standard Morita context. For an R-module M, let R′ = End_{R}M and M′ = Hom_{R}(M, R). Then M is an R−R′-bimodule where R′ acts by α′_{M}(a, F^{o}) = F (a) and for τ^{o} : F^{o} ⟶ G^{o} and f : a ⟶ b, α′_{M}(f, τ^{o}) = G(f) ∘ τ(a) = τ(b) ∘ F (f). It follows that M′ is an R′ −R-bimodule; the R′-action is denoted α_{M′}. Note that M′ is a strict R′-module, so M′⊗_{R}M is also a strict R′-module.

Define φ : M × M′ ⟶ R by φ(a, H) = H(a) and for f : a ⟶ b in M and σ : H_{1} ⟶ H_{2} in M′, φ(f, σ) = H_{2}(f) ∘ σ(a) = σ(b) ∘ H_{1}(f). φ is clearly 2-symmetric monoidal and in fact is R′-biadditive with h = id : φ ∘ (α′_{M} × 1_{M′}) ⟶ φ ∘ (1_{M} × α_{M′}). Thus φ induces an R−R-bimodule morphism E : M⊗_{R′}M′ ⟶ R.

Similarly there is an R-biadditive functor φ′ : M′ × M ⟶ R′ defined on objects by φ′(H, a) = α_{M}(H(−), a)^{o}, and this induces a morphism E′ : M′⊗_{R}M ⟶ R′ of R′−R′- bimodules.

A functor F is called surjective if it is surjective on both objects and morphisms.

5.3. Definition. (i) An R-module M is a generator for R−Mod if for each R-module N there is a category A and a surjection F : M ⊗ S(A) ⟶ N in R − Mod with F permutative.

(ii) M is finitely generated if there is a subcategory A of M such that the composite

R ⊗ S(A) ⟶^{1_{R}⊗S(I)} R ⊗ S(M) ⟶^{ε_{M}} M

is surjective, where I : A ↪ M is the inclusion. If all morphisms of A are identities we say M is generated by a finite subset.

(iii) M is finitely generated G-projective if it is G-projective and is generated by a finite subset A for which s : M ⟶ GM factors as

M ⟶^{σ} R ⊗ S(A) ⟶^{1_{R}⊗S(I)} R ⊗ S(M)

for some σ in R−Mod.

M is a progenerator if it is a generator and is finitely generated G-projective.

Recall the morphism E : M⊗_{R′}M′ ⟶ R and let I : M ⊗ M′ ⟶ M⊗_{R′}M′ be the obvious R-module morphism arising from the morphism S(S^{0}) ⟶ R′ of G-ring categories (Examples 4.8).

5.4. Lemma. The following are equivalent for an R-module M.

(i) M is a generator.

(ii) M ⊗ M′ ⟶^{I} M⊗_{R′}M′ ⟶^{E} R is surjective.

(iii) There is a category A and a surjection F : M ⊗ S(A) ⟶ R in R−Mod with F permutative.

Proof. Suppose M is a generator and let F : M ⊗ S(A) ⟶ R be a permutative surjection. By Proposition 4.18 we have a morphism of S(S^{0})-modules F̃ : S(A) ⟶ Hom_{R}(M, R) and since F is permutative it equals the composite

M ⊗ S(A) ⟶^{1_{M} ⊗_{F}} M ⊗ M′ ⟶^{I} M⊗_{R′}M′ ⟶^{E} R

Since F is surjective it follows E ∘ I is also.

Now assume E ∘ I is surjective, let A = M′ and consider the composite

M ⊗ S(A) ⟶^{1_{M} ⊗G} M ⊗ M′ ⟶^{I} M⊗_{R′}M′ ⟶^{E} R

where G is the permutative extension of id : A ⟶ M′. This composite is surjective since G is surjective.

Suppose given F : M ⊗ S(A) ⟶ R in R−Mod with F permutative and surjective. If N is an R-module, then the composite

M ⊗ S(S(A) ⊗ N) ⟶ M ⊗ (S(A) ⊗ N) ≅ (M ⊗ S(A)) ⊗ N ⟶ R ⊗ N ⟶ N

is permutative and surjective, hence M is a generator.

