Lifting Enhanced Factorization Systems to Functor 2-Categories (Preliminary Version)

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LIFTING ENHANCED FACTORIZATION SYSTEMS TO FUNCTOR 2-CATEGORIES (PRELIMINARY VERSION)

PETER J. HAINE

Abstract. In this paper we give sufficient conditions for lifting an enhanced factorization system (E, M) on a 2-category 푲 to the functor 2-category 푲퐶, where 퐶 is a small category. Due to work of Lack, this result gives coherence results for 2-monads on functor 2- categories. These coherence results are of immediate interest to work in progress of Guillou, May, Merling, and Osorno on equivariant infinite loopspace theory.

0. Overview The motivation for this work comes from the theory of coherence results for 2-monads, specifically 2-monads of functor 2-categories, as this immediate interest to current work in progress of Guillou, May, Merling, and Osorno in equivariant infinite loopspace theory. The basic problem is the following: Given a 2-category 푲 and a 2-monad 푇 on 푲, is every pseudo-푇-algebra equivalent (as a pseudo-푇-algebra) to a strict 푇-algebra? In other words, this question asks if every pseudo-푇-algebra be “strictified” to a strict 푇-algebra. Power [4, Cor. 3.5] proved if 푇 is a 2-monad on the 2-category Cat, of categories, functors, and natural transformations, and 푇 preserves all functors which are bijective-on-objects, then every pseudo-푇-algebra is equivalent to a strict 푇-algebra. In fact, Power’s result is more general: he showed that if 푆 is a small set (regarded as a discrete 2-category) and 푇 is a 2- monad on the functor 2-category Cat푆 that preserves 1-cells in Cat푆 which are 푆-indexed sets of bijective-on-objects functors, then every pseudo-푇-algebra is equivalent to a strict 푇-algebra. There are two ideas employed here; the first is to lift the result pointwise tothe functor 2-category, and the second is that the preservation of a certain class of 1-cells is a sufficient condition for every pseudo-푇-algebra to be “strictified”. Lack [3, §4.2] used enhanced factorization systems, developed by Kelly [1], to generalize Power’s result to an arbitrary 2-category 푲 with an enhanced factorization system. What we are interested in is a generalization of Power’s result to the functor 2-category Cat퐶, or to 푲퐶, where 퐶 is a small category, and 푲 is a 2-category. The goal of this paper is to give conditions to lift an enhanced factorization system ona 2-category 푲 to the functor 2-category 푲퐶, which provides a setting where Lack’s generalized coherence result can be applied. In §1 we review enhanced factorization systems, as well as precisely state Lack’s generalization of Power’s result, which requires an additional condition which we call rigidity of the enhanced factorization system. We also show how to lift the factorization of 1-cells of an enhanced factorization system on a 2-category 푲 to the functor 2-category 푲퐶, where 퐶 is a small category. In §2 we analyze the enhanced factorization system on Cat which Power inherently employed in his coherence result in order to determine sufficient conditions for lifting an enhanced factorization system ona 2 category 푲 to 푲퐶. We conclude §2 by stating the main result of this paper, which says that under mild assumptions, an enhanced factorization system on 푲 can be lifted to 푲퐶, and that if the enhanced factorization system on

Date: November 24, 2015. 1 2 PETER J. HAINE

푲 is rigid, the lifted enhanced factorization system if also rigid. Section 3 is dedicated to proving the “lifting” part of the main result, while §4 is dedicated to showing that the lift of a rigid enhanced factorization system is rigid, under mild hypotheses. Acknowledgments. The author completed much of this work while at the University of Chicago REU program funded by NSF DMS-1344997. The author would like to thank Pe- ter May for suggesting this project, and his many helpful comments and advice. The author would also like to thank Emily Riehl for teaching him most of the category theory he knows and first suggesting that he read about 2-categories, as well as providing many useful comments.

