SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION

Transcription

1 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO Abstract. We give an operadic deinition o a genuine symmetric monoidal G-category, and we prove that its classiying space is a genuine E G-space. We do this by developing some very general categorical coherence theory. We combine results o Corner and Gurski, Power, and Lack, to develop a strictiication theory or pseudoalgebras over operads and monads. It specializes to strictiy genuine symmetric monoidal G-categories to genuine permutative G-categories. All o our work takes place in a general internal categorical ramework that has many quite dierent specializations. When G is a inite group, the theory here combines with previous work to generalize equivariant ininite loop space theory rom strict space level input to considerably more general category level input. It takes genuine symmetric monoidal G-categories as input to an equivariant ininite loop space machine that gives genuine Ω-G-spectra as output. Contents Introduction and statements o results 2 Acknowledgements 5 1. Categorical preliminaries Internal categories Chaotic categories The embedding o Set in V 9 2. Pseudoalgebras over operads and 2-monads Pseudoalgebras over 2-Monads The 2-monads associated to operads Pseudoalgebras over operads Operadic speciication o symmetric monoidal G-categories Enhanced actorization systems Enhanced actorization systems The enhanced actorization system on Cat(V ) Proo o the properties o the EFS on Cat(V ) The Power Lack strictiication theorem The statement o the strictiication theorem The construction o the 2-unctor St Mathematics Subject Classiication. Primary 18D10, 18C15, 55P48; Secondary 55P91, 55U40. B. Guillou was supported by Simons Collaboration Grant and NSF grant DMS M. Merling was supported by NSF grant DMS A.M. Osorno was supported by Simons Foundation Grant , the Woodrow Wilson Career Enhancement Fellowship and NSF grant DMS NSF RTG grant DMS supported several collaborator visits to Chicago. 1

2 2 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO 5.3. The proo o the strictiication theorem Appendix: strongly concrete categories 33 Reerences 35 Introduction and statements o results Symmetric monoidal categories are undamental to much o mathematics, and they provide crucial input to the ininite loop space theory developed in the early 1970 s. There it was very convenient to use the still earlier categorical strictiication theory showing that symmetric monoidal categories are monoidally equivalent to symmetric strict monoidal categories, whose products are strictly associative and unital. Following Anderson [1], topologists call symmetric strict monoidal categories permutative categories. Equivariantly, we take this as inspiration, and in this paper we give a deinition o genuine symmetric monoidal G-categories and prove that they can be strictiied to genuine permutative G-categories, as deined in [11]. These are G-categories with extra structure that ensures that their classiying spaces are genuine E G- spaces, so that ater equivariant group completion they can be delooped by any inite dimensional representation V o G. This theory shows that we can construct genuine G-spectra and maps between them rom genuine symmetric monoidal G- categories and unctors that respect the monoidal structure only up to isomorphism. While this paper is a spin-o rom a large scale ongoing project on equivariant ininite loop space theory, it gives a reasonably sel-contained exposition o the relevant categorical coherence theory. In contrast to its equivariant setting in our larger project, this work is designed to be more widely applicable, and in act the equivariant setting plays no particular role other than providing motivation. We say more about that motivation shortly, but we irst discuss the categorical context in which most o our work takes place. Category theorists have developed a powerul and subtle theory o 2-monads and their pseudoalgebras [5, 19, 26, 31]. It gives just the right ramework and results or our strictiication theorem. Working in an arbitrary ground 2-category K, we briely recall the deinitions o 2-monads T, (strict) T-algebras and T- pseudoalgebras, (strict) T-maps and T-pseudomorphisms, and algebra 2-cells in Section 2.1. With these deinitions, we have the ollowing three 2-categories. 1 T-PsAlg: T-pseudoalgebras and T-pseudomorphisms. T-AlgPs: T-algebras and T-pseudomorphisms. T-AlgSt: T-algebras and (strict) T-maps. In all o them, the 2-cells are the algebra 2-cells. Power discovered [26] and Lack elaborated [19] a remarkably simple way to strictiy structures over a 2-monad. 2 Power s short paper deined the strictiication St on pseudoalgebras, and Lack s short paper (on codescent objects) deined St on 1-cells and 2-cells. The result and its proo are truly beautiul category theory. 1 We shall make no use o the second choice. We include it because it is oten convenient and much o the relevant categorical literature ocuses on it. 2 We are greatly indepted to Power and Lack or correspondence about this result.

3 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 3 Generalizing to our internal categorical context, we obtain the ollowing strictiication theorem in Section 5.1. Theorem 0.1. Let K have a rigid enhanced actorization system (E, M ) and let T be a monad in K which preserves E. Then the inclusion o 2-categories J: T-AlgSt T-PsAlg has a let 2-adjoint strictiication 2-unctor St, and the component o the unit o the adjunction is an internal equivalence in T-PsAlg. As we explain in Remark 5.5, the counit also becomes an internal equivalence once we use J to consider it as a map o pseudo-algebras. We shall take the opportunity to expand on the papers o Power and Lack with a number o new details, and we give a reasonably complete and sel-contained exposition. The hypothesis about rigid enhanced actorization systems (EFS) is developed and specialized to the examples o interest to us in Section 4, and the construction o St and proo o the theorem are given in Section 5. The reader is orgiven i she does not immediately see a connection between this theorem and our motivation in terms o symmetric monoidal G-categories. That is what the rest o the paper provides. Our ocus is on the 2-category K = Cat(V ) o categories internal to a suitable category V. We describe this context in Section 1.1. We speciy a rigid EFS on Cat(V ) in Section 4.2, deerring proos to Section This has nothing to do with operads or monads. As we show in Section 2.2, an operad O in Cat(V ) has an associated 2-monad O deined on Cat(V ). Guided by the monadic theory and largely ollowing Corner and Gurski [8], we deine O-pseudoalgebras, O-pseudomorphisms, and algebra 2- cells (alias O-transormations) in Section 2.3. With these deinitions, we have the three 2-categories O-PsAlg: O-pseudoalgebras and O-pseudomorphisms. O-AlgPs: O-algebras and O-pseudomorphisms. O-AlgSt: O-algebras and (strict) O-maps. In all o them, the 2-cells are the algebra 2-cells. With motivation rom symmetric monoidal categories, our deinitions in Section 2 dier a bit rom those in the literature, in particular adding normality conditions. We have tailored our deinitions so that an immediate comparison gives the ollowing monadic identiications o our 2-categories o operadic algebras in Cat(V ). O-PsAlg = O-PsAlg O-AlgPs = O-AlgPs O-AlgSt = O-AlgSt It requires some work to deine O-PsAlg since Cat(V ) is a 2-category, so that instead o requiring the usual diagrams in the strict context to commute, we must ill them with 2-cells that are required to be coherent and we must make the coherence precise. The monadic orerunner charts the path. 3 We are greatly indepted to Gurski or correspondence about this generalization o the EFT on Cat deined by Power [26].