5.5. Examples. (i) If F : M ⟶ N is a surjection in R−Mod and N is a generator, then so is M.

(ii) The (strict) R-module morphism R⊗S(S^{0}) ⟶ R is surjective, so R and R⊗S(S^{0}) are generators.

(iii) If A is a category and there is a surjection A ⟶ S^{0} (i.e. A has an object not in the component of ∗), then we have a surjection F(A) = R ⊗ S(A) ⟶ R ⊗ S(S^{0}) ⟶ R, so F(A) is a generator.

We will use the notation of Example 5.2 and Definition 5.3 throughout the proofs of 5.6 and 5.7. Let Φ = ε_{M} ∘ (1_{R} ⊗ S(I)) : F(A) ⟶ M and A = {a_{0},…, a_{n}} with a_{0} = ∗. Notice that any two R-module morphisms F, G : M ⟶ N that agree on A are R-naturally isomorphic. For then F ∘Φ and G∘Φ agree on A and hence are R-naturally isomorphic by Lemma 5.1, so composing with σ gives the same for F and G. The functors π_{i}^{A} : A ⟶ R with π_{i}^{A}(a_{i}) = 1 and π_{i}^{A}(a_{k}) = 0, k ≠ i, induce R-module morphisms π̂_{i} : F(A) ⟶ R for i = 1,…, n and we let π_{i} = π̂_{i} ∘ σ. Also let I_{j} = α_{M}(π_{j}(−), a_{j}) ∈ Hom_{R}(M, M) and I_{A} = I_{1} ⊕ ⋯ ⊕ I_{n}. Then I_{A} and 1_{M} are R-naturally isomorphic by Lemma 5.1 since they agree on A.

5.6. Lemma. If M is finitely generated G-projective, then there is a lax 2-natural equivalence τ_{M} : Hom_{R}(M, R)⊗_{R}(−) ⟶ Hom_{R}(M, −) of 2-functors R−Mod ⟶ R′−Mod.

Proof. For an R-module N, τ_{M}^{0}(N) is induced by the R-biadditive morphism φ′_{N} : Hom_{R}(M, R) × N ⟶ Hom_{R}(M, N) with φ′_{N}(H, b) = α_{N}(H(−), b). For G : N ⟶ P in R−Mod, τ_{M}^{1}(G) : G_{∗} ∘ τ_{M}^{0}(N) ⟶ τ_{M}^{0}(P) ∘ (1_{M′}⊗_{R}G) is the R-natural transformation with components induced by h_{G}.

Define a (strict) 2-natural transformation γ_{M} : Hom_{R}(M, −) ⟶ Hom_{R}(M, R)⊗_{R}(−) by

γ_{M}(N)(F) = [e_{n}; (π_{1}, F a_{1}),…, (π_{n}, F a_{n})]

and

γ_{M}(N)(δ) = [1_{e_{n}}; (1_{π_{1}}, δ(a_{1})),…, (1_{π_{n}}, δ(a_{n}))]

for δ : F ⟶ F′ in Hom_{R}(M, N). To see γ_{M}(N) is in R′ − Mod, it suffices by lax naturality to check for N = M. Letting x = [e_{n}; (π_{1}, a_{1}),…, (π_{n}, a_{n})] ∈ M′⊗_{R}M, we have γ_{M}(M)(F) = x · F^{o}, the right R′-action on M′⊗_{R}M, so γ_{M}(M) is a morphism of right R′-modules. It is also a morphism of left R′-modules as follows. In M′⊗_{R}M we have isomorphisms

x · F^{o} ≅ [e_{n}; (π_{1}, I_{1}F (a_{1}) ⊕ ⋯ ⊕ I_{n}F (a_{1})),…, (π_{n}, I_{1}F (a_{n}) ⊕ ⋯ ⊕ I_{n}F (a_{n}))]