1. Enhanced Factorization Systems We begin by reviewing enhanced factorization systems. 1.1. Notation. Throughout this paper, we adopt the following notational conventions. (1.1.a) We write Cat for the 2-category of small categories, functors, and natural transformations. (1.1.b) Throughout 푲 denotes a 2-category and 퐶 denotes an ordinary category. (1.1.c) We denote the objects of an ordinary category 퐶 by lowercase Roman letters 푐, 푐′, etc., and a morphism 푐 푐′ by a lowercase Roman letter such as 푓 or 푔. (1.1.d) Since we are mostly interested in the case of functor 2-categories of the form 푲퐶, unless otherwise specified we write capital Roman letters 퐹, 퐺, etc. for objects, lowercase Greek letters 훼, 훽, etc. for 1-cells, and uppercase Greek letters Ψ, Φ, etc. for 2-cells. (1.1.e) We use calligriphic letters such as E and M to denote a distinguished class of 1-cells in a 2-category. (1.1.f) We use a double-struck arrow “⟹” to denote a 2-cell in a 2-category, or, when noted, a natural transformation. 1.2. Definition ([3, § 4.2]). An enhanced factorization system on a 2-category 푲 consists of a pair of classes of 1-cells (E, M), both containing all isomorphisms, satisfying the following properties. (1.2.a) Every 1-cell 훼 of 푲 factors as a composite ̄훼∘ ̃훼, where ̃훼∈ E and ̄훼∈ M. (1.2.b) For a diagram in 푲 of the form

휀 퐹 퐹′

훼 훼′ ⟸Ψ 퐺 퐺′ 휇 where 휀 ∈ E, 휇 ∈ M, and Ψ is an invertible 2-cell, there is a unique pair (훿, Ψ)̃ consisting of a 1-cell 훿∶ 퐹′ 퐺 and an invertible 2-cell Ψ∶̃ 훼′ ⟹ 휇훿 so that we have a factorization 휀 휀 퐹 퐹′ 퐹 퐹′ 훿 훼 훼′ = 훼 훼′ ⟸ ⟸ Ψ Ψ̃ 퐺 퐺′ 퐺 퐺′ . 휇 휇 By the uniqueness, if Ψ is the identity, then 훾훿 = 훼′ and Ψ̃ is the identity. LIFTING ENHANCED FACTORIZATION SYSTEMS TO FUNCTOR 2-CATEGORIES (PRELIMINARY VERSION) 3

(1.2.c) Suppose that we are given 1-cells 휀∶ 퐹 퐹′ in E, 휇∶ 퐺 퐺′ in M, and two pairs ′ ′ ′ ′ 훼1, 훼2 ∶ 퐹 퐺 and 훼1, 훼2 ∶ 퐹 퐺 in 푲, so that the squares 휀 휀 퐹 퐹′ 퐹 퐹′

′ and ′ 훼1 훼1 훼2 훼2 퐺 퐺′ 퐺 퐺′ 휇 휇 ′ ′ commute, and 2-cells Ψ and Ψ such that 휇Ψ = Ψ 휀. Let 훿1 and 훿2 denote the unique 1-cells so that 훼1 = 훿1휀 and 훼2 = 훿2휀 given by (1.2.b). Then there exists a unique ′ 2-cell Δ∶ 훿1 ⟹ 훿2 so that Δ휀 = Ψ and 휇Δ = Ψ . We say that an enhanced factorization system (E, M) is rigid if the following additional property holds.