4 4 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO O course, this is analogous to the identiication o O-algebras and O-algebras or operads in spaces that motivated the coinage o the word operad in the irst place [21]. The theory o 2-monads gives a ormalism that allows us to treat operad algebras in a context with many other examples. It will be applied to algebras over categories o operators in the sequel [12]. With these identiications, Theorem 5.4 has the ollowing specialization. Theorem 0.2. Let O be an operad in Cat(V ). Then the inclusion o 2-categories has a let 2-adjoint J: O-AlgSt O-PsAlg St: O-PsAlg O-AlgSt, and the components o the unit o the adjunction are internal equivalences in O-PsAlg. Returning to our motivation, we discuss the specialization to symmetric monoidal categories in Section 3. Nonequivariantly, permutative categories are the same thing as P-algebras in Cat, where P is the categorical version o the Barratt- Eccles operad. Formally, the category o permutative categories is isomorphic to the category o P-algebras in Cat [22]. This suggests a generalization in which we replace P by a more general operad and replace Cat by a more general category o (small) categories. The generalization is illuminating nonequivariantly and should have other applications, but it is essential equivariantly, as we now explain. A naive permutative G-category is a permutative category with G-action, that is, a G-category with an action o the operad P, where we think o the categories P(j) as G-categories with trivial G-action. Permutative categories are the input o an operadic ininite loop space machine deined in [22, 30] and axiomatized in [23]. Its output is connective Ω-spectra with zeroth space given by the group completion o the classiying space o the input permutative category. Naive permutative G- categories work the same way. They naturally give rise to naive Ω-G-spectra. However, naive Ω-G-spectra really are naive. They are not even adequate to represent the Z-graded homology theories we see in nature. Naive permutative G-categories are inadequate input to a theory with genuine G-spectra as output. Genuine permutative G-categories are deined in [11] as algebras over an equivariant generalization P G o P, and these give the input or an operadic equivariant ininite loop space machine. We do not know any interpretation o genuine permutative G-categories other than the operadic one. Since the operads P and P G are the ones whose algebras are permutative categories, we call them the permutativity operads henceorward, and we recall their deinitions in 3. Morphisms between symmetric monoidal categories, or even between permutative categories, are rarely strict; they are given by strong and sometimes even lax symmetric monoidal unctors. Classical coherence theory shows how to convert such morphisms o symmetric monoidal categories to symmetric strict monoidal unctors between permutative categories. By irst strictiying and then applying a classical ininite loop space machine to classiying spaces, this allows classical ininite loop space theory to construct morphisms between spectra rom strong symmetric monoidal unctors between symmetric monoidal categories. Our theory will allow us to do the same thing equivariantly, starting rom genuine symmetric monoidal G-categories, but we must irst deine what those are.

5 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 5 A pseudoalgebra over P is a (small) symmetric monoidal category. 4 This suggests the ollowing new deinition. We shall be more precise in 3. Deinition 0.3. A (genuine) symmetric monoidal G-category is a P G -pseudoalgebra. A strong symmetric monoidal unctor o symmetric monoidal G-categories is a pseudomorphism o P G -algebras. A transormation between strong symmetric monoidal unctors is a P G -transormation. Henceorward, when we say symmetric monoidal G-category we always mean genuine. When we talk about naive symmetric monoidal G-categories, we will always explicitly say naive. The same convention applies to permutative G- categories. As we explain in 3, there is a unctor that sends naive permutative G-categories to naively equivalent genuine permutative G-categories and sends naive symmetric monoidal G-categories to naively equivalent genuine symmetric monoidal G-categories. The unctor applies to nonequivariant permutative and symmetric monoidal categories, viewed as G-categories with trivial G-action. This gives a plentitude o examples. We discuss the philosophy behind Deinition 0.3 in 3, where we also indicate relevant categorical questions that have been addressed by Rubin [29, 28] in work complementary to ours. He works concretely in the equivariant context o N G- operads pioneered by Blumberg and Hill [6] and developed urther by Rubin and others [7, 15, 29], and he compares our symmetric monoidal G-categories with the analogous but deinitionally disparate context o G-symmetric monoidal categories o Hill and Hopkins [16]. We shall say a bit more about his work in 3. It is not obvious that (genuine) symmetric monoidal G-categories are equivalent to (genuine) permutative G-categories, but Theorem 0.2 shows that they are. Corollary 0.4. The inclusion o permutative G-categories in symmetric monoidal G-categories has a let 2-adjoint strictiication 2-unctor. For a symmetric monoidal G-category X, the unit X StX o the adjunction is an equivalence o symmetric monoidal G-categories. Combined with the results o [11, Section 4.5], this gives the ollowing conclusion. Theorem 0.5. There is a unctor K G rom symmetric monoidal G-categories to Ω- G-spectra such that Ω K G (A ) is an equivariant group completion o the classiying G-space BA. Thus K G takes P G -pseudoalgebras and P G -pseudomorphism to genuine G- spectra and maps o G-spectra; it even takes algebra 2-cells between P G -pseudomorohisms to homotopies between maps o G-spectra (Remark 1.27). The proos give explicit constructions. Even nonequivariantly, this is a generalization o previous published work, although this specialization has long been understood as olklore. At least on a ormal level, this, coupled with [11, 25], completes the development o additive equivariant ininite loop space theory. Acknowledgements. The essential ideas in this paper come rom the beautiul categorical papers by Power [26] and Lack [19] and rom earlier categorical work o Kelly and Street, or example in [5, 31]. This paper is a testament to the power o ideas in the categorical literature. We owe an enormous debt o gratitude to Steve Lack, John Power, Nick Gurski, and Mike Shulman or all o their help. We also 4 This is true up to minor quibbles explained in 3

