aa r X i v : . [ m a t h . C T ] J a n SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPYTHEORY
JOHN BOURKE
Abstract.
We study Quillen model categories equipped with a monoidalskew closed structure that descends to a genuine monoidal closed structureon the homotopy category. Our examples are 2-categorical and include per-mutative categories and bicategories. Using the skew framework, we adaptEilenberg and Kelly’s theorem relating monoidal and closed structure to thehomotopical setting. This is applied to the construction of monoidal bicate-gories arising from the pseudo-commutative 2-monads of Hyland and Power. Introduction
The notion of a monoidal closed category captures the behaviour of the tensorproduct and internal hom on classical categories such as those of sets and vectorspaces. Some of the basic facts about monoidal closed categories have an intuitivemeaning. For instance, the isomorphism(1.1) C ( A, B ) ∼ = C ( I, [ A, B ])says that elements of the internal hom [
A, B ] are the same thing as morphisms A → B .Recently some new variants have come to light. Firstly, the skew monoidalcategories of Szlach´anyi [36] in which the structure maps such as ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) have a specified orientation and are not necessarily invertible. Shortlyafterwards the dual notion of a skew closed category was introduced by Street[35]. Here one has a canonical map(1.2) C ( A, B ) → C ( I, [ A, B ])but this need not be invertible. Intuitively, we might view this relaxation as say-ing that [
A, B ] should contain the morphisms A → B as elements, but possiblysomething else too.In the present paper a connection is drawn between skew structures and homo-topy theory. We study examples of Quillen model categories C in which thecorrect internal homs [ A, B ] have more general weak maps A B as elements.By the above reasoning these examples are necessarily skew. These skew closedcategories form part of enveloping monoidal skew closed structures that descendto the homotopy category Ho ( C ) where, in fact, they yield genuine monoidalclosed structures. The study of skew structures on a category that induce gen-uine structures on the homotopy category is our main theme. Date : February 17, 2018.2000
Mathematics Subject Classification.
Primary: 18D10, 55U35.
Our examples are 2-categorical in nature – most involve tweaking better knownweak 2-categorical structures to yield not strict, but skew, structures. For ex-ample, we describe a monoidal skew closed structure on the 2-category of per-mutative categories – symmetric strict monoidal categories – and strict maps.This contains, on restricting to the cofibrant objects, a copy of the well knownmonoidal bicategory of permutative categories and strong maps. More generally,we describe a skew structure for each pseudo-commutative 2-monad T on Cat in the sense of [13]. Other examples concern 2-categories and bicategories.The theory developed in the present paper has a future goal, concerning Gray-categories, in mind. It was shown in [4] that there exists no homotopically wellbehaved monoidal biclosed structure on the category of Gray-categories. Theplan is, in a future paper, to use the results developed here to understand thecorrect enriching structure on the category of Gray-categories.Let us now give an overview of the paper. Section 2 is mainly background onskew monoidal, skew closed and monoidal skew closed categories. We recallStreet’s theorem describing the perfect correspondence between skew monoidalstructures ( C , ⊗ , I ) and skew closed structures ( C , [ − , − ] , I ) in the presence ofadjointness isomorphisms C ( A ⊗ B, C ) ∼ = C ( A, [ B, C ]). In Theorem 2.6 we refor-mulate Eilenberg and Kelly’s theorem [8], relating monoidal and closed structure,in the skew language. Finally, we introduce symmetric skew closed categories.It turns out that all the examples of skew closed structures that we meet in thepresent paper can be seen as arising from certain multicategories in a canonicalway. In Section 3 we describe the passage from such multicategories to skewclosed categories.Using the multicategory approach where convenient, Section 4 gives concreteexamples of some of the skew closed structures that we are interested in. We de-scribe the examples of categories with limits, permutative categories, 2-categoriesand bicategories.Section 5 concerns the interaction between skew structures and Quillen modelstructures that lies at the heart of the paper. We begin by describing how askew monoidal structure ( C , ⊗ , I ) can be left derived to the homotopy category.This is the skew version of Hovey’s construction [12]. We call ( C , ⊗ , I ) homotopymonoidal if the left derived structure ( Ho ( C ) , ⊗ l , I ) is genuinely monoidal. Thisis complemented by an analysis of how skew closed structure can be right derived to the homotopy category, and we obtain a corresponding notion of homotopyclosed category. Combining these cases Theorem 5.11 describes how monoidalskew closed structure can be derived to the homotopy category. This is used toprove Theorem 5.12, a homotopical analogue of Eilenberg and Kelly’s theorem,which allows us to recognise homotopy monoidal structure in terms of homotopyclosed structure.Section 6 returns to the examples of categories with limits and permutativecategories in the more general setting of pseudo-commutative 2-monads T on Cat . We make minor modifications to Hyland and Power’s construction [13] ofa pseudo-closed structure on T-Alg to produce a skew closed structure on the2-category T-Alg s of algebras and strict morphisms. For accessible T this formspart of an enveloping monoidal skew closed structure which, using Theorem 5.12, KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 3 we show to be homotopy monoidal. Using this, we give a complete constructionof the monoidal bicategory structure on T-Alg of Hyland and Power – thus solv-ing a problem of [13].Section 7 consists of an in-depth analysis of the skew structure on the categoryof bicategories and strict homomorphisms. Though not particularly interestingin its own right, we regard this example as a preliminary to future work in higherdimensions.
Acknowledgments.
The author thanks the organisers of the Cambridge Cate-gory Theory Seminar and of CT2015 in Aveiro for providing the opportunityto present this work, and thanks Sofie Royeaerd for useful feedback on an earlydraft. Skew monoidal and skew closed categories
Skew monoidal categories.
Skew monoidal categories were introduced bySzlach´anyi [36] in the study of bialgebroids over rings. There are left and rightversions (depending upon the orientation of the associativity and unit maps) andit is the left handed case that is of interest to us.
Definition 2.1.
A (left) skew monoidal category ( C , ⊗ , I, α, l, r ) is a category C together with a functor ⊗ : C × C → C , a unit object I ∈ C , and natural families α A,B,C : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ), l A : I ⊗ A → A and r A : A → A ⊗ I satisfying five axioms [36].There is no need for us to reproduce these five axioms here as we will not usethem. We remark only that they are neatly labelled by the five words abcdaib aib abiii of which the first refers to MacLane’s pentagon axiom.Henceforth the term skew monoidal is taken to mean left skew monoidal. Amonoidal category is precisely a skew monoidal category in which the constraints α, l and r are invertible.2.2. Skew closed categories.
In the modern treatment of monoidal closed cate-gories as a basis for enrichment [16] it is the monoidal structure that is typicallytreated as primitive. Nonetheless, the first major treatment [8] emphasised theclosed structure, presumably because internal homs are often more easily de-scribed than the corresponding tensor products. In the examples of interest tous (see Section 4) this is certainly the case. These examples will not be closedin the sense of ibid. but only skew closed.
Definition 2.2 (Street [35]) . A (left) skew closed category ( C , [ − , − ] , I, L, i, j )consists of a category C equipped with a bifunctor [ − , − ] : C op × C → C and unitobject I together with(1) components L = L AB,C : [
B, C ] → [[ A, B ] , [ A, C ]] natural in
B, C and ex-tranatural in A ,(2) a natural transformation i = i A : [ I, A ] → A , JOHN BOURKE (3) components j = j A : I → [ A, A ] extranatural in A ,satisfying the following five axioms.(C1) [[ A, C ] , [ A, D ]] L * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ [ C, D ] L ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ L (cid:15) (cid:15) [[[ A, B ] , [ A, C ]] , [[ A, B ] , [ A, D ]]] [ L, (cid:15) (cid:15) [[ B, C ] , [ B, D ]] [1 ,L ] / / [[ B, C ] , [[ A, B ] , [ A, D ]]].( C
2) [[
A, A ] , [ A, C ]] [ I, [ A, C ]] ( C
3) [
B, B ] [[
A, B ] , [ A, B ]][
A, C ] [
A, C ] I / / L O O i (cid:15) (cid:15) [ j, / / L / / j a a ❈❈❈❈❈❈❈❈❈ j = = ④④④④④④④④④ ( C
4) [
B, C ] [[
I, B ] , [ I, C ]] ( C I [ I, I ][[
I, B ] , C ] I L / / j / / [ i, (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ [1 ,i ] { { ✇✇✇✇✇✇✇✇✇ " " ❉❉❉❉❉❉❉❉❉❉❉ i | | ①①①①①①①①①① ( C , [ − , − ] , I ) is said to be left normal when the composite function(2.1) C ( A, B ) [ A, − ] / / C ([ A, A ] , [ A, B ]) C ( j, / / C ( I, [ A, B ])is invertible, and right normal if i : [ I, A ] → A is invertible. A closed category is, by definition, a skew closed category which is both left and right normal. Variants 2.3.
We will regularly mention a couple of variants on the above defini-tion and we note them here.(1) We will sometimes consider skew closed 2-categories: the
Cat -enriched ver-sion of the above concept. The difference is that C is now a ,[ − , − ] a and each of the three transformations in eachvariable.(2) We call a structure ( C , [ − , − ] , L ) satisfying C semi-closed category. The original definition of closed category [8] involved an underlying functor to Set . Weare using the modified definition of [33] (see also [26]) which eliminates the reference to
Set . KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 5
The correspondence between skew monoidal and skew closed categories.
A monoidal category ( C , ⊗ , I ) in which each functor − ⊗ A : C → C has aright adjoint [ A, − ] naturally gives rise to the structure ( C , [ − , − ] , I ) of a closedcategory. Counterexamples to the converse statement are described in Section3 of [6]: no closed category axiom ensures the associativity of the correspondingtensor product. An appealing feature of the skew setting is that there is a perfectcorrespondence between skew monoidal and skew closed structure. Theorem 2.4 (Street [35]) . Let C be a category equipped with an object I and a pairof bifunctors ⊗ : C × C → C and [ − , − ] : C op × C → C related by isomorphisms ϕ : C ( A ⊗ B, C ) ∼ = C ( A, [ B, C ]) natural in each variable. There is a bijectionbetween extensions of ( C , ⊗ , I ) to a skew monoidal structure and of ( C , [ − , − ] , I ) to a skew closed structure. Our interest is primarily in the passage from the closed to the monoidal sideand, breaking the symmetry slightly, we describe it now: for the full symmetrictreatment see [35]. • l : I ⊗ A → A is the unique map such that the diagram(2.2) C ( A, B ) v ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ C ( l, / / C ( I ⊗ A, B ) ϕ (cid:15) (cid:15) C ( I, [ A, B ])commutes for all B . Here v = C ( j, ◦ [ A, − ] : C ( A, B ) → C ( I, [ A, B ]) isthe morphism defining left normality.
In particular l is invertible for each A just when v is. • r : A → A ⊗ I is the unique morphism such that the diagram(2.3) C ( A ⊗ I, B ) ϕ (cid:15) (cid:15) C ( r, / / C ( A, B ) C ( A, [ I, B ]) C (1 ,i ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ commutes for all B . In particular r is invertible for each A just when i is. • Transposing the identity through the isomorphism ϕ : C ( A ⊗ B, A ⊗ B ) ∼ = C ( A, [ B, A ⊗ B ]) yields a morphism u : A → [ B, A ⊗ B ] natural in eachvariable. Write t : [ A ⊗ B, C ] → [ A, [ B, C ]] for the composite(2.4) [ A ⊗ B, C ] L / / [[ B, A ⊗ B ] , [ B, C ]] [ u, / / [ A, [ B, C ]] JOHN BOURKE which, we note, is natural in each variable. The constraint α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) is the unique morphism rendering commutative the diagram(2.5) C ( A ⊗ ( B ⊗ C ) , D ) ϕ (cid:15) (cid:15) C ( α, / / C (( A ⊗ B ) ⊗ C, D ) ϕ (cid:15) (cid:15) C ( A ⊗ B, [ C, D ]) ϕ (cid:15) (cid:15) C ( A, [ B ⊗ C, D ]) C (1 ,t ) / / C ( A, [ B, [ C, D ]])for all D . In particular α is invertible just when t is. Definition 2.5.
A monoidal skew closed category consists of a skew monoidalcategory ( C , ⊗ , I, α, l, r ) and skew closed category ( C , [ − , − ] , I, L, i, j ), togetherwith natural isomorphisms ϕ : C ( A ⊗ B, C ) ∼ = C ( A, [ B, C ]) all related by theabove equations.Of course in the presence of the isomorphisms either bifunctor determines theother. Accordingly a monoidal skew closed category is determined by either theskew monoidal or closed structure together with the isomorphisms ϕ .We remark that monoidal skew closed structures on the category of left R -modules over a ring R that have R as unit correspond to left bialgebroids over R . This was the reason for the introduction of skew monoidal categories in [36].The following result – immediate from the above – is, minus the skew monoidalterminology, contained within Chapter 2 and in particular Theorem 5.3 of [8]. Theorem 2.6 (Eilenberg-Kelly) . Let ( C , ⊗ , [ − , − ] , I ) be a monoidal skew closedcategory. Then ( C , ⊗ , I ) is monoidal if and only if ( C , [ − , − ] , I ) is closed and thetransformation t : [ A ⊗ B, C ] → [ A, [ B, C ]] is an isomorphism for all A, B and C . Eilenberg and Kelly’s theorem can be used to recognise monoidal structurein terms of closed structure. However it can be difficult to determine whether t : [ A ⊗ B, C ] → [ A, [ B, C ]] is invertible. This difficulty disappears in the presenceof a suitable symmetry.2.4.
Symmetry.
A symmetry on a skew closed category begins with a naturalisomorphism s : [ A, [ B, C ]] ∼ = [ B, [ A, C ]]. If C is left normal the vertical maps C ( A, [ B, C ]) v (cid:15) (cid:15) s / / C ( B, [ A, C ]) v (cid:15) (cid:15) C ( I, [ A, [ B, C ]]) C ( I,s ) / / C ( I, [ B, [ A, C ]])are isomorphisms, so that we obtain an isomorphism s by conjugating C ( I, s ).If C underlies a monoidal skew closed category this in turn gives rise to a naturalisomorphism C ( A ⊗ B, C ) ∼ = C ( B ⊗ A, C ) KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 7 and so, by Yoneda, a natural isomorphism(2.6) c : B ⊗ A ∼ = A ⊗ B .
Our leading examples of skew closed categories do admit symmetries, but are not left normal: accordingly, the symmetries are visible on the closed side butnot on the monoidal side. However they often reappear on the monoidal sideupon passing to the homotopy category – see Theorem 5.13.
Definition 2.7.
A symmetric skew closed category consists of a skew closedcategory ( C , [ − , − ] , I ) together with a natural isomorphism s : [ A, [ B, C ]] ∼ =[ B, [ A, C ]] satisfying the following four equations.(S1) [ A, [ B, C ]] s ' ' ◆◆◆◆◆◆◆◆◆◆◆ / / [ A, [ B, C ]][ B, [ A, C ]] s ♣♣♣♣♣♣♣♣♣♣♣ (S2) [ A, [ B, [ C, D ]]] [1 ,s ] (cid:15) (cid:15) s / / [ B, [ A, [ C, D ]]] [1 ,s ] / / [ B, [ C, [ A, D ]]] s (cid:15) (cid:15) [ A, [ C, [ B, D ]]] s / / [ C, [ A, [ B, D ]]] [1 ,s ] / / [ C, [ B, [ A, D ]]](S3) [ A, [ B, C ]] s (cid:15) (cid:15) L / / [[ D, A ] , [ D, [ B, C ]]] [1 ,s ] / / [[ D, A ] , [ B, [ D, C ]]] s (cid:15) (cid:15) [ B, [ A, C ]] [1 ,L ] / / [ B, [[ D, A ] , [ D, C ]]](S4)[
A, B ] L / / [[ A, A ] , [ A, B ]] [ j, / / [ I, [ A, B ]] s / / [ A, [ I, B ]] [1 ,i ] / / [ A, B ] C is said to be symmetric closed if its underlying skew closed category is closed. Variants 2.8.
As in Variants 2.3 there are evident notions of symmetric skewclosed 2-categories and symmetric semi-closed categories.
Remark 2.9.