(1.2.d) For any 1-cell 휇∶ 퐹 퐺 in M and any 1-cell 훼∶ 퐺 퐹 in 푲, if 휇훼 ≅ id퐺, then 훼휇 ≅ id퐹. 1.3. Examples. The following classical examples of enhanced factorization systems are es- sentially due to Power [4]. (1.3.a) The 2-category Cat has an rigid enhanced factorization system (B, F) where B is the class of functors which are bijective-on-objects, and F is the class of fully faithful functors. (1.3.b) If 푆 is a small set, the 2-category Cat푆 has an rigid enhanced factorization system (B푆, F 푆) where B푆 is the class of 푆-indexed sets of functors, all of which are bijective- on-objects, and F 푆 is the class of 푆-indexed sets of functors, all of which are fully faithful functors. (1.3.c) More generally, if 푆 is a small set, and 푲 is a 2-category with a rigid enhanced factorization system (E, M), then 푲푆 has an rigid enhanced factorization system (E푆, M 푆) where E푆 is the class of 푆-indexed sets of 1-cells in E, and M 푆 is the class of 푆-indexed sets of 1-cells in M. Now let us explain Lack’scoherence result [3, §4.2]. Weassume familiarity with 2-monads, algebras for 2-monads, and pseudo-algebras for 2-monads. We do not review these notions because the main results and proofs in this paper do not actually require any knowledge of 2- monads, although the motivation for this work does. The unfamiliar reader should consult [2, §§3.1–3.2; 3, §1; 4, §2]. ps 1.4. Notation. Suppose that 푲 is a 2-category and that 푇 is a 2-monad on 푲. Write Alg푇 for the 2-category of pseudo-푇-algebras, morphisms, and algebra 2-cells. 1.5. Definition. Suppose that 푲 is a 2-category with an enhanced factorization system (E, M). We say that a 2-monad 푇 on 푲 preserves E if 휀 ∈ E implies that 푇휀 ∈ E. Lack’s generalization of Power’s result is the following. 1.6. Theorem ([3, Thm. 4.6]). Suppose that 푲 is a 2-category with a rigid enhanced factorization system (E, M), and that 푇 is a 2-monad on 푲. If 푇 preserves E, then every pseudo-푇- ps algebra is equivalent in Alg푇 to a strict 푇-algebra. For the rest of this section we are concerned with showing that an enhanced factorization system (E, M) on a 2-category 푲 can be lifted to the functor 2-category 푲퐶 to classes of morphisms E퐶 and M 퐶 so that every 1-cell in 푲퐶 factors as a composite of a 1-cell in E퐶 followed by a 1-cell in M 퐶, though the other axioms to define an enhanced factorization 4 PETER J. HAINE system on 푲퐶 are not generally satisfied without additional assumptions. We first make the following simplifying observation. 1.7. Observation. Suppose that 퐶 is a small category, and that 푲 is a 2-category. The objects of the functor 2-category 푲퐶 are, a priori, 2-functors 퐶 푲, 1-cells are strict 2-natural transformations, and 2-cells are modifications. However, since 퐶 is an ordinary category, a 2-functor 퐶 푲 is just a functor from 퐶 to the underlying 1-category 푲1 of 푲, and a 2-natural transformation between 2-functors 퐹, 퐺∶ 퐶 푲 is simply a natural transformation. Since modifications do not deal with the 2-naturality of a 2-natural transformation, they have the usual structure [2, §1.4]. 1.8. Proposition. Suppose that 퐶 is a small category and that 푲 is a 2-category with an enhanced factorization system (E, M). Given objects 퐹, 퐺 ∈ 푲퐶 and a 1-cell 훼∶ 퐹 ⟹ 퐺, then 훼 factors as ̄훼∘ ̃훼, where ̃훼푐 ∈ E for each 푐 ∈ 퐶, and ̄훼푐 ∈ M for each 푐 ∈ 퐶. Proof. Using the enhanced factorization system on 푲, for each 푐 ∈ 퐶, choose a factorization

훼 퐹(푐) 푐 퐺(푐)

̃훼푐 ̄훼푐 푅(푐) , where 푅(푐) is an object of 푲, ̃훼푐 ∈ E, and ̄훼푐 ∈ M. Then because (E, M) is an enhanced factorization system on 푲, for each morphism 푓∶ 푐 푐′ in 퐶, we have a unique factorization

̃훼 ̃훼 퐹(푐) 푐 푅(푐) 퐹(푐) 푐 푅(푐)

퐹(푓) ̄훼푐 퐹(푓) ̄훼푐

′ ′ 퐹(푐 ) 퐺(푐) = 퐹(푐 ) 푅(푓) 퐺(푐) (1.8.1)

̃훼푐′ 퐺(푓) ̃훼푐′ 퐺(푓) 푅(푐′) 퐺(푐′) 푅(푐′) 퐺(푐′) , ̄훼푐′ ̄훼푐′ so that each of the triangles in the right-hand diagram commute. We want to show that the assignment [푓∶ 푐 푐′] [푅(푓)∶ 푅(푐) 푅(푐′)] defines a functor, that the components ̃훼푐 and ̄훼푐 assemble into 2-natural transformations. To see this, first notice that by the uniqueness of the factorization1.8.1 ( ), it is clear that 푅(id푐) = id푅(푐). Then by the commutativity of the triangles in the right-hand diagram of (1.8.1), we get a commutative diagram

퐹(푓) 퐹(푔) 퐹(푐) 퐹(푐′) 퐹(푐″)