6 6 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO thank Jonathan Rubin or the nice observation recorded in 6, which helps justiy our ramework o internal rather than just enriched categories. 1. Categorical preliminaries 1.1. Internal categories. We need some elementary category theory to nail down relevant details about our general context. In part to do equivariant work without working equivariantly, we work in a context o internal V -categories, where V is any category with all inite limits. Some obvious examples are the category Set o sets, the category Cat o (small) categories, and the category U o spaces, 5 but there are many others. All examples come with based and equviariant variants, and the latter are o special interest to us. Remark 1.1. The category V has a terminal object, namely the product o the empty set o objects. A based object in V is an object V with a choice o morphism v 0 : V. A based map (V, v 0 ) (W, w 0 ) is a morphism V W that is compatible with the choices o basepoint, and V denotes the category o based objects and based morphisms. Finite limits in V are inite limits in V with the induced map rom given by the universal property. Remark 1.2. Let G be a discrete group. A G-object V in V has an action o G given by automorphisms g : V V satisying the evident unit and composition axioms. A G-map is a morphism V W that is compatible with given group actions, and GV denotes the category o G-objects and G-maps. Finite limits in GV are inite limits in V with the induced action by G. We understand V -categories to mean internal V -categories and we recall the deinition. Deinition 1.3. A V -category C consists o objects Ob C and Mor C o V with source, target, identity, and composition maps S, T, I, and C in V that satisy the axioms o a category. A V -unctor : C C is given by object and morphism maps in V that commute with S, T, I, and C. We write Cat(V ) or the category o V -categories and V -unctors. By contrast, a small category D enriched in V is given by a set o objects and an object D(c, d) o V or each pair (c, d) o objects o D, with composition given by maps in V and identities given by maps D(c, c) in V. Warning 1.4. In the categorical literature, V -categories usually reer to the enriched rather than the internal notion. In the unbased case, we can use the unctor V: Set V o Section 1.3 below to view categories enriched over (V, ) as special cases o internal ones. Example 1.5. A 2-category is a category enriched in Cat, and its enriched unctors are called 2-unctors. A category internal to Cat is a double category, and the internal unctors are double unctors. Remark 1.6. Since V has a terminal object, so does Cat(V ). It is easily checked that the categories Cat(V ) and Cat(V ) are canonically isomorphic. We shall use the notation Cat(V ). 5 As usual, spares are taken to be compactly generated and weak Hausdor.

7 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 7 Remark 1.7. A GV -category is a category internal to GV. Thus G acts on both the object o objects and the object o morphisms via morphisms in V. One can easily check that Cat(GV ) is canonically isomorphic to GCat(V ). We are especially interested in GU. Remark 1.8. One reason to require internal V -categories rather than just enriched ones is that it allows us to deine an inclusion i: V Cat(V ). We simply view an object X o V as a discrete V -category ix with Ob(iX) = Mor(iX) = X, and S, T, and I all identity maps, and C the canonical isomorphism X X X = X. It is straightorward to check that i is ull and aithul and is let adjoint to the object unctor. Thus Cat(V )(ix, A ) = V (X, ObA ). We oten omit i rom the notation, regarding V as a ull subcategory o Cat(V ). Along with the V -categories and V -unctors o Deinition 1.3, we need V -natural transormations, which we abbreviate to V -transormations. Deinition 1.9. A V -transormation α: = g, where and g are V -unctors A B, is a map α: Ob A Mor B in V such that the ollowing two diagrams commute. (1.10) Mor B α (S,T ) Ob A Ob B Ob B (,g) (1.11) Ob A Mor A (T,Id) Mor A (Id,S) Mor A Ob A α g α Mor B ObB Mor B C Mor B Mor B ObB Mor B C Note that the right down and let down composites do indeed land in the pullback, since S α T = T = T and T α S = g S = S g. The vertical composite β α o α: = g and β : g = h is the composite ObA (β,α) MorB ObB MorB C MorB. The identity V -transormation id: = is given by I = I : ObA MorB. We say that α: = g is an isomorphism, or α is invertible, i there is a V - transormation α 1 : g = such that α α 1 = id and α 1 α = id. As in Set, the condition in (1.11) or α 1 ollows rom that or α. The horizontal composite β α o α and β, as in the diagram A α B g β C, g