The notion of symmetric closed category described above coincideswith that of [6], though this may not be immediately apparent. Their invertibleunit map i : X → [ I, X ] points in the opposite direction to ours. Reversingit, their (CC4) is clearly equivalent to our (S4). Their remaining axioms are aproper subset of those above, with (C1), (C3), (C4) and (C5) omitted. But asthey point out in Proposition 1.3 any symmetric closed category in their senseis a closed category and hence satisfies all four of these.I first encountered a result close to the following one as Proposition 2.3 of[6], which shows that a symmetric closed category C gives rise to a symmetric JOHN BOURKE promonoidal one by setting P ( A, B, C ) = C ( A, [ B, C ]). This easily implies thata symmetric closed category gives rise to a symmetric monoidal one on takingadjoints. In a discussion about that result, Ross Street pointed out that a skewmonoidal category with an invertible natural isomorphism A ⊗ B ∼ = B ⊗ A satisfying the braid equation B Theorem 2.10 (Day-LaPlaza, Street) . Let ( C , ⊗ , [ − , − ] , I ) be monoidal skew closed.(1) The transformation t : [ A ⊗ B, C ] → [ A, [ B, C ]] is invertible if ( C , [ − , − ] , I ) is left normal and admits a natural isomorphism s : [ A, [ B, C ]] ∼ = [ B, [ A, C ]] satisfying S3. In particular, if ( C , [ − , − ] , I ) is actually closed and admitssuch a symmetry then ( C , ⊗ , I ) is monoidal.(2) If ( C , [ − , − ] , I, s ) is symmetric closed then ( C , ⊗ , I, c ) is symmetric monoidal.Proof. For (1) we first prove that(2.7) [ A ⊗ B, [ C, D ]] t (cid:15) (cid:15) s / / [ C, [ A ⊗ B, D ]] [1 ,t ] (cid:15) (cid:15) [ A, [ B, [ C, D ]]] [1 ,s ] / / [ A, [ C, [ B, D ]]] s / / [ C, [ A, [ B, D ]]]commutes. From (2.4) we have t = [ u, ◦ L . So the upper path equals[[1 , [ u, ◦ [1 , L ] ◦ s = [1 , [ u, ◦ s ◦ [1 , s ] ◦ L = s ◦ [1 , s ] ◦ [ u, ◦ L = s ◦ [1 , s ] ◦ t by S3, naturality of s twice applied, and the definition of t .By assumption each component v : C ( A, B ) → C ( I, [ A, B ]) is invertible. Ac-cordingly a morphism f : [ A, B ] → [ C, D ] gives rise to a further morphism f : C ( A, B ) → C ( C, D ) by conjugating C ( I, f ). At [
A, g ] : [
A, B ] → [ A, C ] weobtain [
A, g ] = C ( A, g ). At L : [ A, B ] → [[ C, A ] , [ C, B ]] an application of C L = [ C, − ] : C ( A, B ) → C ([ C, A ] , [ C, B ]). Now ( − ) , being definedby conjugating through natural isomorphisms, preserves composition. Combin-ing the two last cases we find that t = [ u, ◦ L = C ( u, ◦ [ B, − ] = ϕ , theadjointness isomorphism. Applying ( − ) to the above diagram, componentwise,then gives the commutative diagram below. C ( A ⊗ B, [ C, D ]) ϕ (cid:15) (cid:15) s / / C ( C, [ A ⊗ B, D ]) C ( C,t ) (cid:15) (cid:15) C ( A, [ B, [ C, D ]]) C ( A,s ) / / C ( A, [ C, [ B, D ]]) s / / C ( C, [ A, [ B, D ]])Since the left vertical path and both horizontal paths are isomorphisms, so is C ( C, t ) for each C . Therefore t is itself an isomorphism. The remainder of (1)now follows from Theorem 2.6.As mentioned, Part 2 follows from Proposition 2.3 of [6]. We note an alternativeelementary argument. Having established the commutativity of (2.7) and that t is an isomorphism, we are essentially in the presence of what De Schipper calls KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 9 a monoidal symmetric closed category . Theorem 6.2 of [7] establishes that amonoidal symmetric closed category determines a symmetric monoidal one, asrequired. (cid:3) From multicategories to skew closed categories
Our examples of skew closed categories in Section 4 can be seen as arising fromclosed multicategories equipped with further structure. In the present sectionwe describe how to pass from such multicategories to skew closed categories.Multicategories were introduced in [25] and have objects
A, B, C . . . togetherwith multimaps ( A , . . . , A n ) → B for each n ∈ N . These multimaps can becomposed and satisfy natural associativity and unit laws. We use boldface C fora multicategory and C for its underlying category of unary maps. A symmetric multicategory C comes equipped with actions of the symmetric group S n on thesets C ( A , . . . A n ; B ) of n -ary multimaps. These actions must be compatible withmultimap composition. For a readable reference on the basics of multicategorieswe refer to [27].3.1. Closed multicategories.
A multicategory C is said to be closed if for all B, C ∈ C there exists an object [ B, C ] and evaluation multimap e : ([ B, C ] , B ) → C with the universal property that the induced function(3.1) C ( A , . . . , A n ; [ B, C ]) → C ( A , . . . , A n , B ; C )is a bijection for all ( A , . . . , A n ) and n ∈ N . We can depict the multimap e : ([ B, C ] , B ) → C as below.(3.2) e ❘❘❘❘❘❘❘ [ B, C ] ❧❧❧❧❧❧❧ B C
Semiclosed structure.
Using the above universal property one obtains abifunctor [ − , − ] : C op × C → C . Given f : B → C the map [ A, f ] : [
A, B ] → [ A, C ]is the unique one such that the two multimaps(3.3) e [ A, f ] [
A, C ] ❙❙❙❙❙❙ [ A, B ] A ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ C = e f CBA ❤❤❤❤❤❤❤❤ [ A, B ] ❱❱❱❱❱❱ This is not exactly the case as De Schipper, following Eilenberg-Kelly, includes a basicfunctor V : C →
Set in his definition of symmetric closed category. However this basic functorplays no role in the proof of the cited result. coincide, whilst [ f, A ] is defined in a similar manner.The natural bijections C ([ B, C ] , [ A, B ] , A ; C ) ∼ = C ([ B, C ] , [ A, B ]; [
A, C ]) ∼ = C ([ B, C ] , [[ A, B ] , [ A, C ]])induce a unique morphism L : [ B, C ] → [[ A, B ] , [ A, C ]] such that the multimaps(3.4) eeL [[ A, B ] , [ A, C ]] ❊❊❊❊❊❊❊ [ B, C ] [
A, C ] ▲▲▲▲▲▲❞❞❞❞❞❞❞❞❞❞❞❞❞ [ A, B ] ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ A C = e eB ♦♦♦♦♦❘❘❘❘❘❘❘ [ A, B ] ❧❧❧❧❧❧❧ A ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ [ B, C ] C coincide.3.1.2. Symmetry. If C is a symmetric multicategory the natural bijections C ([ A, [ B, C ]] , B, A ; C ) ∼ = C ([ A, [ B, C ]] , B ; [ A, C ]) ∼ = C ([ A, [ B, C ]] , [ B, [ A, C ]])induce a unique map s : [ A, [ B, C ]] → [ B, [ A, C ]] such that the multimaps(3.5) ees [ B, [ A, C ]] ❊❊❊❊❊❊❊ [ A, [ B, C ]] [
A, C ] ▲▲▲▲▲▲❞❞❞❞❞❞❞❞❞❞❞❞❞ B ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ A C = ee [ B, C ] ▲▲▲▲▲▲tttttttttttttttt A ❲❲❲❲❲❲❲❲❲❲❲❲❲❲ [ A, [ B, C ]] B C coincide. The multimap above right depicts the image of e ◦ ( e,
1) : ([ A, [ B, C ]] , A, B ) → ([ B, C ] , B ) → C under the action C ([ A, [ B, C ]] , A, B ; C ) ∼ = C ([ A, [ B, C ]] , B, A ; C )of the symmetric group.3.1.3. Nullary map classifiers and units.
Definition 3.1.
A multicategory C has a nullary map classifier if there exists anobject I and multimap u : ( − ) → I such that the induced morphism C ( u ; A ) : C ( I, A ) → C ( − ; A )is a bijection for each A . KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 11
Equivalently, if the functor C ( − ; ?) : C →
Set sending an object A to theset of nullary maps ( − ) → A is representable. In a closed multicategory anullary map classifier I enables the construction of morphisms i : [ I, A ] → A and j : I → [ A, A ]. The former is given by(3.6) u eI ❥❥❥❥❥❥❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ [ I, A ] A For j observe that the identity 1 : A → A corresponds under the isomorphism C ( A, A ) ∼ = C ( − ; [ A, A ]) to a nullary map ˆ1 : ( − ) → [ A, A ]. Now j : I → [ A, A ] isdefined to be the unique map such that(3.7) j ◦ u = ˆ1 . In [29] the term unit for a closed multicategory means something stronger thana nullary map classifier: it consists of a multimap u : ( − ) → I for which (3.6) isinvertible. By Remark 4.2 of ibid. a unit is a nullary map classifier.3.2. The results.
The following result is Proposition 4.3 of [29].
Theorem 3.2 (Manzyuk) . If C is a closed multicategory with unit I then ( C , [ − , − ] , I, L, i, j ) is a closed category. Since we are interested in constructing mere skew closed categories a nullarymap classifier suffices.
Theorem 3.3. If C is a closed multicategory with a nullary map classifier I then ( C , [ − , − ] , I, L, i, j ) is a skew closed category. Furthermore if C is a symmetricmulticategory then ( C , [ − , − ] , I, L, i, j, s ) is symmetric skew closed.Proof. We only outline the proof, which involves routine multicategorical dia-gram chases best accomplished using string diagrams as in (3.2)–(3.6). (Wenote that the deductions of C1 and C3 are given in the proof of Proposition 4.3of [29].) The axioms C1, C2 and C4 each assert the equality of two maps X ⇒ [ Y , . . . , [ Y n − , [ Y n , Z ]] .. ]constructed using [ − , − ], L , i and j . These correspond to the equality of thetransposes ( X, Y , . . . , Y n ) ⇒ Z obtained by postcomposition with the evaluation multimaps. Since [ − , − ], L , i and j are defined in terms of their interaction with the evaluation multimaps onlytheir definitions, together with the associativity and unit laws for a multicategory,are required to verify these axioms. C3 and C5 each concern the equality of twomaps I ⇒ A . Here one shows that the corresponding nullary maps ( − ) ⇒ A coincide. Again this is straightforward. The axioms S1-S4 are verified in asimilar fashion. (cid:3) Theorem 3.3, as stated, will not apply to the examples of interest, none ofwhich quite has a nullary map classifier. What we need is a generalisation thatdeals with combinations of strict and weak maps.
Definition 3.4.
Let C be a multicategory equipped with a subcategory C s ⊆ C of strict morphisms containing all of the identities. We say that a multimap f : ( A , . . . , A n ) → B is strict in i (or A i abusing notation) if for all families ofmultimaps { a j : ( − ) → A j : j ∈ { , . . . , i − , i + 1 , . . . , n }} the unary map f ◦ ( a , . . . a i − , , a i +1 , . . . a n ) : A i → B is strict. Theorem 3.5.
Let C be a closed multicategory equipped with a subcategory C s → C of strict maps containing the identities. Suppose further that(1) A multimap ( A , . . . , A n , B ) → C is strict in A i if and only if its transpose ( A , . . . , A n ) → [ B, C ] is.(2) There is a multimap u : ( − ) → I , precomposition with which induces abijection C ( u, A ) : C s ( I, A ) → C ( − ; A ) for each A .Then ( C , [ − , − ] , L ) is a semi-closed category. Moreover [ − , − ] and L restrict to C s where they form part of a skew closed structure ( C s , [ − , − ] , I, L, i, j ) .Furthermore if C is a symmetric multicategory then ( C , [ − , − ] , L, s ) is symmetricsemi-closed and ( C s , [ − , − ] , I, L, i, j, s ) is symmetric skew closed.Proof. We must show that these assumptions ensure that the bifunctor [ − , − ] : C op × C restricts to C s and that the transformations L, i, j and s have strictcomponents. Beyond this point the proof is identical to that of Theorem 3.3.A consequence of Definition 3.4 is that multimaps strict in a variable are closedunder composition : that is, given f : ( A , A . . . A n ) → B k strict in A i and g : ( B , B . . . B m ) → C strict in B k the composite multimap( B . . . B k − , A . . . A i . . . A n , B k +1 . . . B m ) → C is strict in A i . We use this fact freely in what follows.Observe that since 1 : [ A, B ] → [ A, B ] is strict its transpose, the evaluationmultimap e : ([ A, B ] , A ) → B , is strict in [ A, B ]. It follows that if f : B → C isstrict then the composite multimap f ◦ e : ([ A, B ] , A ) → B → C of (3.3) is strictin [ A, B ]. Accordingly its transpose [
A, f ] : [
A, B ] → [ A, C ] is strict. Likewise[ f, A ] is strict if f is. Hence [ − , − ] restricts to C s .Since evaluation multimaps are strict in the first variable the composite e ◦ (1 , e ) :([ B, C ] , [ A, B ] , A ) → C of (3.4) is strict in [ B, C ]. Since transposing this twiceyields L : [ B, C ] → [[ A, B ] , [ A, C ]] we conclude that L is strict. The composite i = e ◦ (1 , u ) : [ I, A ] → A of (3.6) is strict as e : ([ I, A ] , I ) → A is strict in thefirst variable. Clearly j is strict.In a symmetric multicategory the actions of the symmetric group commute withcomposition. It follows that if F : ( A , . . . A n ) → B is strict in A i and ϕ ∈ Sym ( n ) then ϕ ( F ) : ( A ϕ (1) , . . . , A ϕ ( n ) ) → B is strict in A ϕ ( i ) . Therefore thecomposite ([ A, [ B, C ]] , B, A ) → C on the right hand side of (3.5) is strict in[ A, [ B, C ]]. Since s : [ A, [ B, C ]] ∼ = [ B, [ A, C ]] is obtained by transposing thistwice, it follows that s is strict. (cid:3) KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 13
In the next section we will encounter several examples of
Cat -enriched mul-ticategories, hence 2-multicategories [13]. A 2-multicategory C has categories C ( A , . . . , A n ; B ) of multilinear maps and transformations between, and an ex-tension of multicategorical composition dealing with these transformations. Thereis an evident notion of closed 2-multicategory , in which the bijection (3.1) is re-placed by an isomorphism, and of symmetric 2-multicategory. Theorem 3.5generalises straightforwardly to 2-multicategories as we now record. Theorem 3.6.
Let C be a closed 2-multicategory equipped with a locally full sub2-category C s → C of strict maps containing the identities. Suppose further that(1) A multimap ( A , . . . , A n , B ) → C is strict in A i if and only if its transpose ( A , . . . , A n ) → [ B, C ] is.(2) There is a multimap u : ( − ) → I , precomposition with which induces anisomorphism C ( u, A ) : C s ( I, A ) → C ( − ; A ) for each A .Then ( C , [ − , − ] , L ) is a semi-closed 2-category. Moreover [ − , − ] and L restrictto C s where they form part of a skew closed 2-category ( C s , [ − , − ] , I, L, i, j ) .Furthermore if C is a symmetric 2-multicategory then ( C , [ − , − ] , L, s ) is symmet-ric semi-closed and ( C s , [ − , − ] , I, L, i, j, s ) is a symmetric skew closed 2-category. Remark 3.7.
Theorem 5.1 of [29] shows that the notions of closed multicategorywith unit and closed category are, in a precise sense, equivalent. We do not knowwhether skew closed categories are equivalent to some kind of multicategoricalstructure. 4.
Examples of skew closed structures
The goal of this section is to describe a few concrete examples of the kind ofskew closed structures that we are interested in. All can be seen to arise frommulticategories although sometimes it will be easier to describe the skew closedstructure directly.In each case we meet a category, or 2-category, C of weak maps equipped with asubcategory C s of strict maps. The subcategory of strict maps is well behaved –locally presentable, for instance – whereas C is not. The objects of the internalhom [ A, B ] are the weak maps but these only form part of a skew closed structureon the subcategory C s of strict maps .4.1. Categories with structure.
The following examples can be understood asarising from pseudo-commutative 2-monads in the sense of [13] – this more ab-stract approach is described in Section 6.4.1.1.