̃훼푐 ̃훼푐′ ̃훼푐″ 푅(푓) 푅(푔) 푅(푐) 푅(푐′) 푅(푐″) (1.8.2)

̄훼푐 ̄훼푐′ ̄훼푐″ 퐺(푐) 퐺(푐′) 퐺(푐″) , 퐺(푓) 퐺(푔) LIFTING ENHANCED FACTORIZATION SYSTEMS TO FUNCTOR 2-CATEGORIES (PRELIMINARY VERSION) 5 where each of the sub-squares commutes. Then the commutativity of1.8.2 ( ) along with the functoriality of 퐹 and 퐺 shows that the diagram

퐹(푔푓) 퐹(푐) 퐹(푐″)

̃훼푐 ̃훼푐″ 푅(푔)푅(푓) 푅(푐) 푅(푐″)

̄훼푐 ̄훼푐″ 퐺(푐) 퐺(푐″) 퐺(푔푓) commutes. Thus by the uniqueness of 푅(푔푓), this implies that 푅(푔푓) = 푅(푔)푅(푓). Then, by construction, the components ̃훼푐 and ̄훼푐 assemble into 2-natural transformations ̃훼 and ̄훼, □ respectively. Moreover, 훼 = ̄훼̃훼 because 훼푐 = ̄훼푐 ̃훼푐 for all 푐 ∈ 퐶. The point of this is that the enhanced factorization system on 푲 yields a way of factoring 1-cells in 푲퐶 pointwise. 1.9. Definition. Suppose that 푲 is a 2-category with an enhanced factorization system (E, M), and that 퐶 is a small category. Let E퐶 denote the collection of 1-cells in 푲퐶, all of whose components are in E, and M 퐶 denote the collection of 1-cells in 푲퐶, all of whose components are in M. Given a 1-cell 훼 ∈ 푲퐶, we call the factorization 훼 = ̄훼∘ ̃훼, where ̃훼∈ M 퐶 and ̄훼∈ E퐶 (described in Proposition 1.8) the pointwise factorization of 훼.

2. The Enhanced Factorization System on Cat In this section we analyze the enhanced factorization system (B, F) on Cat of bijective- on-objects and fully faithful functors. Most if not all of the results in this section are well- known, but we provide proofs of them as they are the motivation for the definitions that we give at the end of the section, and we do not have explicit references for them. Moreover, they also provide us with an example of our main result, which can be easily generalized to the 2-category of categories internal to a cartesian monoidal category, which is of interest to Guillou, May, Merling, and Osorno. The following few results will use more classical categorical notation. 2.1. Notation. In Lemmas 2.2 and 2.5 only, we change notation, using classical categorical notation as we find it is more familiar. In Lemmas 2.2 and 2.5, we write: (2.1.a) 퐶, 퐷, and 퐸 for categories, (2.1.b) 퐵 for a functor which is bijective-on-objects, (2.1.c) 퐹 for a fully faithful functor, (2.1.d) 퐺 and 퐻 for arbitrary functors, (2.1.e) and lowercase Greek letters such as 휂 and 휆 for natural transformations. 2.2. Lemma. Suppose that we have categories and functors as displayed below

퐺 퐶 퐷 퐹 퐸 , 퐻 where 퐹 is fully faithful. (2.2.a) If 휂, 휂′ ∶ 퐺 ⟹ 퐻 are natural transformations such that 퐹휂 = 퐹휂′, then 휂 = 휂′. 6 PETER J. HAINE

(2.2.b) A natural transformation 휂∶ 퐹퐺 ⟹ 퐹퐻 lifts to a unique natural transformation ̂휂∶ 퐺 ⟹ 퐻 with the property that 퐹 ̂휂= 휂. (2.2.c) With notation as in (2.2.b), if 휂 is a natural isomorphism, so is .̂휂 (2.2.d) If 퐺(푐) = 퐻(푐) for all 푐 ∈ 퐶 and 퐹퐺 = 퐹퐻, then 퐺 = 퐻. Thus, if 퐵∶ 퐶′ 퐶 is bijective-on-objects, 퐺퐵 = 퐵퐻 and 퐹퐺 = 퐹퐻, then 퐺 = 퐻. Proof. Notice that (2.2.a) follows immediately from the fact that 퐹 is fully faithful and for ′ all 푐 ∈ 퐶 the components 휂푐 and 휂푐 have the same domain and codomain. Second, suppose that we have a natural transformation 휂∶ 퐹퐺 ⟹ 퐹퐻. Since 퐹 is fully faithful, for each 푐 ∈ 퐶, there exists a unique morphism 푐̂휂 ∶ 퐺(푐) 퐻(푐) in 퐷 so that 퐹(푐 ̂휂 ) = 휂푐. To see that the morphisms 푐̂휂 assemble into a natural transformation ̂휂∶ 퐺 퐻, suppose that 푓∶ 푐 푐′ consider the square