8 8 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO is given by the common composite in the commutative diagram Ob B Mor B (,α) Ob A (α,) Mor B Ob B β g Mor C ObC Mor C C g β Mor C Mor C ObC Mor C C In particular, using the same notation as above, the whiskering β is given by the composite Ob A Ob B β Mor C, and similarly, the whiskering g α is given by the composite Ob A α Mor B g Mor C. Notation Let V be a category with inite limits. Then the collection o V -categories, V -unctors, and V -transormations orms a 2-category, which we will also denote by Cat(V ), updating the notation o Deinition 1.3. In particular, we have the updated notations Cat(V ) and Cat(GV ) or the based and equivariant variants viewed as 2-categories Chaotic categories. We recall the deinition o chaotic (or indiscrete) category in the general context o internal categories. Deinition A V -category C is said to be chaotic (or indiscrete) i the map is an isomorphism in V. Mor(C ) (S,T ) Ob(C ) Ob(C ) Chaotic V -categories, despite their simplicity, are important since they lead to natural constructions o operads in V. An ordinary category A is chaotic i each A (x, y) is a point. For a set X there is a canonical chaotic category EX with object set X. This is related to other constructions in [14, Section 1]. We saw in Remark 1.8 that the object unctor Ob: Cat(V ) V has a let adjoint inclusion unctor i; the chaotic category unctor is right adjoint to Ob, as we show in Lemma 1.16 below. To generalize to V -categories, we start with the construction o EX. Deinition Let X be an object o V. The chaotic V -category EX has Ob EX = X and Mor EX = X X. The maps S, T, and I are the projections π 2, π 1, and the diagonal respectively, and the map C is id ε id: (X X) X (X X) = X X X X X, where ε: X ; that is, C is projection onto the irst and third coordinates. Remark When V = Set, every object o EX is initial and terminal, so that is isomorphic to a skeleton o EX. Thereore BEX is contractible. This also applies when V is the category o spaces. Lemma The chaotic V -category unctor E : V Cat(V ) is right adjoint to the object unctor Ob, so that there is a natural isomorphism o sets V (Ob A, X) = Cat(V ) (A, EX).

9 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 9 Moreover, or any two V -unctors E, F : A EX, there exists a unique V - transormation α: E F, necessarily a V -isomorphism. Proo. The V -unctor F : A EX corresponding to a map : Ob A X in V is given by on objects and by Mor A (T,S) Ob A Ob A X X on morphisms. Thus Ob F = by deinition, and a little diagram chase shows that F is the only V -unctor with object map. Given V -unctors E and F and a V -transormation α: E = F, the condition in (1.10) orces α = (F, E). Again, a small diagram chase shows that α so deined is indeed a V -transormation. The ollowing result is a reinterpretation o the second statement o Lemma Corollary The the category o V -unctors and V -natural transormations rom A to EX is isomorphic to the chaotic category on the set o V -maps rom ObA to X. Note that the counit Ob E Id o the adjunction is the identity. Lemma The unit map A E(ObA ) o the adjunction is an isomorphism i and only i the V -category A is chaotic. As a right adjoint, the chaotic category unctor preserves products and other limits and thereore preserves all structures deined in terms o those operations. We can view it as an especially elementary orm o categoriication The embedding o Set in V. Many operads and other constructions are irst deined in the category Set. In the unbased case, assuming that V has coproducts in addition to inite limits, we can use the ollowing deinition to lit such constructions to V. Deinition Deine V: Set V to be the unctor that sends a set S to s S, the coproduct o copies o the terminal object indexed on S. It has a right adjoint U: V Set speciied by letting UX = V (, X). Thus (1.20) V (VS, X) = Set(S, UX). Remark In all o the unbased examples o interest, the unit map Id UV o the adjunction is an isomorphism. This expresses the intuition that a map rom a point into a disjoint union o points is the same as a choice o one o the points. It ensures that V is a ull and aithul unctor. Henceorward, in the unbased case, we assume this and thus regard Set as a subcategory o V, omitting V rom the notation. Remark When the unit Id UV o the adjunction between Sets and V is an isomorphism, the adjunction lits to an adjunction between Sets and V. Indeed, we deine V o a set S with basepoint s 0 to be the based object = V( ) V(s0) VS, and similarly, we deine U o a based object (X, x 0 ) in V to be = UV( ) = U( ) U(x0) UX.

10 10 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO The unit and the counit o the original adjunction then become based maps, giving the desired adjunction. Deinition The adjunction between Sets and V also lits to the equivariant setting in the ollowing way. Deine V: GSet GV to be the unctor that sends a G-set S to the object VS in V with the action o G induced by the unctoriality o V applied to the maps o sets g : S S or g G. Thinking o the action by G on an object X o GV as given by a map VG X X in V and applying U, we obtain an action o G on UX. This gives a orgetul unctor U: GV GSet that is right adjoint to V. Thus (1.24) GV (VS, X) = GSet(S, UX). The ollowing remark applies equally well in the nonequivariant and equivariant contexts. Remark As a let adjoint, V preserves colimits. To ensure that V preserves operads and other structure in Set, we assume henceorward that V also preserves inite limits. As we explain in the brie Section 6, which was provided to us by Jonathan Rubin, this is a very mild assumption that holds in all o our unbased examples. The assumption ensures that the adjunction (V, U), when applied to objects and morphisms, induces an adjunction (1.26) Cat(V )(VA, B) = Cat(A, UB), where A is a category and B is a V -category. The unctor V: Cat Cat(V ) is again ull and aithul, and we regard Cat as a subcategory o Cat(V ), omitting V rom the notation. We end this section by noting that using the unctor V and assuming that V is cartesian closed, one can see that V -transormations can be thought o as analogues o homotopies. Let I be the category with objects [0] and [1] and a unique nonidentity morphism I : [0] [1], and consider it as a V -category via the unctor V. For V -unctors, g : A B, there is a bijection between V -transormations rom to g and V -unctors h: A I B that restrict to on A [0] and to g on A [1]. Indeed, given α: Ob A Mor B, we deine h: A I B on objects as Ob (A I ) = ObA {[0],[1]} = {[0],[1]} Ob A On morphisms, h is given by the V -unctor Mor (A I ) = MorA = {id 0,id 1,I} α {id 0,id 1,I} {[0],[1]} Mor B S,T Ob B. Mor A Mor B speciied on the three components o the coproduct by, g and the common composite in (1.11), respectively. We leave it to the reader to check that this assignment is a bijection. Remark Taking V = GU, taking O to be an E G-operad in Cat(GU ), and using that the classiying space unctor B preserves products and takes I to the unit interval, we can use our ininite loop space machinery [13, 25], in particular [13, Proposition 6.16], to transport GU -transormations between strict maps o O- algebras to homotopies between maps o G-spectra.