Categories with specified limits.
Let D be a set of small categories, thoughtof as diagram types. There is a symmetric 2-multicategory D - Lim whose objects A are categories A equipped with a choice of D -limits. The objects of the cat-egory D - Lim ( A , . . . , A n ; B ) are functors F : A × . . . × A n → B preserving D -limits in each variable, and the morphisms are just natural transformations.For the case n = 0 we have D - Lim ( − ; B ) = B . The morphisms in the 2-category of unary maps D -Lim are the D -limit pre-serving functors, amongst which we have the 2-category D -Lim s of strict D -limit preserving functors and the inclusion j : D -Lim s → D -Lim. Accord-ingly a multimap F : ( A , . . . , A n ) → B is strict in A i just when each functor F ( a , . . . , a i − , − , a i +1 , . . . , a n ) : A i → B preserves D -limits strictly.The functor category [ A, B ] has a canonical choice of D -limits inherited point-wise from B . Since D -limits commute with D -limits the full subcategory j : D -Lim( A , B ) → [ A, B ] is closed under their formation (although D -Lim s ( A , B )is not!) and we write [ A , B ] for D -Lim( A, B ) equipped with this choice of D -limits.It is routine to verify that the objects [ A , B ] exhibit D -Lim as a closed 2-multicategory and moreover that a multimap F : ( A , . . . , A n , B ) → C is strictin A i just when its transpose ( A , . . . , A n ) → [ B , C ] is. With regards units, thekey point is that the forgetful 2-functor U : D -Lim s → Cat has a left 2-adjoint F . This follows from [3] but see also Section 6.2. Accordingly we have a naturalisomorphism D - Lim ( − ; A ) ∼ = Cat (1 , A ) ∼ = D -Lim s ( F , A ). By Theorem 3.6 weobtain the structure of a symmetric semi-closed 2-category ( D -Lim , [ − , − ] , L, s )restricting to a symmetric skew closed 2-category ( D -Lim , [ − , − ] , F , L, i, j, s ).The skew closed structure on D -Lim s fails to extend to D -Lim because the unitmap j : F → [ A , A ] is only pseudo-natural in morphisms of D -Lim. (It does,however, extend to a pseudo-closed structure on D -Lim in the sense of [13]).The skew closed D -Lim s is neither left nor right normal: for example, the canon-ical functor D -Lim s ( A , B ) → D -Lim s ( F , [ A , B ]) is isomorphic to the inclusion D -Lim s ( A , B ) → D -Lim( A , B ) and this is not in general invertible.4.1.2. Permutative categories and so on.
An example amenable to calculationconcerns symmetric strict monoidal – or permutative – categories. The symmet-ric skew closed structure can be seen as arising from a symmetric 2-multicategory,described in [9]. Because the relevant definition of multilinear map is rather long,we treat the skew closed structure directly.Let Perm s and Perm denote the 2-categories of permutative categories with thestrict symmetric monoidal and strong symmetric monoidal functors between. Us-ing the symmetry of B the category Perm( A , B ) inherits a pointwise structure[ A , B ] ∈ Perm. Namely we set ( F ⊗ G ) − = F − ⊗ G − . The structure isomor-phism ( F ⊗ G )( a ⊗ b ) ∼ = ( F ⊗ G ) a ⊗ ( F ⊗ G ) b combines the structure isomorphisms F ( a ⊗ b ) ∼ = F a ⊗ F b and G ( a ⊗ b ) ∼ = Ga ⊗ Gb with the symmetry as below: F ( a ⊗ b ) ⊗ G ( a ⊗ b ) ∼ = F a ⊗ F b ⊗ Ga ⊗ Gb ∼ = F a ⊗ Ga ⊗ F b ⊗ Gb .
The structural isomorphism concerning monoidal units is obvious. The hom ob-jects [
A, B ] extend in the obvious way to a 2-functor [ − , − ] : Perm op × Perm → Perm. Moreover, the functor [ C , − ] : Perm( A , B ) → Perm([ C , A ] , [ C , B ]) liftsto a strict map L = [ C , − ] : [ A , B ] → [[ C , A ] , [ C , B ]] because both domain andcodomain have structure inherited pointwise from B . We omits details of thesymmetry isomorphism s : [ A , [ B , C ]] ∼ = [ B , [ A , C ]].The unit F i : [ F , A ] → A is given by evaluation at 1 whilst j : F → [ A , A ] is the unique symmetric strict monoidal functor with j (1) = 1 A . KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 15
Again the skew closed structure is neither left nor right normal.This example can be generalised to deal with general symmetric monoidal cate-gories. A careful analysis of both tensor products and internal homs on the 2-category SMon of symmetric monoidal categories and strong symmetric monoidalfunctors was given by Schmitt [31].4.2.
Examples of skew closed structures not aris-ing from pseudo-commutative 2-monads, even in the extended sense of [28], in-clude 2-categories and bicategories. We focus upon the more complex case ofbicategories. Let Bicat denote the category of bicategories and homomorphisms(also called pseudofunctors) and Bicat s the subcategory of bicategories and stricthomomorphisms. We describe a symmetric skew closed structure on Bicat s withinternal hom Hom ( A, B ) the bicategory of pseudofunctors, pseudonatural trans-formations and modifications from A to B .This skew closed structure arises from a closed symmetric multicategory. Webegin by briefly recalling the multicategory structure, which was introduced andstudied in depth in Section 1.3 of [37] by Verity, and to which we refer for furtherdetails. The multicategory Bicat – denoted by
Hom s in ibid. – has bicategoriesas objects. The multimaps are a variant of the cubical functors of [11]. Moreprecisely, a multimap F : ( A , . . . , A n ) → B consists of • for each n -tuple ( a , . . . , a n ) an object F ( a , . . . , a n ) of B ; • for each 1 ≤ i ≤ n a homomorphism F ( a , . . . , a i − , − , a i +1 . . . , a n ) extend-ing the above function on objects; • for each pair 1 ≤ i < j ≤ n , n -tuple of objects ( a , . . . , a n ) and morphisms f : a i → a ′ i ∈ A i and f j : a j → a ′ j ∈ A j , an invertible 2-cell: F ( a i , a j ) F ( a ′ i , a j ) F ( a i , a ′ j ) F ( a ′ i , a ′ j ) F ( f i ,a j ) / / F ( f i ,a ′ j ) / / F ( a i ,f j ) (cid:15) (cid:15) F ( a ′ i ,f j ) (cid:15) (cid:15) F ( f i ,f j ) v ~ tttttt where we have omitted to label the inactive parts of the n -tuple ( a , . . . , a n )under the action of F . These invertible 2-cells are required to form thecomponents of pseudonatural transformations both vertically – F ( − i , f j ) : F ( − i , a j ) → F ( − i , a ′ j ) – and horizontally – F ( f i , − j ) : F ( a i , − j ) → F ( a ′ i , − j )– and satisfy a further cubical identity involving trios of morphisms.A nullary morphism ( − ) → B is simply defined to be an object of B . Observethat the category of unary maps of Bicat is simply Bicat. It is establishedin Section 1.3 of ibid. – see Lemma 1.3.4 and the discussion that follows –that the symmetric multicategory
Bicat is closed, with hom-object given by thebicategory
Hom ( A, B ) of homomorphisms, pseudonatural transformations andmodifications from A to B .A multimap F : ( A , . . . , A n ) → B is strict in A i just when each homomorphism F ( a , . . . , a i − , − , a i +1 . . . , a n ) is strict. An inspection of the bijection of Lemma ibid. makes it clear that the natural bijection Bicat ( A , . . . , A n , B ; C ) ∼ = Bicat ( A , . . . , A n ; Hom ( B, C ))respects strictness in A i .Turning to the unit, recall that Bicat ( − ; A ) = A . The forgetful functor ( − ) :Bicat s → Set has a left adjoint F for general reasons – see Section 7 for moreon this. It follows that we have a bijection Bicat s ( F , A ) ∼ = Bicat ( − ; A ) where F F • and a sin-gle generating 1-cell e : • → • . General morphisms are (non-empty) bracketedcopies of e such as (( ee ) e ), and two such morphisms are connected by a unique2-cell, necessarily invertible.By Theorem 3.5 we obtain a symmetric semi-closed category (Bicat s , Hom, L )which restricts to a symmetric skew closed structure (Bicat s , Hom, F , L, i, j ).As in the preceding examples the skew closed structure on Bicat s is not closedand fails to extend to Bicat.In Section 7 we further analyse this symmetric skew closed structure. Accord-ingly we describe a few aspects of it in more detail. Firstly, let us describe theaction of the functor Hom ( − , − ) : Bicat op × Bicat → Bicat. From a homomor-phism f : A B the homomorphism Hom ( f,
1) :
Hom ( B, C ) → Hom ( A, C )obtained by precomposition is always strict, and straightforward to describe.The postcomposition map
Hom (1 , f ) = Hom ( B, C ) → Hom ( B, D ) induced bya strict homomorphism f : C → D is equally straightforward.Though not strictly required in what follows, for completness we mention theslightly more complex case where f is non-strict. At η : g → h ∈ Hom ( B, C ) thepseudonatural transformation f η : f g → f h has components f η a : f ga → f ha at a ∈ B ; at α : a → b the invertible 2-cell ( f η ) α : f hα ◦ f η a λ f + f ( hα ◦ η a ) fη α + f ( η b ◦ gα ) λ f − + f η b ◦ f gα conjugates f η α by the coherence constraints for f . The action of f ∗ on 2-cells is straightforward. The coherence constraints f ( η ◦ µ ) ∼ = f ( η ) ◦ f ( µ ) and f ( id g ) ∼ = id ( f g ) for f ∗ are pointwise those for f .The only knowledge required of L : Hom ( B, C ) → Hom ( Hom ( A, B ) , Hom ( A, C ))is that it has underlying function
Hom ( A, − ) : Bicat( B, C ) → Bicat(
Hom ( A, B ) , Hom ( A, C )) . The unit map i : Hom ( F , A ) → A evaluates at the single object • of F j : F → Hom ( A, A ) is the unique strict homomorphism sending • to theidentity on A .This example can be modified to deal with 2-categories. Let - Cat ⊂ Bicat and 2-Cat s ⊂ Bicat s be the symmetric multicategory and category obtained byrestricting the objects from bicategories to 2-categories. Since Hom ( A, B ) is a2-category if B is, we obtain a closed multicategory - Cat by restriction. In thiscase we have a natural bijection 2-Cat s (1 , A ) ∼ = A = - Cat ( − ; A ). It followsthat we obtain a symmetric skew closed structure (2-Cat s , Hom, L, , i, j ) withthe same semi-closed structure as before, but with the simpler unit 1. KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 17
Lax morphisms.
Each of the above examples describes a symmetric skewclosed structure arising from a symmetric closed multicategory. In each casethere are non-symmetric variants dealing with lax structures, of which we men-tion a few now. These have the same units but different internal homs. In D -Lim s the hom [ A , B ] is the functor category [ A, B ] equipped with D -limitspointwise in B . In Perm s the internal hom [ A , B ] consists of lax monoidal func-tors and monoidal transformations. For Bicat s one can take [ A, B ] to be thebicategory of homomorphisms and lax natural transformations from A to B .5. Skew structures descending to the homotopy category
In the present section we consider categories C equipped with a Quillen modelstructure as well as a skew monoidal or skew closed structure. We describe condi-tions under which the skew structures descend to the homotopy category Ho ( C )and call the skew monoidal/closed structures on C homotopy monoidal/closed ifthe induced structures on Ho ( C ) are genuinely monoidal/closed . Theorem 5.11gives a complete description of how monoidal skew closed structure descendsto the homotopy category. Our analogue of Eilenberg and Kelly’s theorem isTheorem 5.12: it allows us to recognise homotopy monoidal structure in termsof homotopy closed structure.We assume some familiarity with the basics of Quillen model categories, as intro-duced in [30], and covered in Chapter 1 of [12]. Let us fix some terminology andstarting assumptions. We assume that all model categories C have functorial fac-torisations. It follows that C is equipped with cofibrant and fibrant replacementfunctors Q and R together with natural transformations p : Q → q : 1 → R whose components are respectively trivial fibrations and trivial cofibrations. Let j : C c → C and j : C f → C denote the full subcategories of cofibrant and fibrantobjects, through which Q and R respectively factor. The four functors preserveweak equivalences and hence extend to the homotopy category. At that level weobtain adjoint equivalences Ho ( C c ) Ho ( j ) - - Ho ( C ) Ho ( Q ) m m Ho ( C f ) Ho ( j ) - - Ho ( C ) Ho ( R ) m m with counit and unit given by Ho ( p ) and its inverse, and Ho ( q ) and its inverserespectively. If a functor between model categories F : C → D preserves weakequivalences between cofibrant objects we can form its left derived functor F l = Ho ( F Q ) : Ho ( C ) → Ho ( D ), equally Ho ( F j ) Ho ( Q ) : Ho ( C ) → Ho ( C c ) → Ho ( D ). If G preserves weak equivalences between fibrant objects then G r = Ho ( GR ) = Ho ( Gj ) Ho ( R ) is its right derived functor. Skew monoidal structure on the homotopy category.
Let C be a modelcategory equipped with a skew monoidal structure ( C , ⊗ , I, α, l, r ). Our interestis in left deriving this to a skew monoidal structure on Ho ( C ). In the monoidalsetting this was done in [12] and the construction in the skew setting, describedbelow, is essentially identical. Axiom M. ⊗ : C × C → C preserves cofibrant objects and weak equivalencesbetween them and the unit I is cofibrant.The above assumption ensures that the skew monoidal structure on C restrictsto one on C c and that the restricted functor ⊗ : C c ×C c → C c preserves weak equiv-alences. Accordingly we obtain a skew monoidal structure ( Ho ( C c ) , Ho ( ⊗ ) , I )with the same components as before. Transporting this along the adjoint equiv-alence Ho ( j ) : Ho ( C c ) ⇆ Ho ( C ) : Ho ( Q ) yields a skew monoidal structure( Ho ( C ) , ⊗ l , I, α l , l l , r l )on Ho ( C ). We will often refer to ( Ho ( C ) , ⊗ l , I ) as the left-derived skew monoidalstructure since ⊗ l is the left derived functor of ⊗ . On objects we have A ⊗ l B = QA ⊗ QB and an easy calculation shows that the constraints for the skewmonoidal structure are given by the following maps in Ho ( C ).(5.1) Q ( QA ⊗ QB ) ⊗ QC p ⊗ / / ( QA ⊗ QB ) ⊗ QC α (cid:15) (cid:15) QA ⊗ ( QB ⊗ QC ) (1 ⊗ p ) − / / QA ⊗ Q ( QB ⊗ QC )(5.2) QI ⊗ QA p ⊗ / / I ⊗ QA l / / QA p / / A (5.3) A p − / / QA r / / QA ⊗ I (1 ⊗ p ) − / / QA ⊗ QI Definition 5.1.
Let ( C , ⊗ , I ) be a skew monoidal structure on a model category C satisfying Axiom M. We say that C is homotopy monoidal if ( Ho ( C ) , ⊗ l , I ) isgenuinely monoidal. Proposition 5.2.
Let ( C , ⊗ , I ) be a skew monoidal category with a model structuresatisfying Axiom M. The following are equivalent.(1) ( C , ⊗ , I ) is homotopy monoidal.(2) For all cofibrant X, Y, Z the map α : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) is aweak equivalence, and for all cofibrant X both maps r : X → X ⊗ I and l : I ⊗ X → X are weak equivalences.Proof. Observe that the constraints (5.1),(5.2) and (5.3) are α QA,QB,QC , l QA and r QA conjugated by isomorphisms in Ho ( C ). It follows that ( Ho ( C ) , ⊗ l , I )is genuine monoidal just when for α QA,QB,QC , l QA and r QA are isomorphismsin Ho ( C ) for all A, B and C . Axiom M ensures for that cofibrant A, B, C thatwe have isomorphisms α A,B,C ∼ = α QA,QB,QC , l A ∼ = l QA and r A ∼ = r QA in Ho ( C ) so that the former maps are isomorphisms just when the latter ones are. Thisproves the claim. (cid:3) Notation 5.3.