퐺(푓) 퐺(푐) 퐺(푐′) (2.2.1) 푐̂휂 푐̂휂 ′ 퐻(푐) 퐻(푐′) . 퐻(푓) To see that (2.2.1) commutes, notice that

퐹(푐 ̂휂 ′ ∘ 퐺(푓)) = 휂푐′ ∘ 퐹퐺(푓) = 퐹퐻(푓) ∘ 휂푐 = 퐹(퐻(푓) ∘푐 ̂휂 ), by the functoriality of 퐹 and naturality of 휂. Since 퐹 is fully faithful and 푐̂휂 ′ ∘ 퐺(푓) and 퐻(푓) ∘푐 ̂휂 have the same source and target, this implies that 푐̂휂 ′ ∘ 퐺(푓) = 퐻(푓) ∘푐 ̂휂 , so (2.2.1) commutes. Hence the morphisms 푐̂휂 assemble into a natural transformation ̂휂∶ 퐺 ⟹ 퐻 with the property that 퐹 ̂휂= 휂. Notice that the claim (2.2.c) follows immediately from (2.2.b) and the fact that fully faithful functors reflect isomorphisms. To prove (2.2.d) all that needs to be verified is that 퐺(푓) = 퐻(푓) for all morphisms 푓 of 퐶. Since 퐺(푐) = 퐻(푐) for all 푐 ∈ 퐶, the morphisms 퐺(푓) and 퐻(푓) have the same source and target. Since 퐹퐺(푓) = 퐹퐻(푓), and 퐹 is fully faithful, this implies that 퐺(푓) = 퐻(푓). □

The following definition is a generalization of2.2 items( .b) and (2.2.c) of Lemma 2.2. 2.3. Definition. Suppose that 푲 is a 2-category and suppose that 휇 is a 1-cell in 푲. We say that post-comosition with 휇 creates invertible 2-cells if whenever we have 1-cells 훼, 훽 ∈ 푲 with the same source and target equipped with an invertible 2-cell Ψ∶ 휇훼 ⟹ 휇훽, there exists a unique invertible 2-cell Ψ∶̂ 훼 ⟹ 훽 so that 휇Ψ̂ = Ψ. Suppose that M is a class of 1-cells in a 2-category 푲. We say that post-composition with 1-cells in M creates invertible 2-cells, if post-composition with every 1-cell 휇 ∈ M creates invertible 2-cells. The following definition is a generalization of item(2.2.d) of Lemma 2.2. 2.4. Definition. We say that an enhanced factorization system (E, M) on a 2-category 푲 separates parallel pairs if whenever we are given a parallel pair 훼, 훼′ ∶ 퐹 퐺 so that there exists a 1-cell 휀 ∈ E such that 훼휀 = 훼′휀 and a 1-cell 휇 ∈ M such that 휇훼 = 휇훼′, we have 훼 = 훼′. Now for a result regarding functors which are bijective-on-objects. LIFTING ENHANCED FACTORIZATION SYSTEMS TO FUNCTOR 2-CATEGORIES (PRELIMINARY VERSION) 7

2.5. Lemma. Suppose that we have categories, functors, and natural transformations as displayed below 퐺

퐵 ⇒ ⇒ 퐶 퐷 휂 휆 퐸 , 퐻 where 퐵 is bijective-on-objects. If 휂퐵 = 휆퐵, then 휂 = 휆.