11 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION Pseudoalgebras over operads and 2-monads 2.1. Pseudoalgebras over 2-Monads. Deinition 2.1. A 2-monad on a 2-category K is a Cat-enriched monad in K. Precisely, it is a 2-unctor T: K K together with 2-natural transormations ι: I T and : TT T satisying the evident unit and associativity laws: the ollowing diagrams o 2-natural transormations must commute. T ιt T 2 Tι T T 3 T T 2 T T T 2 T Deinition 2.2. A (strict) T-algebra (X, ) is an object X o K together with an action 1-cell : X such that the ollowing diagrams commute. X T 2 X T ι X X X X. In particular, is a T-algebra with action map or any X K. A T-pseudoalgebra (X,, ϕ, υ) requires the same two diagrams to commute up to invertible 2-cells υ : id = ι X and ϕ: T = X, satisying three coherence axioms ([26, 2.4]). One deines lax T-algebras similarly, but not requiring υ and ϕ to be invertible. We shall not consider them. A T-pseudoalgebra is normal i the irst diagram commutes, so that υ is the identity. We restrict attention to normal pseudoalgebras henceorward. With this restriction, the irst two coherence axioms translate to requiring that the whiskerings ϕ ι and ϕ Tι X are both the identity transormation =. The remaining coherence axiom requires the equality o diagrams T 3 X T 2 X T 2 T 2 X T ϕ T ϕ X = T 3 X T 2 X T 2 T 2 X ϕ T T Tϕ T 2 T X Deinition 2.3. A T-pseudomorphism (, ζ): (X,, ϕ) (Y, ξ, ψ) o T-pseudoalgebras is given by a 1-cell : X Y and an invertible 2-cell ζ : ξ T =. T TY X ζ X Y ξ

12 12 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO satisying two coherence axioms ([26, 2.5]). I ζ is the identity, is said to be a strict T-map. One deines lax T-maps by not requiring ζ to be invertible, but we shall not consider those. Restricting X and Y to be normal, we require the whiskering ζ ι X to be the identity transormation =. This makes sense since the naturality o ι and the normality equalities ι X = id X and ξ ι Y = id Y show that the domain and target o ζ ι X are both. There is then only one remaining coherence axiom. It requires the equality o diagrams T 2 X T 2 TζT2 Y Tξ T TY ϕ X T ζ Y ξ = T 2 X T 2 T ζ X T 2 Y TY Tξ ψ ξ TY Y ξ Deinition 2.4. An algebra 2-cell λ: (, ζ) = (g, κ) is given by a 2-cell λ: = g in K, not necessarily invertible, such that X T Tλ Tg κ g TY ξ = Y X T ζ g TY λ Y ξ With these deinitions, we have the three 2-categories T-PsAlg, T-AlgPs, and T-AlgSt promised in the introduction The 2-monads associated to operads. To construct a monad rom an operad, we must assume that V and thereore Cat(V ) has colimits in addition to having inite limits. The construction o the monad associated to an operad requires equivariance and base object identiications, which are examples o colimits. Since colimits o categories are oten notoriously ill-behaved, we oer a philosophical comment on how we use the 2-monads associated to operads in topology. Remark 2.5. We are interested in O-G-categories X and their classiying G-spaces X = BX. No monads need play any role in the statements o the theorems we are proving about them, but we are using 2-monads on categories o G-categories or the proos. With some exceptions, we neither know nor care about any commutation properties o B relating these 2-monads to monads on categories o G-spaces. Such relations would be suspect since we cannot expect the relevant colimits to commute with B. That is, we are using 2-monads purely ormally to obtain inormation about the underlying categories o O-G-algebras. Operads are deined in any symmetric monoidal category and in particular in any cartesian monoidal category. An operad O in Cat(V ) consists o V -categories

13 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 13 O(j) or j 0 with right actions o the symmetric groups Σ j, a unit V -unctor 1: O(1), where is the trivial V -category, and structure V -unctors γ : O(k) O(j 1 ) O(j k ) O(j j k ) that are equivariant, unital, and associative in the sense that is prescribed in [21, Deinition 1.1]. Assumption 2.6. We assume throughout that operads O are taken to be reduced operads in Cat(V ). Reduced means that O(0) is the terminal object, so that an O-algebra A has a base object 0, namely the image o under the action. We write 0 or the identity V -unctor O(0). For the most useul contexts, we must also assume that O is Σ-ree, meaning that the symmetric group Σ j acts reely on the jth object O(j) or all j, but we do not restrict to Σ-ree operads in this paper. We shall be especially interested in chaotic operads. Deinition 2.7. An operad O in Cat(V ) is chaotic i each o its V -categories O(n) is chaotic. We will shortly deine strict algebras and pseudoalgebras over an operad in Cat(V ). For an operad O in any symmetric monoidal category (W, ), we have an isomorphism o categories between (strict) O-algebras and O + -algebras, where O + is the monad on W that is constructed rom O by deining (2.8) O + X = n 0 O(n) Σn X n. Note that Σ n acts on the right o O(n) and on the let o X n. Intuitively, we are identiying aρ x with a ρx or σ Σ n and elements a O(n) and x X n. As explained in [24, Section 4], i W is cartesian monoidal and O is reduced, there is a monad O on W whose (strict) algebras are the same as those o O +. The dierence is that O + -algebras acquire base objects via their actions, whereas O-algebras have preassigned base objects that must agree with those assigned by their actions; O is constructed rom O + using base object identiications. We can adjoin disjoint base objects by taking X + = X, and then O + (X) = O(X + ). In all topological applications, the monad O is o considerably greater interest than the monad O +, and we shall restrict attention to it. We need a preliminary deinition to deine O in our context. Deinition 2.9. Let O be an operad in Cat(V ) and let A be a based V -category. In line with Assumption 2.6, let 0 denote the base object o A. Let 1 r n. Deine σ r : O(n) O(n 1) to be the composite V -unctor (2.10) O(n) = O(n) n id 1 r n r O(n) O(1) r 1 O(0) O(1) n r γ O(n 1).