We call ( C , ⊗ , I ) homotopy symmetric monoidal if ( Ho ( C ) , ⊗ l , I )admits the further structure of a symmetric monoidal category, but emphasisethat this refers to a symmetry on Ho ( C ) not necessarily arising from a symmetryon C itself. KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 19
Skew closed structure on the homotopy category.
Let C be a model cate-gory equipped with a skew closed structure ( C , [ − , − ] , I, L, i, j ). Our intentionis to right derive the skew closed structure to the homotopy category. This con-struction, more complex than its monoidal counterpart, is closely related to theconstruction of a skew closed category ( C , [ Q − , − ] , I ) from a closed comonad Q [35]. Axiom C.
For cofibrant X the functor [ X, − ] preserves fibrant objects and trivialfibrations. For fibrant Y the functor [ − , Y ] preserves weak equivalences betweencofibrant objects. The unit I is cofibrant. It follows from Axiom C that if A is cofibrant then [1 , p B ] : [ A, QB ] → [ A, B ]is a trival fibration. Accordingly we obtain a lifting k A,B as below.(5.4) Q [ A, B ] p [ A,B ] $ $ ❏❏❏❏❏❏❏❏❏ k A,B / / [ A, QB ] [1 ,p B ] z z ttttttttt [ A, B ]Because I is cofibrant we also have a lifting e as below.(5.5) I (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ e / / QI p I ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ I Lemma 5.4.
Let ( C , [ − , − ] , I ) satisfy Axiom C. Then each of the following fourdiagrams (5.6) Q [ QB,C ] k (cid:15) (cid:15) QL / / Q [[ QA,QB ] , [ QA,C ]] Q [ k, / / Q [ Q [ QA,B ] , [ QA,C ]] k / / [ Q [ QA,B ] ,Q [ QA,C ]][1 ,k ] (cid:15) (cid:15) [ QB,QC ] L / / [[ QA,QB ] , [ QA,QC ]] [ k, / / [ Q [ QA,B ] , [ QA,QC ]] (5.7) Q [ I, B ] Qi ❍❍❍❍❍❍❍❍❍ k / / [ I, QB ] i { { ✈✈✈✈✈✈✈✈✈ QB (5.8) I j e / / QI Qj / / Q [ A, A ] Q [ p, / / Q [ QA, A ] k / / [ QA, QA ] We could weaken Axiom C by requiring that [ X, − ] preserves only weak equivalencesbetween fibrant objects, rather than all trivial fibrations. This is still enough to construct theskew closed structure of Theorem 5.5 though the proof becomes slightly longer. Because weneed the stronger Axiom MC in the crucial monoidal skew closed case anyway, we emphasisethe convenient Axiom C. (5.9) Q [ B, C ] Q [ f,g ] (cid:15) (cid:15) k B,C / / [ B, QC ] [ f,Qg ] (cid:15) (cid:15) Q [ A, D ] k A.D / / [ A, QD ] commutes up to left homotopy. Moreover, if X is fibrant then the image of eachdiagram under [ − , X ] commutes in Ho ( C ) . Note that in (5.9) A and B are cofibrant and the morphisms f : B → A and g : C → D are arbitrary. Proof.
In each case we are presented with a pair of maps f, g : U ⇒ V with U cofibrant. To prove that f and g are left homotopic it suffices, by Proposition1.2.5(iv) of [12], to show that there exists a trivial fibration h : V → W with h ◦ f = h ◦ g . We take the trivial fibrations [1 , p [ QA,QC ] ], p B , [1 , p A ] and [1 , p D ]respectively. Each diagram, postcomposed with the relevant trivial fibration, iseasily seen to commute.For the second point observe that any functor C → D sending weak equivalencesbetween cofibrant objects to isomorphisms identifies left homotopic maps - thisfollows the proof of Corollary 1.2.9 of ibid . Applying this to the composite of[ − , X ] : C → C op and C op → Ho ( C ) op gives the result. (cid:3) Axiom C ensures that the right derived functor[ − , − ] r : Ho ( C ) op × Ho ( C ) → Ho ( C )exists with value [ A, B ] r = [ QA, RB ]. The unit for the skew closed structurewill be I . Using Axiom C we form transformations L r , i r and j r on Ho ( C ) asbelow.(5.10)[ QA, RB ] [ Qq, − (cid:15) (cid:15) [ Q [ QC, RA ] , R [ QC, RB ]][
QRA, RB ] L / / [[ QC, QRA ] , [ QC, RB ]] [ k, / / [ Q [ QC, RA ] , [ QC, RB ]] [1 ,q ] O O (5.11) [ QI, RA ] [ e, / / [ I, RA ] i / / RA q − / / A (5.12) I j / / [ A, A ] [ p, / / [ QA, A ] [1 ,q ] / / [ QA, RA ] Theorem 5.5.
Let C be a model category equipped with a skew closed structure ( C , [ − , − ] , I ) satisfying Axiom C. Then Ho ( C ) admits a skew closed structure ( Ho ( C ) , [ − , − ] r , I ) with constraints as above.Proof. In order to keep the calculations relatively short we will first describea slightly simpler skew closed structure on Ho ( C f ). We then obtain the skewclosed structure on Ho ( C ) by transport of structure.So our main task is to construct a suitable skew closed structure on Ho ( C f ). NowAxiom C ensures that [ QA, B ] is fibrant whenever B is. The restricted bifunctor KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 21 [ Q − , − ] : C opf × C f → C f then preserves weak equivalences in each variable andso extends to a bifunctor Ho ([ Q − , − ]) on Ho ( C f ). For the unit on Ho ( C f ) wetake RI .The constraints are given by the following three maps.(5.13) [ QA, B ] L / / [[ QC, QA ] , [ QC, B ]] [ k, / / [ Q [ QC, A ] , [ QC, B ]](5.14) [
QRI, A ] [ Qq, / / [ QI, A ] [ e, / / [ I, A ] i / / A (5.15) RI q − / / I j / / [ A, A ] [ p, / / [ QA, A ]We should explain why the above components are natural on Ho ( C f ) – in theappropriate variance – since consideration of extraordinary naturality is perhapsnon-standard.Given F, G : A ⇒ B and a family of maps { η A : F A → GA : A ∈ A} we canconsider the class of morphisms N at ( η ) ⊆ M or ( A ) with respect to which η isnatural. N at ( η ) is closed under composition and inverses in A . If ( A , W ) is acategory equipped with a collection of weak equivalences W then each arrow of Ho ( A ) is composed of morphisms in A together with formal inverses w − where w ∈ W . It follows that the family { η A : F A → GA : A ∈ A} is natural in Ho ( A )just when it is natural where restricted to A . Similarly given S : A op × A → B and a family of morphisms { θ A : X → S ( A, A ) : A ∈ A} we can consider theclass Ex ( θ ) ⊆ M or ( A ) with respect to which θ is extranatural. This has thesame closure properties as before. It follows that the family θ A : X → S ( A, A )is extranatural in Ho ( A ) just when it is extranatural when restricted to mapsin A .Using this reasoning we deduce that (5.14) and (5.15) are natural. We likewiseobtain the naturality of the L -component of (5.13) in each variable. So it sufficesto show that ( k C,A ,
1) : [[
QC, QA ] , [ QC, B ]] → [ Q [ QC, A ] , [ QC, B ]] is natural ineach variable. This follows from Diagram (5.9) of Lemma 5.4.We verify the diagrams (C1-C5) below. Each involves an instance of the corre-sponding diagram (C1-C5) for the skew closed structure on C itself, an applica-tion of Lemma 5.4 and straightforward applications of naturality.(C2) [ Q [ QA,A ] , [ QA,C ]][ Q [ p, , / / [ Q [ A,A ] , [ QA,C ]][
Qj, / / [ QI, [ QA,C ]] 1 ' ' ❖❖❖❖❖❖❖❖❖❖❖ [ Qq, − / / [ QRI, [ QA,C ]][
Qq, (cid:15) (cid:15) [ QI, [ QA,C ]][ e, (cid:15) (cid:15) [[ QA,QA ] , [ QA,C ]][ k, O O [ j, / / [ I, [ QA,C ]] i (cid:15) (cid:15) [ QA,C ] L O O / / [ QA,C ] C3)
RI q − / / I j % % ▲▲▲▲▲▲▲▲▲▲▲ j / / [ B,B ] L (cid:15) (cid:15) [ p, / / [ QB,B ] L (cid:15) (cid:15) [[ QA,B ] , [ QA,B ]] [ p, ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ [[1 ,p ] , / / [[ QA,QB ] , [ QA,B ]][ k, (cid:15) (cid:15) [ Q [ QA,B ] , [ QA,B ]] (C4) [[ QRI,QB ] , [ QRI,C ]] [ k, / / [1 , [ Qq, ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ [ Q [ QRI,B ] , [ QRI,C ]] [1 , [ Qq, ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ [[ QRI,QB ] , [ QI,C ]] [ k, / / [ Q [ QRI,B ] , [ QI,C ]][1 , [ e, (cid:15) (cid:15) [ QB,C ] L ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ [ Qi, (cid:15) (cid:15) [ i, & & L ' ' ❖❖❖❖❖❖❖❖❖❖❖ L / / [[ QI,QB ] , [ QI,C ]][[
Qq, , ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ [1 , [ e, (cid:15) (cid:15) [ k, / / [ Q [ QI,B ] , [ QI,C ]][1 , [ e, (cid:15) (cid:15) [ Q [ Qq, , ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ [[ I,QB ] , [ I,C ]][1 ,i ] (cid:15) (cid:15) [[ e, , / / [[ QI,QB ] , [ I,C ]][1 ,i ] (cid:15) (cid:15) [ k, / / [ Q [ QI,B ] , [ I,C ]][1 ,i ] (cid:15) (cid:15) [ Q [ Qq, , / / [ Q [ QRI,B ] , [ I,C ]][1 ,i ] (cid:15) (cid:15) [[ I,QB ] ,C ][ k, w w ♦♦♦♦♦♦♦♦♦♦♦ [[ e, , / / [[ QI,QB ] ,C ] [ k, / / [ Q [ QI,B ] ,C ] [ Q [ Qq, , / / [ Q [ QRI,B ] ,C ][ Q [ I,B ] ,C ] [ Q [ e, , (C5) RI q − / / I (cid:31) (cid:31) j (cid:15) (cid:15) j / / [ RI,RI ][ q, (cid:15) (cid:15) [ p, / / [ QRI,RI ][ Qq, (cid:15) (cid:15) [ I,I ] i (cid:15) (cid:15) [1 ,q ] / / [ I,RI ] i (cid:15) (cid:15) % % ❑❑❑❑❑❑❑❑❑❑ [ p, / / [ QI,RI ][ e, (cid:15) (cid:15) I q / / RI [ I,RI ] i o o K E W S T R U CT U R E S I N - C A T E G O R Y T H E O R YAN D H O M O T O P Y T H E O R Y (C1) [ Q [ QA,C ] , [ QA,D ]] L / / [[ Q [ QA,B ] ,Q [ QA,C ]] , [ Q [ QA,B ] , [ QA,D ]]][ k, (cid:15) (cid:15) [ QC,D ] L (cid:15) (cid:15) L / / [[ QA,QC ] , [ QA,D ]] L (cid:15) (cid:15) L / / [ k, ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ [[[ Q [ QA,B ] , [ QA,QC ]] , [ Q [ QA,B ] , [ QA,D ]]] [[1 ,k ] , ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ [[ k, , (cid:15) (cid:15) [ Q [ Q [ QA,B ] , [ QA,C ]] , [ Q [ QA,B ] , [ QA,D ]]][ Q [ k, , (cid:15) (cid:15) [[[ QA,QB ] , [ QA,QC ]] , [[ QA,QB ] , [ QA,D ]]] [1 , [ k, / / [ L, (cid:15) (cid:15) [[[ QA,QB ] , [ QA,QC ]] , [ Q [ QA,B ] , [ QA,D ]]][ L, (cid:15) (cid:15) [[ QB,QC ] , [ QB,D ]][ k, (cid:15) (cid:15) [1 ,L ] / / [[ QB,QC ] , [[ QA,QB ] , [ QA,D ]]] [ k, ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ [1 , [ k, / / [[ QB,QC ] , [ Q [ QA,B ] , [ QA,D ]]] [ k, ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ [ Q [[ QA,QB ] , [ QA,C ]] , [ Q [ QA,B ] , [ QA,D ]]][
QL, (cid:15) (cid:15) [ Q [ QB,C ] , [ QB,D ]] [1 ,L ] / / [ Q [ QB,C ] , [[ QA,QB ] , [ QA,D ]]] [1 , [ k, / / [ Q [ QB,C ] , [ Q [ QA,B ] , [ QA,D ]]]
We now transport the skew closed structure along the adjoint equivalence Ho ( j ) : Ho ( C f ) ⇆ Ho ( C ) : Ho ( R ). The skew closed structure obtained in this way hasbifunctor Ho ( C ) op × Ho ( C ) Ho ( R ) op × Ho ( R ) / / Ho ( C f ) op × Ho ( C f ) Ho ([ Q − , − ]) / / Ho ( C f ) Ho ( j ) / / Ho ( C ) and unit given by I RI / / Ho ( C f ) Ho ( j ) / / Ho ( C ) . So Ho ([ QR − , R − ]) and RI respectively. Neither is quite as claimed. We fi-nally obtain the skew closed structure stated in the theorem by transferringthis last skew closed structure along the isomorphisms of bifunctors [ Qq,
1] : Ho ([ QR − , R − ]) → Ho ([ Q − , R − ]) and of units q − : RI → I . (cid:3) We often refer to ( Ho ( C ) , [ − , − ] r , I ) as the right derived skew closed structuresince [ − , − ] r is the right derived functor of [ − , − ]. Definition 5.6.
Let C be a model category with a skew closed structure ( C , [ − , − ] , I )satisfying Axiom C. We say that ( C , [ − , − ] , I ) is homotopy closed if the right de-rived skew closed structure ( Ho ( C ) , [ − , − ] r , I ) is genuinely closed. Proposition 5.7.
Let ( C , [ − , − ] , I ) be a skew closed category satisfying Axiom C.Then ( C , [ − , − ] , I ) is homotopy closed if and only if the following two conditionsare met.(1) For all cofibrant A and fibrant B the map v = C ( j, ◦ [ A, − ] : C ( A, B ) → C ( I, [ A, B ]) is a bijection on homotopy classes of maps.(2) For all fibrant A the map i : [ I, A ] → A is a weak equivalence.Proof. We will show that (1) and (2) amount to left and right normality of( Ho ( C ) , [ − , − ] r , I ) respectively. Now Ho ( C ) is left normal just when Ho ( C )( A, B ) Ho ( C )( j r , ◦ [ A, − ] r / / Ho ( C )( I, [ A, B ] r )is a bijection for all A and B . As in any skew closed category this map is naturalin both variables. Since we have isomorphisms QA → A and B → RB in Ho ( C )the above map will be an isomorphism for all A, B just when it is so for allcofibrant A and fibrant B . For such A and B we consider the diagram C ( A, B ) C ( j, ◦C ([ p,q ] , ◦ [ QA,R − ] / / (cid:15) (cid:15) C ( I, [ QA, RB ]) (cid:15) (cid:15) Ho ( C )( A, B ) [ Ho ( C )( j r , ◦ [ A, − ] r / / Ho ( C )( I, [ QA, RB ])which is commutative by definition of [ A, − ] r and j r . The left and right verticalmorphisms are surjective and identify precisely the homotopic maps. It follows KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 25 that the bottom row is invertible just when the top row induces a bijection onhomotopy classes. By naturality of p and q we can rewrite the top row as C ( A, B ) C ( j, ◦ [ A, − ] / / C ( I, [ A, B ]) C (1 , [ p,q ]) / / C ( I, [ QA, RB ]) . Axiom C ensures that [ p, q ] : [
A, B ] → [ QA, RB ] is a weak equivalence betweenfibrant objects. Since I is cofibrant it follows that C (1 , [ p, q ]) is a bijection onhomotopy classes. Therefore the above composite is a bijection on homotopyclasses just when its left component is.Now Ho ( C ) is right normal just when i r = q − ◦ i RA ◦ [ e,
1] : [
QI, RA ] → [ I, RA ] → RA → A is invertible; equally just when i RA : [ I, RA ] → RA is invertible foreach A . For fibrant A we have i A ∼ = i RA in Ho ( C ) and the result follows. (cid:3) The symmetric case.