Proof. Since 휂퐵 = 휆퐵, for all 푐 ∈ 퐶 we have 휂퐵(푐) = 휆퐵(푐). Since 퐵 is bijective-on-objects, this □ says that 휂푑 = 휆푑 for each 푑 ∈ 퐷, so 휂 = 훽. Representable epimorphisms express the property conclusion of Lemma 2.5 about functors which are bijective-on-objects. Representable monomorphism express the dual property of fully faithful functors, which is (2.2.d) of Lemma 2.2. 2.6. Definition. We say that a 1-cell 휀∶ 퐹 퐺 in a 2-category 푲 is a representable epim- porphism if whenever we have objects, 1-cells, and 2-cells as displayed below

휀 ⇒ ⇒ 퐹 퐺 Ψ Φ 퐻 , if Ψ휀 = Φ휀, then Ψ = Φ. Representable monomorphisms are defined dually. Given an enhanced factorization system (E, M) on a 2-category 퐾, we say that E consists of representable epimorphisms if all 1-cells in E are representable epimorphisms, and M consists of representable monomorphisms if all of the 1-cells in M are representable monomorphisms. If E consists of representable epimorphisms and M consists of representable monomorphisms, we simply say that the enhanced factorization system (E, M) on 푲 is representable. Now we are ready to state the main results of this paper. 2.7. Proposition. Suppose that 푲 is a 2-category with an enhanced factorization system (E, M) and that 퐶 is a small category. If (E, M) separates parallel pairs and E consists of representable epimorphisms, then the pointwise factorization (E퐶, M 퐶) produced in Proposi- tion 1.8 defines an enhanced factorization system on 푲퐶. 2.8. Theorem (Main Result). Suppose that 퐶 is a small category, and that 푲 is a 2-category with an enhanced factorization system (E, M). If (E, M) is representable, separates parallel pairs, and post-composition with 1-cells in M creates invertible 2-cells, then the pointwise factorization (E퐶, M 퐶) on 푲퐶 defines an rigid enhanced factorization system on 푲퐶. Sections 3 and 4 are dedicated to proving Proposition 2.7 and Theorem 2.8. The point of the conditions stated in Proposition 2.7 and Theorem 2.8 is the following: When we were proving Proposition 1.8, we could use the properties of higher cells in 푲 to show that a collection of 1-cells defined pointwise were “coherent” in the sense that they assembled into a natural transformation. However, when given a collection of 2-cells in 푲, there is not a higher structure to ensure that the 2-cells are “coherent” in the sense that they assemble themselves into a modification. Hence, in order to have such a condition hold, we need some extra underlying structure in the enhanced factorization system. The following corollary of Theorem 2.8 is a direct application of Lack’s result Theorem 1.6. 2.8.1. Corollary. Suppose that 퐶 is a small category, 푲 is a 2-category with an enhanced factorization system (E, M), and that 푇 is a 2-monad on 푲퐶. Also suppose that (E, M) is representable, separates parallel pairs, and post-composition with 1-cells in M creates invertible ps 2-cells. Then every pseudo-푇-algebra is equivalent in Alg푇 to a strict 푇-algebra. 8 PETER J. HAINE

2.9. Example. For any small category 퐶, the pointwise factorization (B퐶, F 퐶) defines a rigid enhanced factorization system on Cat퐶. Moreover, if 푇 is a 2-monad on Cat퐶 which preserves the natural transformations whose components are bijective-on-objects functors, then every pseudo-푇-algebra is equivalent to a strict 푇-algebra. In particular, this implies Power’s result [4, Cor. 3.5].

3. Lifting Enhanced Factorization Systems This section is dedicated to proving Proposition 2.7. The first step in this is to show that, under these hypotheses, the pointwise factorization satisfies the property Definition 1.2.b defining an enhanced factorization system. 3.1. Lemma. Suppose that 푲 is a 2-category with an enhanced factorization system (E, M) and that 퐶 is a small category. If (E, M) separates parallel pairs and E consists of representable epimorphisms, then for every diagram

휀 퐹 퐹′

훼 훼′ ⟸Ψ 퐺 퐺′ 휇 in 푲퐶, where Ψ is an invertible 2-cell, 휀 ∈ E퐶, and 휇 ∈ M 퐶, the 2-category 푲퐶 has a unique 1-cell 훿∶ 퐹′ 퐺 in and a unique modification Ψ∶̃ 훼′ ⟹ 휇훿 so that 훿휀 = 훼 and Ψ휀̃ = Ψ. Moreover, Ψ̃ is necessarily invertible. Proof. Since 휀 ∈ E퐶, 휇 ∈ M 퐶, and (E, M) is an enhanced factorization system on 푲, for each 푐 ∈ 퐶, we have a factorization