14 14 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO Deine σ r : A n 1 A n to be the insertion o base object V -unctor (2.11) σ r = id r 1 0 id n r : A n 1 A n. Construction Let O be a (reduced) operad in Cat(V ). We construct a 2-monad O in the 2-category Cat(V ) o based V -categories. Let Λ be the subcategory o injections and permutations in the category o inite based sets n. Then O is a contravariant unctor on Λ via the symmetric group actions and the degeneracy unctors σ r. For a based V -category A, the powers A n give a covariant unctor A on Λ via permutations and the insertions o base object unctors σ r. Deine (2.13) O(A ) = O Λ A. The unit ι: A OA is induced by the V -map O(1) determined by id O(1) and the product : O 2 O is induced by the structural maps γ o the operad Pseudoalgebras over operads. We deine pseudoalgebras over an operad O in Cat(V ), largely ollowing Corner and Gurski [8]. 6 The deinition can be extended to operads in any 2-category with products. Deinition An O-pseudoalgebra A = (A,, ϕ) is a V -category A together with action V -unctors = n : O(n) A n A and invertible composition V -transormations ϕ = ϕ(n; m 1,, m n ) (2.15) O(n) ( r O(m r) A mr ) id ( r mr ) O(n) A n n π ϕ A O(n) ( r O(m r)) A m γ id O(m) A m. m Here 1 r n, m = m m n, and π is the shule that moves the variables O(m r ) to the let and identiies A m1 A mn with A m. These data must satisy the ollowing axioms. When we say that an instance o (2.15) commutes, we mean that the corresponding component o ϕ is the identity. Axiom 2.16 (Equivariance). The ollowing diagram commutes or ρ Σ n. O(n) A n id ρ O(n) A n ρ id O(n) A n n A This means that the induce a map : O + A A. n 6 They only consider V = Set, but the generalization is immediate.

15 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 15 Axiom 2.17 (Unit Object). The ollowing whiskering o an instance o the diagram (2.15) commutes or 1 r n; that is, the whiskering o ϕ along the composite o the irst map o (2.10) and an instance o π 1 is the identity 2-cell, giving the ollowing commutative diagram. O(n) A n 1 id σ r O(n) A n σ r id O(n 1) A n 1 n 1 A This means that the induce a map : OA A. Axiom 2.18 (Operadic Identity). The ollowing diagram commutes. n A 1 id O(1) A = We require coherence axioms or the V -transormations ϕ. These are dictated by compatibility with the monadic axioms in 2.1 and we use those to abbreviate the statements o the operadic axioms. Axiom [Equivariance] When the diagram (2.15) is obtained rom another such diagram by precomposing with a permutation, we require ϕ to be the whiskering o ϕ in the original diagram by the permutation. Precisely, given ρ Σ n and τ r Σ mr, there are equalities o whiskerings and A ϕ(n; m 1,..., m r ) = ϕ(n; m ρ(1),..., m ρ(n) ) (ρ ρ 1 ) ϕ(n; m 1,..., m r ) = ϕ(n; m 1,..., m r ) id (τ r τr 1 ). r This means that the ϕ pass to orbits to deine an invertible 2-cell σ in the diagram 1 O 2 +A T O + A ϕ O + A A. Using the unit object axiom, it ollows that ϕ then passes through base object identiications to deine an invertible 2-cell σ in the diagram (2.20) O 2 A T OA OA ϕ A. Axiom [Operadic Identity] The whiskering o ϕ(1; n) along 1 id: O(n) A n O(1) O(n) A n ( )

16 16 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO is the identity, and the whiskering o ϕ(n; 1 n ) along is the identity. id (1 id) n : O(n) A n O(n) (O(1) A ) n Axiom [Operadic Composition] Writing = (γ id) π, m = r m r, p r = s p rs, and p = r,s p rs, we require the ollowing two pasting diagrams to be equal. O(n) ( r O(mr) s (O(prs) A prs ) ) id r (id s prs ) O(n) r (O(mr) A mr ) id r mr O(n) A n O(m) r,s (O(prs) A prs ) id r,s prs O(m) A m ϕ n m ϕ O(p) A p A p O(n) ( r O(mr) s (O(prs) A prs ) ) id r (id s prs ) O(n) r (O(mr) A mr ) id O(n) r (O(pr) A pr ) id ϕ id r id r mr O(n) A n O(m) r,s (O(prs) A prs ) ϕ n O(p) A p p A. This axiom is the translation o the equality o pasting diagrams speciied in Deinition 2.2. I the transormations ϕ are all the identity, then all axioms are satisied automatically, and A is a (strict) O-algebra as originally deined in [21, Section 1]. It is clear rom the deinition that A is an O-pseudoalgebra i and only i it is a normal O-pseudoalgebra. The two Operadic Identity properties are precisely what is needed to give the normality. Deinition An O-pseudomorphism (, ζ): (A,, ϕ) and (B, ξ, ψ) o O-pseudoalgebras is given by a V -unctor : A B and a sequence o invertible V - transormations ζ n O(n) A n n A id n ζ n O(n) B n We require to preserve 0 and 1, so that ζ 0 and the whiskering o ζ 1 with the map 1 id: A = A O(1) A are the identity. Then is a based map, and B. n