= [e_{n}; (π_{1}, α_{M}(π_{1}F (a_{1}), a_{1}) ⊕ ⋯ ⊕ α_{M}(π_{n}F (a_{1}), a_{n})),…,

(π_{n}, α_{M}(π_{1}F (a_{n}), a_{1}) ⊕ ⋯ ⊕ α_{M}(π_{n}F (a_{n}), a_{n}))]
≅ ⊕_{i,j} [e_{1}; (π_{i}, α_{M}(π_{j}F (a_{i}), a_{j}))]
≅ ⊕_{i,j} [e_{1}; (π_{i}(−) · π_{j}F (a_{i}), a_{j})]
≅ ⊕_{j=1}^{n} [e_{1}; (π_{1}(−) · π_{j}F (a_{1}) ⊕ ⋯ ⊕ π_{n}(−) · π_{j}F (a_{n}), a_{j})]
≅ ⊕_{j=1}^{n} [e_{1}; (π_{j}F, a_{j})]

≅ [e_{n}; (π_{1}F, a_{1}),…, (π_{n}F, a_{n})] = F^{o} · x

which can be used to give γ_{M}(M) a left R′-module morphism structure; in fact γ_{M}(M) is an R′−R′-bimodule morphism with this structure.

Define modifications ρ : τ_{M} ∘γ_{M} ⟶ id and λ : γ_{M} ∘τ_{M} ⟶ id with ρ(N) and λ(N) the R-natural transformations with components the following chains of isomorphisms, where G ∈ Hom_{R}(M, N) and y = [e_{m}; (H_{1}, b_{1}),…, (H_{m}, b_{m})] ∈ M′⊗_{R}M.

ρ(N)(G) : (τ_{M}^{0}(N) ∘ γ_{M}^{0}(N))(G) = α_{N}(π_{1}(−), G(a_{1})) ⊕ ⋯ ⊕ α_{N}(π_{n}(−), G(a_{n})) ≅ Gα_{M}(π_{1}(−), a_{1}) ⊕ ⋯ ⊕ Gα_{M}(π_{n}(−), a_{n})

≅ G(α_{M}(π_{1}(−), a_{1}) ⊕ ⋯ ⊕ α_{M}(π_{n}(−), a_{n}))
= Gτ_{M}^{0}(M)(x) ≅ G

λ(N)(y) : (γ_{M}^{0}(N) ∘ τ_{M}^{0}(N))(y) = [e_{n}; (π_{1}, α_{N}(H_{1}(a_{1}), b_{1}) ⊕ ⋯ ⊕ α_{N}(H_{m}(a_{1}), b_{m})),…,

(π_{n}, α_{N}(H_{1}(a_{n}), b_{1}) ⊕ ⋯ ⊕ α_{N}(H_{m}(a_{n}), b_{m}))]

≅ ⊕_{i,j} [e_{1}; (π_{i}, α_{N}(H_{j}(a_{i}), b_{j}))]
≅ ⊕_{i,j} [e_{1}; (π_{i}(−) · H_{j}(a_{i}), b_{j})]
≅ ⊕_{j=1}^{m} [e_{1}; (π_{1}(−) · H_{j}(a_{1}) ⊕ ⋯ ⊕ π_{n}(−) · H_{j}(a_{n}), b_{j})]
= y

This completes the proof.

5.7. Proposition. (i) If M is finitely generated G-projective, then E′ : M′⊗_{R}M ⟶ R′ is an equivalence of R′−R′-bimodules.

(ii) If M is a progenerator, then E : M⊗_{R′}M′ ⟶ R is an equivalence of R −Rbimodules.

Proof. (i). E′ is τ_{M}^{0}(M) composed with the anti-isomorphism of A_{∞}-rings

ψ : Hom_{R}(M, M) ≅ R′

(an isomorphism of R′−R′-bimodules). Let Σ′ : R′ ⟶ M′⊗_{R}M be γ_{M}(M) ∘ ψ^{−1}. From the proof of Lemma 5.6, Σ′ is an R′−R′-bimodule inverse equivalence of E′.