Consider a skew closed structure ( C , [ − , − ] , I ) satis-fying Axiom C. By Theorem 5.5 we may form the right derived skew closed struc-ture ( Ho ( C ) , [ − , − ] r , I ). Then from a symmetry isomorphism s : [ A, [ B, C ]] ∼ =[ B, [ A, C ]] we can define a symmetry isomorphism s r : [ A, [ B, C ] r ] r ∼ = [ B, [ A, C ] r ] r on the right derived internal hom as below.(5.16) [ QA,R [ QB,RC ]][1 ,q ] − / / [ QA, [ QB,RC ]] s / / [ QB, [ QA,RC ]] [1 ,q ] / / [ QB,R [ QA,RC ]] Proposition 5.8.
Consider ( C , [ − , − ] , I ) satisfying Axiom C, so we have the skewclosed structure ( Ho ( C ) , [ − , − ] r , I ) of Theorem 5.5.If a natural isomorphism s : [ A, [ B, C ]] ∼ = [ B, [ A, C ]] satisfies any of S − S then so does s r : [ A, [ B, C ] r ] r ∼ = [ B, [ A, C ] r ] r . In particular, if ( C , [ − , − ] , I, s ) issymmetric skew closed then so is ( Ho ( C ) , [ − , − ] r , I, s r ) .Proof. As in the proof of Theorem 5.5 we transport the structure from Ho ( C f ).We can extend the skew closed structure ( Ho ( C f ) , Ho [ Q − , − ] , RI ) describedtherein by a symmetry transformation s r whose component at ( A, B, C ) is[
QA, [ QB, C ]] s QA,QB,C / / [ QB, [ QA, C ]]Since the components of s r are just those of s it follows that S S Ho ( C f ) if they do so in C . The diagrams for S S [ QA, [ QB,C ]] s (cid:15) (cid:15) L / / [[ QD,QA ] , [ QD, [ QB,C ]]][1 ,s ] (cid:15) (cid:15) [ k, / / [ Q [ QD,A ] , [ QD, [ QB,C ]]][1 ,s ] (cid:15) (cid:15) [[ QD,QA ] , [ QB, [ QD,C ]]] s (cid:15) (cid:15) [ k, / / [ Q [ QD,A ] , [ QB, [ QD,C ]]] s (cid:15) (cid:15) [ QB, [ QA,C ]] [1 ,L ] / / [ QB, [[ QD,QA ] , [ QD,C ]]] [1 , [ k, / / [ QB, [ Q [ QD,A ] , [ QD,C ]]]6 JOHN BOURKE (S4) [ QA,B ]1 (cid:15) (cid:15) L / / [[ QA,QA ] , [ QA,B ]][ j, (cid:15) (cid:15) [ k, / / [ Q [ QA,A ] , [ QA,B ]][ Q [ p, , / / [ Q [ A,A ] , [ QA,B ]][
Qj, (cid:15) (cid:15) [ QI, [ QA,B ]]1 v v ❧❧❧❧❧❧❧❧❧❧❧❧❧ [ Qq, − (cid:15) (cid:15) [ I, [ QA,B ]] s (cid:15) (cid:15) [ QI, [ QA,B ]] s (cid:15) (cid:15) [ e, o o [ QRI, [ QA,B ]][
Qq, o o s (cid:15) (cid:15) [ QA,B ] [
QA, [ I,B ]][1 ,i ] o o [ QA, [ QI,B ]][1 , [ e, o o [ QA, [ QRI,B ]][1 , [ Qq, o o Therefore ( Ho ( C f ) , Ho [ Q − , − ] , RI, s r ) satisfies any of S − S C , [ − , − ] , I, s )does so. The desired structure on Ho ( C ) is obtained by transporting from theequivalent Ho ( C f ) as in Theorem 5.5. (cid:3) Monoidal skew closed structure on the homotopy category and the homo-topical version of Eilenberg and Kelly’s theorem.
Let C be a model categoryequipped with a monoidal skew closed structure ( C , ⊗ , [ − , − ] , I ). In order toderive the tensor-hom adjunctions − ⊗ QA ⊣ [ QA, − ] to the homotopy category,we will make use of the concept of a Quillen adjunction.An adjunction F : C ⇆ D : U of model categories is said to be a Quillenadjunction if the right adjoint U : D → C preserves fibrations and trivial fibra-tions. One says that U is right Quillen . This is equivalent to asking that F preserves cofibrations and trivial cofibrations, in which case F is said to be leftQuillen . The derived functors F l and U r then exist and form an adjoint pair F l : Ho ( C ) ⇆ Ho ( D ) : U r . This works as follows. At cofibrant A and fibrant B the isomorphisms ϕ A,B : D ( F A, B ) ∼ = C ( A, U B ) are well defined on homotopyclasses and so give natural bijections ϕ A,B : Ho ( D )( F A, B ) ∼ = Ho ( C )( A, U B )– we use the same labelling. At
A, B ∈ Ho ( C ) the hom-set bijections ϕ dA,B : Ho ( D )( F l A, B ) ∼ = Ho ( C )( A, U r B )are then given by conjugating the ϕ A,B as below: Ho ( D )( F QA,B ) (1 ,q ) / / Ho ( D )( F QA,RB ) ϕ A,B / / Ho ( C )( QA,URB )( p, − / / Ho ( C )( A,URB ) It follows that the unit of the derived adjunction – the derived unit – is given by A p − A / / QA η QA / / U F QA Uq F QA / / U RF QA whilst the derived counit admits a dual description.
Proposition 5.9.
For C a model category and ( C , ⊗ , [ − , − ] , I ) monoidal skewclosed the following are equivalent.(1) For cofibrant X the functor − ⊗ X is left Quillen and for each cofibrant Y the functor Y ⊗ − preserves weak equivalences between cofibrant objects. KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 27 (2) For cofibrant X the functor [ X, − ] is right Quillen and for fibrant Y thefunctor [ − , Y ] preserves weak equivalences between cofibrant objects.Proof. Certainly − ⊗ X is left Quillen just when [ X, − ] is right Quillen. Let f : A → B be a weak equivalence between cofibrant objects. Then the naturaltransformation of left Quillen functors − ⊗ f : − ⊗ A → − ⊗ B and the naturaltransformation of right Quillen functors [ f, − ] : [ B, − ] → [ A, − ] are mates . ByCorollary 1.4.4(b) of [12] it follows that Y ⊗ f is a weak equivalence for allcofibrant Y if and only if [ f, Y ] is a weak equivalence for all fibrant Y . (cid:3) Axiom MC. ( C , ⊗ , [ − , − ] , I ) satisfies either of the equivalent conditions of Propo-sition 5.9 and the unit I is cofibrant. Proposition 5.10.
Axiom MC implies Axioms M and C.Proof.
When Axiom MC holds Ken Brown’s lemma (1.1.12 of [12]) ensures thatfor cofibrant X the functor X ⊗ − preserves weak equivalences between cofibrantobjects. So Axiom MC is a strengthening of Axiom M. That it implies AxiomC is clear: if the right adjoint [ X, − ] preserves fibrations it preserves fibrantobjects. (cid:3) For ( C , ⊗ , [ − , − ] , I, ϕ ) satisfying Axiom MC it follows that we can form theleft derived skew monoidal and right derived skew closed structures on Ho ( C ).The following result establishes, as expected, that these form part of a monoidalskew closed structure on Ho ( C ). Theorem 5.11.
Let ( C , ⊗ , [ − , − ] , I, ϕ ) be a monoidal skew closed category satis-fying Axiom MC. Then the left derived skew monoidal structure ( Ho ( C ) , ⊗ l , I ) and the right derived skew closed structure ( C , [ − , − ] r , I ) together with the iso-morphisms ϕ d : Ho ( C )( QA ⊗ QB, C ) ∼ = Ho ( C )( A, [ QB, RC ]) form a monoidal skew closed structure on Ho ( C ) . The straightforward but long proof is deferred until the appendix. The fol-lowing result is the homotopical version of Eilenberg and Kelly’s theorem. Notethat by Proposition 5.7 conditions (1) and (2) amount to ( C , [ − , − ] , I ) beinghomotopy closed. Theorem 5.12.
Let ( C , ⊗ , [ − , − ] , I ) be a monoidal skew closed category satisfyingAxiom MC. Then ( C , ⊗ , I ) is homotopy monoidal if and only if the followingthree conditions are satisfied.(1) For all cofibrant A and fibrant B the function v : C ( A, B ) → C ( I, [ A, B ]) isa bijection on homotopy classes of maps,(2) For all fibrant A the map i : [ I, A ] → A is a weak equivalence,(3) The transformation t : [ A ⊗ B, C ] → [ A, [ B, C ]] is a weak equivalence when-ever A and B are cofibrant and C is fibrant.Proof. Combining Theorems’ 5.11 and 2.6 we have that ( Ho ( C ) , ⊗ l , I ) is monoidaljust when ( Ho ( C ) , [ − , − ] r , I ) is closed and the induced transformation t d : [ QA ⊗ QB, RC ] → [ QA, R [ QB, RC ]] is an isomorphism in Ho ( C ). By Proposition 5.7 closedness amounts to (1) and (2) above. From the proof of Theorem 5.11 thetransformation t d is given by the composite [ Q ( QA ⊗ QB ) ,RC ] [ p, − / / [ QA ⊗ QB,RC ] t / / [ QA, [ QB,RC ]] [1 ,q ] / / [ QA,R [ QB,RC ]] and therefore is invertible just when the central component t QA,QB,RC : [ QA ⊗ QB, RC ] → [ QA, [ QB, RC ]]is so for all
A, B and C . For A, B cofibrant and C fibrant t QA,QB,RC is isomorphicto t A,B,C : [ A ⊗ B, C ] → [ A, [ B, C ]]. Since QA and QB are cofibrant and RC isfibrant it follows that t QA,QB,RC is invertible for all
A, B, C just when t A,B,C isinvertible for all cofibrant A , B and fibrant C . (cid:3) Again we have a symmetric variant.
Theorem 5.13.
Let ( C , ⊗ , [ − , − ] , I ) be a monoidal skew closed category satisfyingAxiom MC.(1) If ( C , [ − , − ] , I ) is homotopy closed and admits a natural symmetry isomor-phism s : [ A, [ B, C ]] ∼ = [ B, [ A, C ]] satisfying S then ( C , ⊗ , I ) is homotopymonoidal.(2) If, in addition to (1), ( C , [ − , − ] , I, s ) is symmetric skew closed then ( C , ⊗ , I ) is homotopy symmetric monoidal.Proof. By Proposition 5.8 if s satisfies S3 then so does s r with respect to( Ho ( C ) , [ − , − ] r , I ). From Theorem 2.10 it follows that ( Ho ( C ) , ⊗ l , I ) is monoidal.The second part follows again by application of Proposition 5.8 and 2.10. (cid:3) Pseudo-commutative 2-monads and monoidal bicategories
In the category
CM on of commutative monoids the set
CM on ( A, B ) forms acommutative monoid [
A, B ] with respect to the pointwise structure of B . Thisis the internal hom of a symmetric monoidal closed structure on CM on whosetensor product represents functions A × B → C that are homomorphisms in eachvariable. From the monad-theoretic viewpoint the enabling property is that thecommutative monoid monad on Set is a commutative monad .Extending this to dimension 2, Hyland and Power [13] introduced the notionof a pseudo-commutative 2-monad T on Cat . Examples include the 2-monadsfor categories with a class of limits, permutative categories, symmetric monoidalcategories and so on. For such T they showed that the 2-category of strict alge-bras and pseudomorphisms admits the structure of a pseudo-closed 2-category –a slight weakening of the notion of a closed category with a 2-categorical element.Theorem 2 of ibid. described a bicategorical version of Eilenberg and Kelly’s the-orem, designed to produce a monoidal bicategory structure on T-Alg. Howeverthey did not give the details of the proof, which involved lengthy calculations ofa bicategorical nature, and expressed their dissatisfaction with the argument. From [13]:“Naturally, we are unhappy with the proof we have just outlined. Since the datawe start from is in no way symmetric we expect some messy difficulties: but the calculationswe do not give are very tiresome, and it would be only too easy to have made a slip. Hence wewould like a more conceptual proof.”
KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 29
In this section we take a slightly different route to the monoidal bicategory struc-ture on T-Alg. We begin by making minor modifications to Hyland and Power’sconstruction to produce a skew closed structure on the 2-category T-Alg s of al-gebras and strict morphisms. This is simply the restriction of the pseudo-closed2-category structure on T-Alg. We then obtain a monoidal skew closed struc-ture on T-Alg s and, using Theorem 5.12, establish that it is homotopy monoidal.The monoidal bicategory structure on T-Alg is obtained by transport of struc-ture from the full sub 2-category of T-Alg s containing the cofibrant objects.6.1. Background on commutative monads. If V is a symmetric monoidal closedcategory and T an endofunctor of V then enrichments of T to a V -functorcorrespond to giving a strength : that is, a natural transformation t A,B : A ⊗ T B → T ( A ⊗ B ) subject to associativity and identity conditions. One obtainsa costrength t ∗ A,B : T A ⊗ B → T ( A ⊗ B ) related to the strength by means of thesymmetry isomorphism c A,B : A ⊗ B → B ⊗ A .If ( T, η, µ ) is a V -enriched monad then we can consider the following diagram(6.1) T A ⊗ T B T ( T A ⊗ B ) T ( A ⊗ B ) T ( A ⊗ T B ) T ( A ⊗ B ) T ( A ⊗ B ) t / / T t ⋆ / / µ (cid:15) (cid:15) t ⋆ (cid:15) (cid:15) T t / / µ / / and if this commutes for all A and B then T is said to be a commutative monad [17].Now if T is commutative and V sufficiently complete and cocomplete then thecategory of algebras V T is itself symmetric monoidal closed [18, 15]. Both ten-sor product and internal hom represent T -bilinear maps – this perspective wasexplored in [19] and more recently in [32]. More generally, a T -multilinear mapconsists of a morphism f : A ⊗ . . . ⊗ A n → B which is a T -algebra map in eachvariable. This means that the diagram A ⊗ . . . ⊗ T A i ⊗ . . . ⊗ A n T ( A ⊗ . . . ⊗ A n ) T BA ⊗ . . . ⊗ A i ⊗ . . . ⊗ A n B t / / T f / / b (cid:15) (cid:15) ⊗ ... ⊗ a i ⊗ ... ⊗ (cid:15) (cid:15) f / / is commutative for each i where the top row t : A ⊗ . . . ⊗ T A i ⊗ . . . ⊗ A n → T ( A ⊗ . . . ⊗ A n ) is the unique map constructible from the strengths and costrengths. T -multilinear maps form the morphisms of a multicategory of T -algebras. Sur-prisingly, the multicategory perspective appears to have first been explored inthe more general 2-categorical setting of [13].6.2. Background on 2-monads.