17 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 17 hence it induces a map O : OA OB. We moreover require ζ n = ζ n (ρ ρ 1 ) or all ρ Σ n. This implies that ζ induces an invertible V -transormation OX O OY ζ X Y. ξ We require the ollowing two pasting diagrams to be equal. O(n) r (O(mr) A mr ) id r (id mr ) O(n) r (O(mr) Bmr ) id mr id ζ mr id ξ mr O(n) A n id n O(n) B n O(m) A m ϕ m ζ n ξ n m A B O(n) r (O(mr) A mr ) id r (id mr ) O(n) r (O(mr) Bmr ) id ξ mr O(n) B n O(m) A m id m ψ O(m) B m ξ n m A ζ m ξ m B The equality o these diagrams is equivalent to that o the pasting diagrams speciied in Deinition 2.3. I the ζ n are identity V -unctors, then is a (strict) O-map. Deinition An algebra 2-cell λ: (, ζ) = (g, κ) is given by a V -transormation λ: = g, not necessarily invertible, such that the pasting diagrams speciied

18 18 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO in Deinition 2.3 are equal. Explicitly, or all n O n A n n A id n id λ n id g n κ n g O B n ξ n = O n A n n B A id n g ζ n O n B n λ B ξ n As promised in the introduction, with these deinitions, we have the three 2- categories O-PsAlg, O-AlgPs, and O-AlgSt, and a comparison o deinitions identiies them with their monadic analogs O-PsAlg, O-AlgPs, and O-AlgSt. Remark Since V is cartesian monoidal, we have a diagonal map o operads : O O O. Use o shows that the 2-category o O-pseudoalgebras is again cartesian monoidal, and it is also bicomplete. Remark We comment on paths not taken. As in [9], we can deine pseudooperads by allowing the associativity diagram or the composition unctor γ to commute only up to V -isomorphism. We can then deine pseudoalgebras over pseudo-operads. Similarly, ollowing [4, 9], we can deine lax or op-lax O-algebras by not requiring the ϕ to be isomorphisms. For example, taking the operad to be the permutativity operad P (see below), this deines lax symmetric monoidal categories. Lax monoidal categories are studied in [4, 9] and are called lax multitensors in [3]. The papers [4, 9] show that lax monoidal categories are strict algebras over an appropriate operad, and the same is also true o lax symmetric monoidal categories. In the absence o applications, we preer to ignore these urther weakenings and this orm o strictiication. 3. Operadic speciication o symmetric monoidal G-categories Except in Remark 3.5, we specialize to the case V = Set in this section. However, we can use the unctor V: Set V rom Deinition 1.19 or its equivariant variant rom Deinition 1.23 to generalize the basic deinitions. Since V preserves inite limits (see Remark 1.25), it preserves groups and operads. Applying V to the operads deined below gives the corresponding operads in V or GV, and their algebras speciy the analogues in V or GV o the algebraic structures we discuss. We irst recall the deinition o the permutativity operad P, which is chaotic by deinition. We start with the associativity operad 7 Assoc in Set, where Assoc(j) = Σ j as a right Σ j -set. We write e j or the identity element o Σ j. We have block sum o permutations homomorphisms : Σ i Σ j Σ i+j. I j = j j k and σ Σ k, we deine σ(j 1,, j k ) Σ j to be the element that permutes the k blocks o letters as σ permutes k letters. With these notations, the structure maps γ are given by 8 γ(σ; τ 1,, τ j ) = σ(j 1,, j k )(τ 1 τ k ). 7 Always denoted M in previous work o the senior author. 8 This corrects an incorrect ormula on [22, p. 82].

19 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 19 This is orced by γ(e k ; e j1,, e jk ) = e j and the equivariance ormulas or ν s Σ js and γ(σ; ν 1 τ 1,, ν k τ k ) = γ(σ; ν 1,, ν j )(τ 1 τ k ) γ(σ; τ 1,, τ k ) = γ(; τ σ 1 (1),, τ σ 1 (k))σ(j 1,, j k ) or Σ k in the deinition o an operad. To see this, take = e k and ν s = e js and use these ormulas in order. Algebras over Assoc are monoids in Set. Deinition 3.1. Let G be a discrete group. Let G act by right multiplication on G and diagonally on G G. With these actions on objects and morphisms, EG is a right G-category. It also has a let action via let multiplication, making it a G-category. As shown in [14], BEG is a universal principal G-bundle. The permutativity operad P is obtained by applying the product-preserving unctor E( ) to Assoc. Deinition 3.2. The permutativity operad P is the chaotic categoriication o Assoc, so that P(j) is the right Σ j -category EΣ j. Clearly P(0) and P(1) are trivial, the latter with unique object e 1 = 1. The structure map γ is induced rom that o Assoc by application o E( ). There is a product-preserving unctor Cat(EG, ) rom the category o G- categories to itsel. It is considered in detail in [11, 14]. Deinition 3.3. Let A be a G-category. Deine Cat(EG, A ) to be the G-category whose objects and morphisms are all (not necessarily equivariant) unctors EG A and all natural transormations between them. The (let) action o G on Cat(EG, A ) is given by conjugation. Note that, by Corollary 1.17, when A is chaotic then so is Cat(EG, A ). Since the unctor Cat(EG, ) preserves products, it also preserves structures deined in terms o products. In particular, it takes G-operads to G-operads. The trivial G-unctor EG induces a G-unctor ι: A Cat(EG, A ). Upon taking classiying spaces, ι induces a nonequivariant homotopy equivalence. Deinition 3.4. The permutativity operad P G in Cat(G-Set) is the chaotic operad P G = Cat(EG, P), where G acts trivially on P. Thus P G (j) is the G-category Cat(EG, EΣ j ). The operad structure is induced rom that o P. Remark 3.5. Returning to a general V, recall the category GV o G-objects in V rom Remark 1.2 and the unctor V: G-Set GV rom Deinition Applying V, we regard P G as an operad in Cat(GV ). In the case V = U, V just gives a G-set the discrete topology. Thus, our notion o a symmetric monoidal G-category immediately extends to G-categories internal to G-spaces. Clearly P G is reduced and P G (1) is trivial with unique object 1. When G is the trivial group, P G = P. The unctor ι speciies a map P P G o G-operads. Application o B gives a weak equivalence BP BP G o nonequivariant operads. The operad BP G is an equivariant E operad, meaning that BP G (j) is a universal (G, Σ j )-bundle (see [14, Theorem 0.4]).