(ii). E is surjective by Lemma 5.4, so there is y = [e_{m}; (b_{1}, H_{1}),…, (b_{m}, H_{m})] ∈ M⊗_{R}M′ with E(y) = 1. Define Σ : R ⟶ M⊗_{R}M′ by Σ(r) = r · y, and note that E ∘ Σ = 1_{R}. As in the proof of Lemma 5.6 one can show r · y ≅ y · r and obtain that Σ is an R−R-bimodule morphism. The diagram

[figure: R′⊗_{R′}M′; E′⊗_{R′}1_{M′}; M′⊗_{R}M⊗_{R′}M′; ^{\overline{α}^{M′}}; 1_{M′}⊗_{R}E; M′; ^{\overline{β}^{M′}}; M′⊗_{R}R]

commutes up to R′ −R-natural isomorphism, where α_{M′} is the left R′-action and β_{M′} is the right R-action. Since (1_{M′}⊗_{R}E) ∘ (1_{M′}⊗_{R}Σ) = 1_{M′⊗_{R}R} we see that 1_{M′}⊗_{R}Σ is an R′ − R-bimodule inverse equivalence of 1_{M′}⊗_{R}E. By Lemma 5.6, we have that Σ_{∗} : Hom_{R}(M, R) ⟶ Hom_{R}(M, M⊗_{R′}M′) is an R′ − R-bimodule inverse equivalence of E_{∗} : Hom_{R}(M, M⊗_{R′}M′) ⟶ Hom_{R}(M, R). Applying Hom_{R′}(M′, −) to this pair of equivalences and using adjointness we now have

Σ_{∗} : Hom_{R}(M⊗_{R′}M′, R) ⟶ Hom_{R}(M⊗_{R′}M′, M⊗_{R′}M′)

and

E_{∗} : Hom_{R}(M⊗_{R′}M′, M⊗_{R′}M′) ⟶ Hom_{R}(M⊗_{R′}M′, R)

are R−R-bimodule inverse equivalences. This gives an R−R-natural isomorphism δ : Σ_{∗} ∘E_{∗} ⟶ Id and hence an R−R-natural isomorphism δ(1_{M⊗_{R′}M′}) : Σ∘E ⟶ 1_{M⊗_{R′}M′}.

5.8. Theorem. If M is a progenerator in R−Mod, then M⊗_{R′}(−) is a lax 2-equivalence of the 2-categories R−Mod and R′−Mod.

Proof. We show M′⊗_{R}(−) is an inverse 2-equivalence of M⊗_{R′}(−). For an R-module N, let \overline{α}_{N} and I_{N} be the equivalences of Proposition 4.17. Define lax 2-natural transformations σ : M⊗_{R′}(−) ∘ M′⊗_{R}(−) ⟶ Id and τ : M′⊗_{R}(−) ∘ M⊗_{R′}(−) ⟶ Id by:

σ^{0}(N) : M⊗_{R′}(M′⊗_{R}N) ≅^{U_{N}} (M⊗_{R′}M′)⊗_{R}N ⟶^{E⊗_{R}1_{N}} R⊗_{R}N ⟶^{\overline{α}_{N}} N

τ^{0}(P) : M′⊗_{R}(M⊗_{R′}P) ≅^{V_{P}} (M′⊗_{R}M)⊗_{R′}P ⟶^{E′⊗_{R′}1_{P}} R′⊗_{R′}P ⟶^{\overline{α}_{P}} P

and for F : N ⟶ L in R−Mod, G : P ⟶ Q in R′−Mod, let σ^{1}(F) = h_{F} ∗_{0} 1_{E⊗_{R}1_{N}}, τ^{1}(G) = h_{G} ∗_{0} 1_{E′⊗_{R′}1_{P}}, horizontal composition of 2-cells.

σ and τ have “inverses” \overline{σ} and \overline{τ}, the (strict) 2-natural transformations with \overline{σ}(N) = U_{N}^{−1} ∘ (Σ⊗_{R}1_{N}) ∘ I_{N} and \overline{τ}(P) = V_{P}^{−1} ∘ (Σ′⊗_{R′}1_{P}) ∘ I_{P}. It is straightforward to write down the required modifications (\overline{σ} ∘ σ ⟶ id, etc.) and this is left to the reader.