The category of small categories
Cat is cartesianclosed and hence provides a basis suitable for enriched category theory. Inparticular one has the notions of
Cat -enriched category – hence 2-category –and of
Cat -enriched monads – hence 2-monads. The appendage “2-” will always refer to strict
Cat -enriched concepts.Given a 2-monad T = ( T, η, µ ) on a 2-category C one has the Eilenberg-Moore 2-category T-Alg s of algebras. In T-Alg s everything is completely strict. There arethe usual strict algebras A = ( A, a ) satisfying a ◦ T a = a ◦ µ A and a ◦ η A = 1. The strict morphisms f : A → B of T-Alg s satisfy the usual equation b ◦ T f = f ◦ a on the nose, whilst the 2-cells α : f ⇒ g ∈ T-Alg s ( A , B ) satisfy b ◦ T α = α ◦ a .T-Alg s is just as well behaved as its Set -enriched counterpart. Important factsfor us are the following ones. • The usual (free, forgetful)-adjunction lifts to a 2-adjunction F ⊣ U where U : T-Alg s → C is the evident forgetful 2-functor. • Suppose that C is a locally presentable 2-category: one, like Cat , that is co-complete in the sense of enriched category theory [16] and whose underlyingcategory is locally presentable [1]. If T is accessible – preserves λ -filteredcolimits for some regular cardinal λ – then T-Alg s is also locally presentable.There are accessible 2-monads T on Cat whose strict algebras are categories with D -limits, permutative categories, symmetric monoidal categories and so on. Inparticular the examples of skew closed 2-categories from Section 4.1 reside on2-categories of the form T-Alg s for T an accessible 2-monad on Cat .So far we have discussed strict aspects of two-dimensional monad theory. Thoughthere are several possibilities, the only weak structures of interest here are pseu-domorphisms of strict T -algebras. A pseudomorphism f : A B consists ofa morphism f : A → B and invertible 2-cell f : b ◦ T f ∼ = f ◦ a satisfying twocoherence conditions [3]. These are the morphisms of the 2-category T-Alg intowhich T-Alg s includes via an identity on objects 2-functor ι : T-Alg s → T-Alg.The inclusion commutes with the forgetful 2-functors(6.2) T-Alg sU " " ❋❋❋❋❋❋❋❋❋ ι / / T-Alg U | | ②②②②②②②②② C to the base. Pseudomorphisms of T -algebras capture functors preserving cat-egorical structure up to isomorphism. For example, in the case that T is the2-monad for categories with D -limits or permutative categories we obtain the2-categories D -Lim and Perm as T-Alg.An important tool in the study of pseudomorphisms are pseudomorphism classi-fiers . If T is a reasonable 2-monad – for instance, an accessible 2-monad on Cat – then by Theorem 3.3 of [3] the inclusion ι : T-Alg s → T-Alg has a left 2-adjoint Q . We call Q A the pseudomorphism classifier of A since each pseudomorphism f : A B factors uniquely through the unit q A : A Q A as a strict morphism Q A → B . The counit p A : Q A → A is a strict map with homotopy theoreticcontent – see Section 6.4.1 below.6.3. From pseudo-commutative 2-monads to monoidal skew closed 2-categories.
Given a 2-monad T on Cat we have, in particular, the corresponding strengths t : T ( A × B ) → A × T B and costrengths T ( A × B ) → T A × B and can enquireas to whether T is commutative. For those structures – such as categories with KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 31 finite products or symmetric monoidal categories – that involve an aspect ofweakness in their definitions the relevant diagram (6.1) rarely commutes on thenose, but often commutes up to natural isomorphism. This leads to the notion ofa pseudo-commutative 2-monad T which is a 2-monad T equipped with invertible2-cells T A × T B T ( T A × B ) T ( A × B ) T ( A × T B ) T ( A × B ) T ( A × B ) t / / T t ⋆ / / µ (cid:15) (cid:15) t ⋆ (cid:15) (cid:15) T t / / µ / / α A,B (cid:11) (cid:19) subject to axioms (see Definition 5 of [13]) asserting the equality of composite 2-cells built from the above ones. If α commutes with the symmetry isomorphism –in the sense that α B,A = T c
A,B ◦ α A,B ◦ c T B,T A – then T is said to be a symmetric pseudo-commutative 2-monad.The 2-monad for categories with D -limits is symmetric pseudo-commutative [28]as are the 2-monads for permutative and symmetric monoidal categories [13].An example of a pseudo-commutative 2-monad which is not symmetric is the2-monad for braided strict monoidal categories [5].6.3.1. The 2-multicategory of algebras.
For T pseudo-commutative one can de-fine T -multilinear maps. A T -multilinear map f : ( A , . . . , A n ) → B consists ofa functor f : A × . . . A n → B together with a family of invertible 2-cells f i : A × . . . × T A i × . . . × A n T ( A × . . . × A n ) T BA × . . . × A i × . . . × A n B t / / T f / / c (cid:15) (cid:15) × ... × a i × ... × (cid:15) (cid:15) f / / f i (cid:11) (cid:19) satisfying indexed versions of the pseudomorphism equations, and a compatibil-ity condition involving the pseudo-commutativity. A nullary map ( − ) → B isdefined to be an object of the category B .There are transformations of multilinear maps and these are the morphisms ofa category T - Alg ( A , A . . . A n ; B ). Proposition 18 of ibid. shows that these arethe hom-categories of a 2-multicategory of T -algebras T - Alg and that, moreover,if T is symmetric pseudo-commutative then T - Alg is a symmetric 2-multicategory.T-Alg is itself recovered as the 2-category of unary maps.Of course we can speak of multimaps ( A , . . . , A n ) → B which are strict in A i :those for which the natural transformation f i depicted above is an identity. Notethat this agrees with the formulation given in Definition 3.4. Theorem 6.1 (Hyland-Power [13]) . The 2-multicategory T - Alg is closed. More-over a multimap ( A , A . . . A n , B ) → C is strict in A i just when the correspond-ing map ( A , A . . . A n ) → [ B , C ] is so. The skew closed structure.
By definition T - Alg ( − ; A ) = A . Since we havea natural isomorphism T-Alg s ( F , A ) ∼ = Cat (1 , A ) ∼ = A and a suitable closed 2-multicategory T - Alg we can apply Theorem 3.6 to obtain a skew closed structureon T-Alg s . Theorem 6.2.
Let T be a pseudo-commutative 2-monad on Cat .(1) Then (T-Alg , [ − , − ] , L ) is a semi-closed 2-category. Moreover [ − , − ] and L restrict to T-Alg s where they form part of a skew closed 2-category (T-Alg s , [ − , − ] , F , L, i, j ) .(2) If T is symmetric then (T-Alg , [ − , − ] , L ) is symmetric semi-closed and (T-Alg s , [ − , − ] , F , L, i, j ) is symmetric skew closed. The skew closed 2-category (T-Alg s , [ − , − ] , F , L, i, j ) has components as con-structed in Section 3. Let us record, for later use, some further information aboutthese components.(1) The underlying category of [ A , B ] is just T-Alg( A , B ). More generally U ◦ [ − , − ] = T-Alg( − , − ) : T-Alg op × T-Alg → Cat .(2) The underlying functor of L : [ A , B ] → [[ C , A ] , [ C , B ]] is given by [ C , − ] A , B :T-Alg( A , B ) → T-Alg([ C , A ] , [ C , B ]) . (3) The underlying functor of i : [ F , A ] → A is the compositeT-Alg( F , A ) U F , A / / Cat ( T , A ) Cat ( η ,A ) / / Cat (1 , A ) ev • / / A whose last component is the evaluation isomorphism.(4) j : F → [ A , A ] is the transpose of the functor ˆ1 : 1 → T-Alg( A , A ) selectingthe identity on A .(1) follows from the construction of the hom algebra [ A , B ] in [13] as a 2-categorical limit in T-Alg created by U : T-Alg → Cat . Theorem 11 of ibid. gives a full description of the isomorphisms T - Alg ( A , . . . , A n , B ; C ) ∼ = T - Alg ( A . . . A n ; [ B , C ]). From this, it follows that the evaluation multimap ev : ([ A , B ] , A ) → B has underlying functor T-Alg( A , B ) × A → B acting byapplication, which is what is required for (3). (2) follows from the analysis,given in Proposition 21 of ibid , of how the same adjointness isomorphisms be-have with respect to underlying maps. (4) is by definition.6.3.3. The monoidal skew closed structure on
T-Alg s . We now describe left 2-adjoints to the 2-functors [ A , − ] : T-Alg s → T-Alg s . For this let us furthersuppose that T is an accessible 2-monad. We must show that each B admits areflection B ⊗ A along [ A , − ]. Since T-Alg s is cocomplete the class of algebrasadmitting such a reflection is closed under colimits; because each algebra is acoequaliser of frees it therefore suffices to show that each free algebra admits a This construction is accomplished in three stages by firstly forming an iso-inserter andthen a pair of equifiers and amounts to the construction of a descent object.
KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 33 reflection. With this is mind observe that the triangle(6.3) T-Alg s T-Alg( A ,ι − ) $ $ ❍❍❍❍❍❍❍❍ [ A , − ] / / T-Alg sU z z ✈✈✈✈✈✈✈✈ Cat commutes. Because T is accessible we have the 2-adjunction Q ⊣ ι and corre-sponding isomorphism T-Alg s ( Q A , − ) ∼ = T-Alg( A , ι − ). Now the representableT-Alg s ( Q A , − ) has a left adjoint − .Q A given by taking copowers. It follows thatat C ∈ Cat the reflection
F C ⊗ A is given by C.Q A . We conclude: Proposition 6.3. If T is an accessible pseudo-commutative 2-monad on Cat theneach [ A , − ] : T-Alg s → T-Alg s has a left 2-adjoint − ⊗ A . In particular (T-Alg s , ⊗ , [ − , − ] , F is a monoidal skew closed 2-category. From
T-Alg s as a monoidal skew closed 2-category to T-Alg as a monoidalbicategory.
Our next goal is to show that (T-Alg s , ⊗ , [ − , − ] , F
1) is homotopymonoidal. In order to do so requires understanding the Quillen model structureon T-Alg s and its relationship with pseudomorphisms. We summarise the keypoints below and refer to the original source [22] for further details.6.4.1. Homotopy theoretic aspects of 2-monads.
Thought of as a mere category,
Cat admits a Quillen model structure in which the weak equivalences are theequivalences of categories. The cofibrations are the injective on objects functorsand the fibrations are the isofibrations: functors with the isomorphism liftingproperty. It follows that all objects are cofibrant and fibrant.Equipped with the cartesian closed structure,
Cat is a monoidal model category[12]. Therefore one can speak of model 2-categories, of which
Cat is the leadingexample. It was shown in Theorem 4.5 of [22] that for an accessible 2-monad T on Cat the model structure lifts along U : T-Alg s → Cat to a model 2-categorystructure on T-Alg s : a morphism of T-Alg s is a weak equivalence or fibrationjust when its image under U is one. It follows immediately that the adjunction F ⊣ U : T-Alg s ⇆ Cat is a Quillen adjunction.Since F preserves cofibrations each free algebra is cofibrant. In fact, the cofibrantobjects are the flexible algebras of [3] and were studied long before the connectionwith model categories was made in [22]. Another source of cofibrant algebrascomes from pseudomorphism classifiers: each Q A is cofibrant. In fact the counit p A : Q A → A of the adjunction Q ⊣ ι : T-Alg s ⇆ T-Alg is a trivial fibration inT-Alg s ; thus Q A is a cofibrant replacement of A .Theorem 4.7 of [3] ensures that if A is flexible then, for all B , the fully faithfulinclusion ι A , B : T-Alg s ( A , B ) → T-Alg( A , B )is essentially surjective on objects: that is, an equivalence of categories . Thisimportant fact can also be deduced from the model 2-category structure: theinclusion ι A , B is isomorphic to T-Alg s ( p A , B ) : T-Alg s ( A , B ) → T-Alg s ( Q A , B )which is an equivalence since p A : Q A → A is a weak equivalence of cofibrantobjects. Finally we note that a parallel pair of algebra morphisms f, g : A ⇒ B are righthomotopic just when they are isomorphic in T-Alg s ( A , B ). This follows fromthe fact that for each algebra B the power algebra [ I, B ] is a path object where I is the walking isomorphism. In particular, if A is cofibrant then f and g arehomotopic just when they are isomorphic.6.4.2. Homotopical behaviour of the skew structure.
Theorem 6.4.
Let T be an accessible pseudo-commutative 2-monad on Cat .(1) Then the monoidal skew closed 2-category (T-Alg s , ⊗ , [ − , − ] , F satisfiesAxiom M C and (T-Alg s , ⊗ , F is homotopy monoidal.(2) If T is symmetric then (T-Alg s , ⊗ , F is homotopy symmetric monoidal.Proof. We observed above that each free algebra is cofibrant. Therefore the unit F A the 2-functor [ A , − ] is right Quillen and that[ − , A ] preserves all weak equivalences. For the first part consider the equality U ◦ [ A , − ] = T-Alg s ( A , ι − ) of (6.3). Since U reflects weak equivalences andfibrations it suffices to show that T-Alg( A , ι − ) : T-Alg s → Cat is right Quillen.Now T-Alg( A , ι − ) ∼ = T-Alg s ( Q A , − ) : T-Alg s → Cat and this last 2-functor isright Quillen since Q A is cofibrant and T-Alg s a model 2-category.For the second part we use the commutativity U ◦ [ − , A ] ∼ = T-Alg( ι − , A ). Argu-ing as before it suffices to show that T-Alg( ι − , A ) : T-Alg s → Cat preserves allweak equivalences or, equally, that the isomorphic T-Alg s ( Qι − , A ) does so. Nowif f : B → C is a weak equivalence then Qιf is a weak equivalence of cofibrantobjects. As A , like all objects, is fibrant and T-Alg s a model 2-category thefunctor T-Alg s ( Qιf, A ) is an equivalence.We now apply Theorem 5.12 to establish that T-Alg s is homotopy monoidal.To verify the three conditions requires only the information on the underlyingfunctors of [ − , − ], L and i given in Section 6.3.2. Firstly we must show that theunderlying function of v A , B : T-Alg s ( A , B ) → T-Alg s ( F , [ A , B ])induces a bijection on homotopy classes of maps for cofibrant A . Since mor-phisms with cofibrant domain are homotopic just when isomorphic it will sufficeto show that v A , B is an equivalence of categories. To this end consider thecomposite:T-Alg s ( A , B ) T-Alg s ( F , [ A , B ]) Cat (1 , T-Alg( A , B )) T-Alg( A , B ) v A , B / / ϕ / / ev • / / in which ϕ is the adjointness isomorphism – recall that U ◦ [ − , − ] = T-Alg( − , − )– and in which ev • is the evaluation isomorphism. It suffices to show that thecomposite is an equivalence. v A , B sends f : A → B to [ A , f ] ◦ j : F → [ A , A ] → [ A , B ], whose image under ϕ is the functor T-Alg( A, f ) ◦ ˆ1 A : 1 → T-Alg( A , A ) → T-Alg( A , B ). Evaluating at • thus returns f viewed as a pseudomap. Theaction on 2-cells is similar and we conclude that the composite is the inclusion ι A , B : T-Alg s ( A , B ) → T-Alg( A , B ). As per Section 6.4.1 this is an equivalence KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 35 since A is cofibrant.Secondly we show that i A : [ F , A ] → A is a weak equivalence for all A : that its underlying functor i A : T-Alg( F , A ) → A is an equivalence of categories. Since F i A ◦ ι F , A : T-Alg s ( F , A ) → T-Alg( F , A ) → A is an equivalence. An easy calculation shows that this is equally the composite ev • ◦ ϕ : T-Alg s ( F , A ) → Cat (1 , A ) → A of the canonical adjunction andevaluation isomorphisms. Hence i A is an equivalence for all A .Let u : A → [ B , A ⊗ B ] denote the unit of the adjunction − ⊗ B ⊣ [ B , − ]. Weare to show that the morphism t A , B , C given by the composite[ u, ◦ L : [ A ⊗ B , C ] → [[ B , A ⊗ B ] , [ B , C ]] → [ A , [ B , C ]]is a weak equivalence for cofibrant A and B . Now the underlying functor of thiscomposite is just the top row below.T-Alg( A ⊗ B , C ) [ B , − ] / / T-Alg([ B , A ⊗ B ] , [ B , C ]) T-Alg( u, / / T-Alg( A , [ B , C ])T-Alg s ( A ⊗ B , C ) ι O O [ B , − ] / / T-Alg s ([ B , A ⊗ B ] , [ B , C ]) ι O O T-Alg s ( u, / / T-Alg s ( A , [ B , C ]) ι O O In this diagram the left square commutes since [ B , − ] restricts from T-Alg toT-Alg s and the right square since u is a strict algebra map. The outer verticalarrows are equivalences since both A ⊗ B and A are cofibrant: the former usingAxiom MC and the latter by assumption. The bottom row is the adjointnessisomorphism so that the top row is an equivalence by two from three.Finally if T is symmetric then, by Theorem 6.2, the skew closed 2-category(T-Alg s , [ − , − ] , F
1) is symmetric skew closed. It now follows from Theorem 5.13that (T-Alg s , ⊗ , F
1) is homotopy symmetric monoidal. (cid:3)
The monoidal bicategory
T-Alg . A monoidal bicategory is a bicategory C equipped with a tensor product C × C C and unit I together with equivalences α : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ), l : I ⊗ A → A and r : A → A ⊗ I pseudonaturalin each variable, and satisfying higher dimensional variants of the axioms for amonoidal category [10]. Note that here we mean equivalences in the , as opposed to weak equivalences.In particular, each skew monoidal 2-category in which the components α , l and r are equivalences provides an example of a monoidal bicategory. Theskew monoidal 2-category (T-Alg s , ⊗ , F
1) is not itself a monoidal bicategory. However (T-Alg s , ⊗ , F
1) satisfies Axiom MC and hence, by Proposition 5.10, itsatisfies Axiom M. Therefore the skew monoidal structure restricts to the fullsub 2-category (T-Alg s ) c of cofibrant objects. Since (T-Alg s , ⊗ , F
1) is homotopymonoidal each component α , l and r is a weak equivalence of cofibrant objects. In fact l : F ⊗ A → A is an equivalence just when A is equivalent to a flexible algebra.Such algebras are called semiflexible [3]. Since all objects are fibrant such weak equivalences are homotopy equivalences:thus equivalences in the 2-categorical sense. We conclude:
Proposition 6.5.