20 20 B. GUILLOU, J.P. MAY, M. MERLING, AND A.M. OSORNO It has been known since [22] that P-algebras are the same as permutative categories, and in [11] we deined a genuine permutative G-category to be a P G - algebra. In principle, or an operad O, O-algebras give unbiased algebraic structures. 9 Products A n A are given or each object o O(n). Biased algebraic structures are deined more economically, usually starting rom a binary product : A A A. Ignoring the associativity isomorphism or cartesian products, the associativity axiom or permutative categories then states that ( id) = (id ). When permutative categories are deined by actions o P, we are given a canonical 3-old product A 3 A, and the associativity axiom now says that both ( id) and (id ) are equal to that 3-old product. The biased deinition o a permutative category requires use o only A n or n 3. Similarly, the biased deinition o a symmetric monoidal category requires use o only A n or n 4. Use o our variables is necessary to state the pentagon axiom in the absence o strict associativity. Just as permutative categories are the same as P-algebras, we claim that symmetric monoidal categories are essentially the same as P-pseudoalgebras. We have required the strict Operadic Identity Axiom on P-pseudoalgebras because that is both natural and necessary to our claim: symmetric monoidal categories A come with the identity operation A A, and there is nothing that might correspond to an isomorphism to the identity operation. More substantially, our Unit Object Axiom requires that 0 be a strict unit object or the product on an O-pseudoalgebra. This is o course not true or symmetric monoidal categories in general. The more precise claim is that symmetric monoidal categories with a strict unit object correspond bijectively to P-pseudoalgebras as we have deined them. This requires proo, which in one direction amounts to deriving the pentagon and hexagon axioms rom the equivariance and associativity properties o the transormations ϕ that appear in the deinition o P-pseudoalgebras, and in the other direction amounts to proving that, conversely, all the properties o the transormations ϕ can be derived rom those at lower levels. Although not in the literature as ar as we know, this is well-known categorical olklore and is let as an exercise. See chapter 3 o [20] or a discussion o the nonsymmetric case. O course, a symmetric monoidal category is monoidally equivalent to a symmetric monoidal category with a strict unit since it is monoidally equivalent to a permutative category, but the ormer equivalence is much easier to prove. It is a categorical analogue o growing a whisker to replace a based space by an equivalent based space with nondegenerate basepoint [10, Section 5]. Just as we require basepoints to be nondegenerate in topology, we require our symmetric monoidal categories to have strict unit objects. In Deinition 0.3, we deined genuine symmetric monoidal G-categories to be P G -pseudoalgebras, implicitly requiring them to satisy our axioms. The operadic deinitions o genuine permutative and symmetric monoidal G-categories give unbiased algebraic structure, and here the biased notions have yet to be determined. Problem 3.6. Determine biased speciications o genuine permutative and symmetric monoidal G-categories. That is, it is desirable to determine explicit additional structure on a naive permutative or symmetric monoidal G-category that suices to give it a genuine 9 Biased versus unbiased algebraic structures are discussed in [20, Section 3.1], or example.

21 SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION 21 structure. As shown in [2] by the ourth author and her collaborators, this problem cannot be solved. More precisely, they show that i G is a nontrivial inite group, the operad P G is not initely generated. This means that in order to speciy the structure o a P G -algebra, one needs to speciy an ininite amount o inormation, subject to an ininite amount o axioms. Using ideas rom Rubin [28], one can produce a initely generated suboperad Q G o P G that is equivariantly equivalent, in the sense that it is also an E G-operad. Bangs et al. solve in [2] the problem o identiying biased speciications or algebras over Q G or G = C p when p = 2, 3. Rubin [28] has solved this problem in a closely related but not identical context. He proves a coherence theorem o just the sort requested or algebras over the N operads that he constructs. Despite the close similarity o context, there is hardly any overlap between his work and ours. His work in progress promises to establish the precise relationship between our symmetric monoidal G-categories and commutative monoids in the relevant G-symmetric monoidal categories o Hill and Hopkins [16]. Precisely, his normed symmetric monoidal categories are intermediate between these and will be compared to each in orthcoming papers o his. Since naive permutative and symmetric monoidal G-categories are just nonequivariant structures with G acting compatibly on all structure in sight, the nonequivariant equivalence between biased and unbiased deinitions applies verbatim to them. This has the ollowing implication, which shows that naive structures can be unctorially extended to naively equivalent genuine structures. Proposition 3.7. The unctor Cat(EG, ) induces unctors rom naive to genuine permutative G-categories and rom naive to genuine symmetric monoidal G- categories. In both cases, the constructed genuine structures are naively equivalent via ι to the given naive structures. In particular, we can apply this to nonequivariant input categories or to categories with G-action. Thus examples o genuine permutative and symmetric monoidal G-categories are ubiquitous. 4. Enhanced actorization systems 4.1. Enhanced actorization systems. In this section, we establish the context or the strictiication theorem by deining enhanced actorization systems. We let K be an arbitrary 2-category. Deinition 4.1. An enhanced actorization system, abbreviated EFS, on K consists o a pair (E, M ) o classes o 1-cells o K, both o which contain all isomorphisms, that satisy the ollowing properties. (i) Every 1-cell actors as a composite where m M and e E. (ii) For a diagram in K o the orm X e I m Y, A e X v ϕ u B m Y,