5.9. Theorem. If M is a progenerator in R−Mod, then (M⊗_{R′}(−), Hom_{R}(M, −)) is a lax 2-adjoint, lax 2-equivalence of the 2-categories R−Mod and R′−Mod.

Proof. (M⊗_{R′}(−), Hom_{R}(M, −)) is a lax 2-adjoint pair by Proposition 4.18. The unit η and counit ε are the composites

η : Id ⟶^{\overline{τ}} M′⊗_{R}(−) ∘ M⊗_{R′}(−) ⟶ Hom_{R}(M, −) ∘ M⊗_{R′}(−)

ε : M⊗_{R′}(−) ∘ Hom_{R}(M, −) ⟶ M⊗_{R′}(−) ∘ M′⊗_{R}(−) ⟶^{σ} Id

where σ and \overline{τ} are as in the proof of Theorem 5.8 and the unlabelled morphisms arise from the lax 2-natural equivalence of Lemma 5.6. η and ε are clearly lax 2-natural equivalences.

5.10. Remark. A_{∞} rings R and T are Morita equivalent if R−Mod and T −Mod are lax 2-equivalent. In [5] we will show that Morita equivalent A_{∞} rings have the same K-theory.

References

[1] M. Barr and J. Beck, Homology and Standard Constructions, in B. Eckmann (ed.) Seminar on Triples and Categorical Homology Theory, Lecture Notes in Math. 80, Springer, Berlin, 1969.

[2] V. G. Drinfel’d, Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations, Res. Rep. Phys., 1-13, Springer, Berlin, 1989.

[3] G. Dunn, E_{n}-Ring Categories, J. Pure Appl. Algebra (to appear).

[4] G. Dunn, K-theory of Braided Tensor Ring Categories with Higher Commutativity, K-Theory 9: 591-605, 1995.

[5] G. Dunn, Localization for the K-theory of A_{∞}-Rings (in preparation).

[6] Z. Fiedorowicz, Symmetric and Braided Bar Constructions (unpublished notes, 1991).

[7] A. Joyal and R. Street, An Introduction to Tannaka Duality and Quantum Groups, in Category Theory, Lecture Notes in Math. 1488, Springer, Berlin, 1991.

[8] M.M. Kapranov and V.A. Voevodsky, 2-Categories and Zamolodchikov Tetrahedra Equations, Proc. Sympos. Pure Math., 56, part 2, 177-259, Amer. Math. Soc., 1994.

[9] G.M. Kelly, M. LaPlaza, G. Lewis and S. MacLane, Coherence in Categories, Lecture Notes in Math. 281, Springer, Berlin, 1972.

[10] G.M. Kelly and R. Street, Review of the Elements of 2-Categories, in Category Seminar, Sydney 1972/73, Lecture Notes in Math. 420, Springer, Berlin, 1974.

[11] J.P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Math. 271, Springer, Berlin, 1972.

[12] J.P. May, E_{∞}-spaces, Group Completions and Permutative Categories, in G. Segal (ed.) New Developments in Topology, London Math. Soc. Lecture Note Series 11, Cambridge University Press, 1974.

[13] J.P. May (with contributions by N. Ray, F. Quinn, and J. Tornehave), E_{∞} Ring Spaces and E_{∞} Ring Spectra, Lecture Notes in Math. 577, Springer, Berlin, 1977.

[14] R. Street, The Algebra of Oriented Simplexes, J. Pure Appl. Algebra 49 (1987) 283-335.

Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003 U.S.A.

Email: gdunn@nmsu.edu

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@triples.math.mcgill.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, University of Genoa: carboni@vmimat.mat.unimi.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.su.oz.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@gar.union.edu Robert Paré, Dalhousie University: pare@cs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@macadam.mpce.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.su.oz.au R. J. Wood, Dalhousie University: rjwood@cs.da.ca