The skew monoidal structure (T-Alg s , ⊗ , F restricts to a skewmonoidal 2-category ((T-Alg s ) c , ⊗ , F which is a monoidal bicategory. In fact (T-Alg s ) c is biequivalent to the 2-category T-Alg of algebras and pseu-domorphisms. Lemma 6.6.
The 2-adjunction Q ⊣ ι : T-Alg s ⇆ T-Alg restricts to a 2-adjunction Q ⊣ ι : (T-Alg s ) c ⇆ T-Alg whose unit and counit are pointwise equivalences. Inparticular, the composite inclusion ι : (T-Alg s ) c → T-Alg is a biequivalence.Proof.
Because each Q A is cofibrant/flexible the adjunction restricts. The unit q : A Q A is an equivalence by Theorem 4.2 of [3]. Since A is flexible thecounit p A : Q A → A is an equivalence in T-Alg s by Theorem 4.4 of ibid . (cid:3) Just as monoidal structure can be transported along an adjoint equivalence ofcategories, so the structure of a monoidal bicategory may be transported alongan adjoint biequivalence. And we obtain the following result: see Theorem 14 of[13]. The present argument has the advantage of dealing solely with the strictconcepts of
Cat -enriched category theory until the last possible moment.
Theorem 6.7.
For T an accessible pseudo-commutative 2-monad on Cat the 2-category
T-Alg admits the structure of a monoidal bicategory. Bicategories
We now return to the skew closed category (Bicat s , Hom, F
1) of Section 4.2and show that it forms part of a monoidal skew closed category that is homotopysymmetric monoidal. A similar, but simpler, analysis yields the correspondingresult for the skew structure on 2-Cat s discussed in Section 4.2 – this is omitted.7.1. Preliminaries on
Bicat s . To begin with, it will be helpful to discuss somegeneralities concerning homomorphism classifiers and the algebraic nature ofBicat s .To this end, let us recall that the category Cat -Gph of
Cat -enriched graphs is nat-urally a – called CG in [23]. CG is locally presentable as a 2-category:that is, cocomplete as a 2-category and its underlying category Cat -Gph is locallypresentable. Section 4 of ibid. describes a filtered colimit preserving 2-monad T on CG whose strict algebras are the bicategories, and whose strict morphismsand pseudomorphisms are the strict homomorphisms and homomorphisms re-spectively. The algebra 2-cells are called icons [24]. We write Icon s and Icon p for the corresponding extensions of Bicat s and Bicat to 2-categories with iconsas 2-cells. It follows from [3] that the inclusion ι : Icon s → Icon p has a left2-adjoint Q : this assigns to a bicategory A its homomorphism classifier QA .As mentioned Cat -Gph is locally presentable. Since T preserves filtered colimitsit follows that the category of algebras Bicat s is locally presentable too, and thatthe forgetful right adjoint U : Bicat s → Cat -Gph preserve limits and filtered col-imits. Now the three functors from
Cat -Gph to
Set sending a
Cat -graph to its set
KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 37 of (0/1/2)-cells respectively are represented by finitely presentable
Cat -graphs.It follows that the composite of each of these with U – the functors( − ) , ( − ) , ( − ) : Bicat s → Set sending a bicategory to its set of (0/1/2)-cells – preserves limits and filteredcolimits. Now a functor between locally presentable categories has a left adjointjust when it preserves limits and is accessible: preserves λ -filtered colimits forsome regular cardinal λ . See, for instance, Theorem 1.66 of [1]. It follows thateach of the above three functors has a left adjoint – we used the adjoint F to( − ) to construct the unit F The monoidal skew closed structure on
Bicat s . Our goal now is to showthat
Hom ( A, − ) : Bicat s → Bicat s has a left adjoint for each A . We willestablish this by showing that Hom ( A, − ) preserves limits and is accessible. Aspointed out above, the functors ( − ) , ( − ) , ( − ) : Bicat s → Set preserve limitsand filtered colimits. Since they jointly reflect isomorphisms they also jointlyreflect limits and filtered colimits. Accordingly it will be enough to show thatthe three functors
Hom ( A, − ) , Hom ( A, − ) , Hom ( A, − ) : Bicat s → Set preserve limits and are accessible. We argue case by case.(1)
Hom ( A, B ) is the set of homomorphisms from A to B . Hence Hom ( A, − ) is naturally isomorphic to Bicat s ( QA, − ) where Q is the homomorphismclassifier. Like any representable functor Hom ( A, − ) preserves limits andis accessible.(2) Hom ( A, B ) is the set of pseudonatural transformations between homomor-phisms. Let Cyl ( B ) denote the following bicategory – first constructed, inthe lax case, in [2] . The objects of Cyl ( B ) are the morphisms of B whilstmorphisms ( r, s, θ ) : f → g are diagrams as below left(7.1) a bc d r / / f (cid:15) (cid:15) s / / g (cid:15) (cid:15) θ + a bc d r / / f (cid:28) (cid:28) f ′ (cid:2) (cid:2) s / / g ′ (cid:2) (cid:2) α + θ ′ + = a bc d r / / f (cid:28) (cid:28) s / / g (cid:28) (cid:28) g ′ (cid:2) (cid:2) θ + β + in which θ is invertible. 2-cells of Cyl ( B ) consist of pairs of 2-cells ( α, β )satisfying the equality displayed above right. Note that here are strict pro-jection homomorphisms d, c : Cyl ( B ) ⇒ B which, on objects, respectivelyselect the domain and codomain of an arrow.It is straightforward to see that we have a natural isomorphism of func-tors Hom ( A, − ) ∼ = Bicat( A, Cyl ( − )). Combining this with Bicat( A, − ) ∼ =Bicat s ( QA, − ) gives an isomorphism Hom ( A, − ) ∼ = Bicat s ( QA, Cyl ( − )).Since this is the composite Bicat s ( QA, − ) ◦ Cyl ( − ) : Bicat s → Bicat s → Set whose second component is representable, it will suffice to show that
Cyl ( − )preserves limits and is accessible.For this, arguing as before, it is enough to show that each of Cyl ( − ) , Cyl ( − ) and Cyl ( − ) preserves limits and is accessible. Certainly we have Cyl ( B ) ∼ = B naturally in B and ( − ) preserves limits and filtered colim-its. We construct Cyl ( − ) as a finite limit in four stages. These stagescorrespond to the sets constructed below: Opp ( B ) = { ( x, y ) : x, y ∈ B , sx = ty, tx = sy } Comp ( B ) = { ( a, b ) : a, b ∈ B , ta = sb } Iso ( B ) = { ( x, y ) ∈ Opp ( B ) : y ◦ x = i ty , x ◦ y = i tx } Cyl ( B ) = { ( a, b, c, d, x, y ) : ( x, y ) ∈ Iso ( B ) , ( a, b ) , ( c, d ) ∈ Comp ( B ) , sx = b ◦ a, tx = d ◦ c } the first three of which, in turn, define the sets of pairs of 2-cells pointingin the opposite direction, of composable pairs of 1-cells, and of invertible2-cells. Each stage corresponds to the finite limit in CAT (Bicat s , Set ) below.
Opp ( B ) / / ( B ) sx,tx ) , , ( ty,sy ) ( B ) Comp ( B ) / / (cid:15) (cid:15) B s (cid:15) (cid:15) Iso ( B ) / / (cid:15) (cid:15) Opp ( B ) ( y ◦ x,x ◦ y ) (cid:15) (cid:15) Cyl ( B ) / / (cid:15) (cid:15) Iso ( B ) ( sx,tx ) (cid:15) (cid:15) B t / / B ( B ) i / / ( B ) Comp ( B ) b ◦ a,d ◦ c ) / / ( B ) Now each of the functors ( − ) ,( − ) and ( − ) preserve finite limits and fil-tered colimits. Since finite limits commute with limits and filtered colimitsin Set it follows that each constructed functor, and in particular,
Cyl ( − ) preserves limits and filtered colimits. Another pullback followed by anequaliser constructs Cyl ( − ) and shows it to have the same preservationproperties: we leave this case to the reader.(3) Finally observe that we can express Hom ( A, B ) as the equaliser of the twofunctions Hom ( A, Cyl ( B )) ⇒ Bicat(
A, B ) sending an element α : f ⇒ g of Hom ( A, Cyl ( B )) to the pair ( df, cf ) and( dg, cg ) respectively. This is natural in B . Therefore Hom ( A, − ) is a finitelimit of functors, each of which has already been shown to preserve limitsand be accessible. Since finite limits in Set commute with limits and with λ -filtered colimits for each regular cardinal λ it follows that Hom ( A, − ) preserves limits and is itself accessible.We conclude: Proposition 7.1.
For each bicategory A the functor Hom ( A, − ) : Bicat s → Bicat s has a left adjoint − ⊗ A . In particular we obtain a monoidal skew closed category (Bicat s , ⊗ , Hom, F . Homotopical behaviour of the skew structure.
We turn to the homotopicalaspects of the skew structure.
KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 39
The model structure on
Bicat s . A 1-cell f : X → Y in a bicategory A issaid to be an equivalence if there exists g : Y → X and isomorphisms 1 X ∼ = gf and 1 Y ∼ = f g . Now a homomorphism of bicategories F : A B is said to be abiequivalence if it is essentially surjective up to equivalence (given Y ∈ B thereexists X ∈ A and an equivalence Y → F X ) and locally an equivalence: eachfunctor F X,Y : A ( X, Y ) → B ( F X, F Y ) is an equivalence of categories.The relevant model structure on Bicat s was constructed in [21]. The weak equiv-alences are those strict homomorphisms that are biequivalences. A strict ho-momorphism F : A → B is said to be a fibration if it has the following twoproperties (1) if f : Y → F X is an equivalence then there exists an equivalence f ⋆ : Y ⋆ → X with F f ⋆ = f and (2) each F X,Y : A ( X, Y ) → B ( F X, F Y ) is anisofibration of categories. We note that all objects are fibrant.The only knowledge that we require of the cofibrant objects is that each homo-morphism classifier QA is cofibrant. To see this observe that if f : A → B is atrivial fibration then there exists a homomorphism g : B A with f ◦ g = 1.Since the inclusion ι : Bicat s → Bicat sends each trivial fibration to a split epi-morphism, and since split epis can be lifted through any object, an adjointnessargument applied to Q ⊣ ι shows that each QA is cofibrant. By Theorem 4.2 of[3] the counit p A : QA → A is a surjective equivalence – equivalence plus splitepi – in the 2-category Icon p . Therefore p A is a trivial fibration and so exhibits QA as a cofibrant replacement of A .The right homotopy relation on Bicat s ( A, B ) is equivalence in the bicategory
Hom ( A, B ). Where needed, we will use the term pseudonatural equivalence forclarity. We note that a morphism η : F → G ∈ Hom ( A, B ) is an equivalence justwhen each component η X : F X → GX is an equivalence in B . That pseudonatu-ral equivalence coincides with right homotopy follows from the fact, used in ibid. ,that the full sub-bicategory P B of Cyl ( B ), with objects the equivalences, is apath object for B . In particular, if A is a cofibrant bicategory then F, G : A ⇒ B are homotopic just when they are equivalent in Hom ( A, B ).7.3.2.
Homotopy monoidal structure.
Finally, we are in a position to prove themain theorem of this section.
Theorem 7.2.
The monoidal skew closed structure (Bicat s , ⊗ , Hom, F satisfiesAxiom MC and is homotopy symmetric monoidal.Proof. Firstly we show that the unit F F is left adjointto ( − ) : Bicat s → Set . Since ( − ) sends trivial fibrations to surjective functions,and since surjective functions can be lifted through 1, it follows by adjointnessthat F F : A B is a biequivalence if andonly if there exists G : B A and equivalences 1 A → GF and 1 B → F G . Aconsequence is that if F : A → B is a biequivalence then so is Hom ( C, F ) and
Hom ( F, D ) for all C and D .To verify Axiom MC, it remains to show that if C is cofibrant and F a fibration,then Hom ( C, F ) is a fibration: in fact, we will show that this is true for all C .To see that Hom ( C, F ) :
Hom ( C, A ) → Hom ( C, B ) is locally an isofibration, consider α : G → H ∈ Hom ( C, A ) and θ : β ∼ = F α . Then each component θ X isinvertible in B and so lifts along F as depicted below. GX HX β ⋆X ' ' α X θ ⋆X (cid:11) (cid:19) ✤ F / / F GX F HX β X ' ' F α X θ X (cid:11) (cid:19) The components β ⋆X : GX → HX admit a unique extension to a pseudonat-ural transformation β ⋆ such that θ ⋆ : β ⋆ → α is a modification: at f : X → Y the 2-cell β ⋆f is given by: Hf ◦ β ⋆X Hf ◦ α X α Y ◦ Gf β ∗ Y ◦ Gf Hf ◦ θ ⋆X + α f + ( θ ⋆Y ) − ◦ Gf + Then
F β ⋆ = β and we conclude that Hom ( C, F ) is locally an isofibration.It remains to show that
Hom ( C, F ) has the equivalence lifting property. Soconsider G : C → A and an equivalence α : H → F G ∈ Hom ( C, B ): a pseudo-natural transformation with each component α X : HX → F GX an equivalencein B . Since F is a fibration there exists an equivalence β X : H ⋆ X → GX ∈ A with F β X = α X . Each such equivalence forms part of an adjoint equivalence( η x , β X ⊣ ρ x , ǫ x ) and at f : X → Y we define H ⋆ ( f ) : H ⋆ X → H ⋆ Y as theconjugate H ⋆ X β X / / GX Gf / / GY ρ Y / / H ⋆ Y in which we take, as a matter of convention, this to mean ( ρ Y ◦ Gf ) ◦ β X . Withthe evident extension to 2-cells H ⋆ becomes a homomorphism. Moreover themorphisms β X naturally extend to an equivalence β : H ⋆ → G ∈ Hom ( C, A ).Although
F H ⋆ X = HX for all X it is not necessarily the case that Hf = F H ⋆ f .Rather, we only have invertible 2-cells ϕ f : Hf ∼ = F H ⋆ f corresponding to thepasting diagram below. HX HYF GX F GY HY Hf ❧❧❧❧❧❧❧❧❧❧ α Y ) ) ❘❘❘❘❘❘❘❘❘❘ F Gf ❧❧❧❧❧❧❧❧❧ α X ) ) ❘❘❘❘❘❘❘❘❘ F ρ Y / / Y $ $ ( α f ) − (cid:11) (cid:19) F η Y (cid:11) (cid:19) Indeed ϕ : H ∼ = F H ⋆ is an invertible icon in the sense of [24]. Since F is locallyan isofibration these lift to invertible 2-cells ϕ ⋆ ( f ) : H ⋆⋆ ( f ) ∼ = H ⋆ f . Moreover H ⋆⋆ becomes a homomorphism, unique such that the above 2-cells yield an in-vertible icon ϕ ⋆ : H ⋆⋆ ∼ = H ⋆ . Composing ϕ ⋆ : H ⋆⋆ ∼ = H ⋆ and β : H ⋆ → G givesthe sought after lifted equivalence. This completes the verification of AxiomMC.From Section 4.2 we know that (Bicat s , Hom, F
1) forms a symmetric skew closedcategory. According to Theorem 5.13 the skew monoidal (Bicat s , ⊗ , F
1) willform part of a homotopy symmetric monoidal category so long as (Bicat s , Hom, F i = ev • : Hom ( F , A ) → A is a biequivalence for each A . KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 41
As pointed out in Section 4.2 F F → Hom (! , A ) : Hom (1 , A ) → Hom ( F , A ) is a biequivalencewhereby it suffices to show that the composite ev • ◦ Hom (! , A ) is a biequivalence.This is just ev • : Hom (1 , A ) → A . It is straightforward, albeit tedious, to verifythat this last map is a biequivalence directly. For a quick proof we can use thefact that for each bicategory A there is a strict 2-category st ( A ) and biequiv-alence p : A st ( A ). Since evaluation is natural in all homomorphisms thesquare below left commutes Hom (1 , A ) ev • (cid:15) (cid:15) Hom (1 ,p ) / / /o/o/o/o/o/o/o Hom (1 , st ( A )) ev • (cid:15) (cid:15) P s (1 , st ( A )) ι o o ev • v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ A p / / /o/o/o/o/o/o/o/o/o/o/o/o/o st ( A )and since both horizontal arrows are biequivalences it suffices to show that ev • : Hom (1 , st ( A )) → st ( A ) is a biequivalence. Now let P s (1 , st ( A )) → Hom (1 , st ( A )) be the full sub 2-category containing the 2-functors. It is easy tosee that ι is essentially surjective up to equivalence – 1 is a cofibrant 2-category!– and hence a biequivalence. Therefore we need only show that the composite ev • : P s (1 , st ( A )) → st ( A ) is a biequivalence. It is an isomorphism.Finally we show that the function v : Bicat s ( A, B ) → Bicat s ( F , Hom ( A, B ))given by the compositeBicat s ( A, B ) Bicat s ( Hom ( A, A ) , Hom ( A, B )) Bicat s ( F , Hom ( A, B )) Hom ( A, − ) / / Bicat s ( j, / / is a bijection on homotopy classes of maps for each cofibrant A . Firstly considerthe strict homomorphism Hom ( A, B ) L / / Hom ( Hom ( A, A ) , Hom ( A, B )) Hom ( j, / / Hom ( F , Hom ( A, B ))of bicategories. By (C2) it composes with i : Hom ( F , Hom ( A, B )) → Hom ( A, B )to give the identity. Since this last map is a biequivalence so too is
Hom ( j, ◦ L by two from three. It follows that its underlying function Bicat( j, ◦ Hom ( A, − ) :Bicat( A, B ) → Bicat( F , Hom ( A, B )) induces a bijection on equivalence classesof objects: pseudonatural equivalence classes of homomorphisms.Now we have a commutative diagramBicat s ( A, B ) ι (cid:15) (cid:15) Bicat s ( j, ◦ Hom ( A, − ) / / Bicat s ( F , Hom ( A, B )) ι (cid:15) (cid:15) Bicat(
A, B ) Bicat( j, ◦ Hom ( A, − ) / / Bicat( F , Hom ( A, B ))in which the vertical functions are the inclusions. Each of the four functions iswell defined on pseudonatural equivalence classes: it follows, by two from three,that the top function will determine a bijection on pseudonatural equivalenceclasses if the two vertical inclusions do so. More generally, if X is a cofibrantbicategory the inclusion ι X,Y : Bicat s ( X, Y ) → Bicat(
X, Y ) induces a bijection on pseudonatural equivalence classes. For we can identify this inclusion, up toisomorphism, with the functionBicat s ( p X ,
1) : Bicat s ( X, Y ) → Bicat s ( QX, Y )where p X : QX → X is the counit of the adjunction Q ⊣ ι . Since p X : QX → X exhibits QX as a cofibrant replacement of X , and so is a weak equivalence be-tween cofibrant objects, it follows – see, for instance, Proposition 1.2.5 of [12] –that Bicat s ( p X ,
1) induces a bijection on homotopy classes, that is, pseudonatu-ral equivalence classes, of morphisms.From Section 4.2 we know that (Bicat s , Hom, F
1) forms a symmetric skew closedcategory. Since it is homotopy closed we conclude from Theorem 5.13 that theskew monoidal (Bicat s , ⊗ , F
1) is homotopy symmetric monoidal. (cid:3) Appendix
Proof of Theorem 5.11.
The isomorphism ϕ d : Ho ( C )( A ⊗ l B, C ) ∼ = Ho ( C )( A, [ B, C ] r )given by the composite Ho ( C )( QA ⊗ QB,C ) (1 ,q ) / / Ho ( C )( QA ⊗ QB,RC ) ϕ / / Ho ( C )( QA, [ QB,RC ])( p, − / / Ho ( C )( A, [ QB,RC ]) is natural in each variable in Ho ( C ) since each component is natural in C .Now the left and right derived structures have components ( Ho ( C ) , ⊗ l , I, α l , l l , r l )and ( Ho ( C ) , [ − , − ] r , I, L r , i r , j r ) respectively. We must prove that these compo-nents are related by the equations (2.2), (2.3) and (2.5) of Section 2.3.For (2.2) we must show that the diagram(8.1) Ho ( C )( A, B ) ( l l , / / v r ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Ho ( C )( QI ⊗ QA, B ) ϕ d (cid:15) (cid:15) Ho ( C )( I, [ QA, RB ])commutes for all A and B . By naturality it suffices to verify commutativity inthe case that A is cofibrant and B is fibrant. By definition v r is the composite Ho ( C )( A,B ) Ho ([ QA,R − ]) / / Ho ( C )([ QA,RA ] , [ QA,RB ])([ p,q ] , / / Ho ( C )([ A,A ] , [ QA,RB ]) ( j, / / Ho ( C )( I, [ QA,RB ]) Since A is cofibrant and B fibrant we can identify Ho ( C )( A, B ) with the set ofhomotopy classes [ f ] : A → B of morphisms f : A → B ; then v r ([ f ]) is thehomotopy class of I j / / [ A, A ] [ p,q ] / / [ QA, RA ] [1 ,Rf ] / / [ QA, RB ]which, by naturality of p and q , coincides with the homotopy class of I j / / [ A, A ] [1 ,f ] / / [ A, B ] [ p,q ] / / [ QA, RB ] . KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 43
Therefore the shorter path in the diagram below is v r . The longer path below is,by definition, the longer path of the triangle (8.1). Accordingly we must showthat the following diagram commutes.(8.2) Ho ( C )( A,B ) ( p, / / ( p,q ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ v (cid:15) (cid:15) Ho ( C )( QA,B ) ( l, / / (1 ,q ) (cid:15) (cid:15) Ho ( C )( I ⊗ QA,B )(1 ,q ) (cid:15) (cid:15) ( p ⊗ , / / Ho ( C )( QI ⊗ QA,B )(1 ,q ) (cid:15) (cid:15) Ho ( C )( QA,RB ) ( l, / / v ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ Ho ( C )( I ⊗ QA,RB ) ( p ⊗ , / / ϕ (cid:15) (cid:15) Ho ( C )( QI ⊗ QA,RB ) ϕ (cid:15) (cid:15) Ho ( C )( I, [ A,B ]) (1 , [ p,q ]) / / Ho ( C )( I, [ QA,RB ]) ( p, / / ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Ho ( C )( QI, [ QA,RB ])( p − , (cid:15) (cid:15) Ho ( C )( I, [ QA,RB ] Each object above is of the form Ho ( C )( X, Y ) for X cofibrant and Y fibrant andwe view each Ho ( C )( X, Y ) as the set of homotopy classes of maps from X to Y .The morphisms are of two kinds. Firstly there are those of the form Ho ( C )( f, Ho ( C )(1 , f ) for f a morphism of C . Such morphisms respect the homotopyrelation and we view them as acting on homotopy classes. The other morphismsare of the form v or ϕ and, because A is cofibrant and B fibrant, each occurence iswell defined on homotopy classes. Accordingly, to verify that the above diagramcommutes it suffices to verify that each sub-diagram commutes. Now apartfrom the commutative triangle on the bottom right, each sub-diagram of (8.2)consists of a diagram involving the hom-sets of C , but with components viewedas acting on homotopy classes. Since in C itself these sub-diagrams commute, bynaturality or (2.2), they certainly commute when viewed as acting on homotopyclasses. Therefore (8.2) commutes.According to (2.3) we must show that(8.3) Ho ( C )( QA ⊗ QI, B ) ϕ d (cid:15) (cid:15) ( r l , ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Ho ( C )( A, [ QI, RB ]) (1 ,i r ) / / Ho ( C )( A, B )commutes for each A and B . Note that in Ho ( C ) the morphism 1 ⊗ e : QA ⊗ I → QA ⊗ QI is inverse to 1 ⊗ p : QA ⊗ QI → QA ⊗ I . Accordingly we can rewrite r l as A p − / / QA r / / QA ⊗ I ⊗ e / / QA ⊗ QI y substituting 1 ⊗ e for (1 ⊗ p ) − . The following diagram then establishes thecommutativity of (8.3). Ho ( C )( QA ⊗ QI,B ) (1 ⊗ e, / / (1 ,q ) (cid:15) (cid:15) Ho ( C )( QA ⊗ I,B )(1 ,q ) (cid:15) (cid:15) ( r, ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ Ho ( C )( QA ⊗ QI,RB ) ϕ (cid:15) (cid:15) (1 ⊗ e, / / Ho ( C )( QA ⊗ I,RB ) ( r, ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ ϕ (cid:15) (cid:15) Ho ( C )( QA,B ) ( p − , ( ( PPPPPPPPPPPP (1 ,q ) (cid:15) (cid:15) Ho ( C )( QA, [ QI,RB ])( p − , (cid:15) (cid:15) (1 , [ e, / / Ho ( C )( QA, [ I,RB ])( p − , (cid:15) (cid:15) (1 ,i ) / / / / Ho ( C )( QA,RB )( p − , ( ( PPPPPPPPPPPP Ho ( C )( A,B )(1 ,q ) (cid:15) (cid:15) ' ' ❖❖❖❖❖❖❖❖❖❖❖ Ho ( C )( A, [ QI,RB ]) (1 , [ e, / / Ho ( C )( A, [ I,RB ]) (1 ,i ) / / Ho ( C )( A,RB ) (1 ,q − / / Ho ( C )( A,B ) Next we calculate that t d : [ Q ( QA ⊗ QB ) , RC ] → [ QA, R [ QB, RC ]] as con-structed in (2.4) has value: [ Q ( QA ⊗ QB ) ,RC ] [ p, − / / [ QA ⊗ QB,RC ] t / / [ QA, [ QB,RC ]] [1 ,q ] / / [ QA,R [ QB,RC ]] . This calculation is given overleaf by the commutative diagram (8.4). All sub-diagrams of (8.4) commute in a routine manner. Apart from basic naturalities weuse the defining equation t = [ u, ◦ L of (2.4), the equation [1 , p ] ◦ k = p of (5.4)and naturality of k as in (5.9). Furthermore, on the bottom right corner, we usethat the morphisms [ Qp A , , [ p QA ,
1] : [
QA, R [ QB, RC ]] ⇒ [[ QQA, R [ QB, RC ]]coincide in Ho ( C ). To see that this is so we argue as in the proof of Lemma 5.4.Namely, p QA and Qp A are left homotopic because they are coequalised by thetrivial fibration p A in C , and, since R [ QB, RC ] is fibrant, the desired equalityfollows.Finally, we use the calculation of t d to prove that the diagram Ho ( C )( QA ⊗ Q ( QB ⊗ QC ) , D ) ϕ d (cid:15) (cid:15) Ho ( C )( α l , / / Ho ( C )( Q ( QA ⊗ QB ) ⊗ QC, D ) ϕ d (cid:15) (cid:15) Ho ( C )( QA ⊗ QB, [ QC, RD ]) ϕ d (cid:15) (cid:15) Ho ( C )( A, [ QB ⊗ QC, RD ]) Ho ( C )(1 ,t d ) / / Ho ( C )( A, [ QB, R [ QC, RD ]])instantiating (2.5) commutes for all A , B , C and D . This is established overleafin the large, but straightforward, commutative diagram (8.5) whose only non-trivial step is an application of (2.5) in C itself. K E W S T R U CT U R E S I N - C A T E G O R Y T H E O R YAN D H O M O T O P Y T H E O R Y (8.4) [ Q ( QA ⊗ QB ) ,RC ] 1 ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ [ p, − (cid:15) (cid:15) [ Qq, − / / [ QR ( QA ⊗ QB ) ,RC ] L / / [ Qq, (cid:15) (cid:15) [[ QB,QR ( QA ⊗ QB )] , [ QB,RC ]][[1 ,Qq ] , (cid:15) (cid:15) [ k, / / [ Q [ QB,R ( QA ⊗ QB )] , [ QB,RC ]][ Q [1 ,q ] , (cid:15) (cid:15) [1 ,q ] / / [ Q [ QB,R ( QA ⊗ QB )] ,R [ QB,RC ]][ Q [1 ,q ] , (cid:15) (cid:15) [ Q ( QA ⊗ QB ) ,RC ] L / / [[ QB,Q ( QA ⊗ QB )] , [ QB,RC ]] [ k, / / [ Q [ QB,QA ⊗ QB ] , [ QB,RC ]] [1 ,q ] / / [ Qu, (cid:15) (cid:15) [ Q [ QB,QA ⊗ QB ] ,R [ QB,RC ]][
Qu, (cid:15) (cid:15) [[ QB,QA ⊗ QB ] , [ QB,RC ]][ u, (cid:15) (cid:15) [[1 ,p ] , O O [ p, ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ [ QQA, [ QB,RC ]] [1 ,q ] / / [ QQA,R [ QB,RC ]][ QA ⊗ QB,RC ] [ p, D D ✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠✠ L ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ t / / [ QA, [ QB,RC ]] [ p, ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ [1 ,q ] / / [ QA,R [ QB,RC ]][ pQA,
QpA, O O Ho ( C )( QA ⊗ Q ( QB ⊗ QC ) ,D )(1 ,q ) (cid:15) (cid:15) ((1 ⊗ p ) − , / / Ho ( C )( QA ⊗ ( QB ⊗ QC ) ,D )(1 ,q ) (cid:15) (cid:15) ( α, / / Ho ( C )(( QA ⊗ QB ) ⊗ QC,D )(1 ,q ) (cid:15) (cid:15) ( p ⊗ , / / Ho ( C )( Q ( QA ⊗ QB ) ⊗ QC,D )(1 ,q ) (cid:15) (cid:15) Ho ( C )(( QA ⊗ QB ) ⊗ QC,RD ) ϕ (cid:15) (cid:15) ( p ⊗ , / / Ho ( C )( Q ( QA ⊗ QB ) ⊗ QC,RD ) ϕ (cid:15) (cid:15) Ho ( C )( QA ⊗ Q ( QB ⊗ QC ) ,RD ) ϕ (cid:15) (cid:15) ((1 ⊗ p ) − , / / Ho ( C )( QA ⊗ ( QB ⊗ QC ) ,RD ) ϕ (cid:15) (cid:15) ( α, ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ Ho ( C )( Q ( QA ⊗ QB ) , [ QC,RD ])( p − , (cid:15) (cid:15) Ho ( C )( QA ⊗ QB, [ QC,RD ]) ϕ (cid:15) (cid:15) ( p, ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ / / (1 ,q ) + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Ho ( C )( QA ⊗ QB, [ QC,RD ])(1 ,q ) (cid:15) (cid:15) Ho ( C )( QA, [ Q ( QB ⊗ QC ) ,RD ])(1 , [ p, − ) / / ( p − , (cid:15) (cid:15) Ho ( C )( QA, [ QB ⊗ QC,RD ])( p − , (cid:15) (cid:15) (1 ,t ) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Ho ( C )( QA ⊗ QB,R [ QC,RD ]) ϕ (cid:15) (cid:15) Ho ( C )( QA, [ QB, [ QC,RD ]]) (1 , [1 ,q ]) / / ( p − , (cid:15) (cid:15) Ho ( C )( QA, [ QB,R [ QC,RD ]])( p − , (cid:15) (cid:15) Ho ( C )( A, [ Q ( QB ⊗ QC ) ,RD ])(1 , [ p, − ) / / Ho ( C )( A, [ QB ⊗ QC,RD ]) (1 ,t ) / / Ho ( C )( A, [ QB, [ QC,RD ]]) (1 , [1 ,q ]) / / Ho ( C )( A, [ QB,R [ QC,RD ]])KEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 47
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Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,Brno 60000, Czech Republic
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