휀 휀 퐹(푐) 푐 퐹′(푐) 퐹(푐) 푐 퐹′(푐)

훿푐 훼 훼′ = 훼 훼′ 푐 ⟸ 푐 푐 ⟸ 푐 Ψ푐 ̃ Ψ푐 퐺(푐) 퐺′(푐) 퐺(푐) 퐺′(푐) . 휇푐 휇푐

The goal is to show that the 1-cells 훿푐 assemble into a 2-natural transformation 훿 and that ̃ ̃ ′ the 2-cells Ψ푐 assemble into an invertible modification Ψ∶ 훼 ⟹ 휇훿. A priori, the 1-cells 훿푐 for 푐 ∈ 퐶 are not related to one-another, so to show that the 1-cells 훿푐 are natural, we exploit the fact that the enhanced factorization system on 푲 separates parallel pairs. Since the enhanced factorization system separates parallel pairs, it suffices to show that 푓∶ 푐 푐′ in 퐶 we have ′ ′ 훿푐′ 퐹 (푓) ∘ 휀푐 = 퐺(푓)훿푐 ∘ 휀푐 and 휇푐′ ∘ 훿푐′ 퐹 (푓) = 휇푐′ ∘ 퐺(푓)훿푐 . ′ First let us show that 훿푐′ 퐹 (푓) ∘ 휀푐 = 퐺(푓)훿푐 ∘ 휀푐. Since 훿푐휀푐 = 훼푐 for each 푐 ∈ 퐶, and 훼 and 휀 are 2-natural transformations, for all morphisms 푓∶ 푐 푐′ in 퐶 we have

퐹(푓) 퐹(푐) 퐹(푐′) 퐹(푐) 퐹(푓) 퐹(푐) 퐹(푐′) 휀푐 휀푐′ 휀푐 ′ = ′ ′ ′ = ′ 퐹 (푓) ′ ′ 훼푐 훼푐′ 퐹 (푐) 퐹 (푐 ) 퐹 (푐) 퐹 (푐 )

′ 훿 훿 ′ 훿 훿 ′ 퐺(푐) 퐺(푐 ) 푐 푐 푐 푐 퐺(푓) 퐺(푐) 퐺(푐′) 퐺(푐) 퐺(푐′) . 퐺(푓) 퐺(푓) LIFTING ENHANCED FACTORIZATION SYSTEMS TO FUNCTOR 2-CATEGORIES (PRELIMINARY VERSION) 9

′ ′ Hence for all 푓∶ 푐 푐 in 퐶 we have 훿푐′ 퐹 (푓) ∘ 휀푐 = 퐺(푓)훿푐 ∘ 휀푐. Now let us show that the diagram

퐹′(푓) 퐹′(푐) 퐹′(푐′) (3.1.1) 훿푐 훿푐′ 휇 ′ 퐺(푐) 퐺(푐′) 푐 퐺′(푐′) . 퐺(푓) commutes. To see this consider the invertible 2-cell ̃ ′ ′ ̃−1 ′ ′ Ψ푐′ 퐹 (푓)퐺 (푓)Ψ푐 ∶ 퐺 (푓)휇푐훿푐 ⟹ 휇푐′ 훿푐′ 퐹 (푓). ̃ ′ ′ ̃−1 A priori, we do not know that that Ψ푐′ 퐹 (푓)퐺 (푓)Ψ푐 is the identity. To see that this is true we use the fact that E consists of representable epimorphisms. Whiskering with 휀푐 we see that ′ ̃−1 ′ −1 −1 퐺 (푓)Ψ푐 휀푐 = 퐺 (푓)Ψ푐 = Ψ푐′ 퐹(푓) as Ψ−1 is a modification. Similarly, notice that by the naturality of 휀, we have ̃−1 ′ ̃−1 −1 Ψ푐′ 퐹 (푓)휀푐 = Ψ푐′ 휀푐′ 퐹(푓) = Ψ푐′ 퐹(푓).

Then since 휀푐 is a representable epimorphism, we see that ̃−1 ′ ′ ̃−1 Ψ푐′ 퐹 (푓) = 퐺 (푓)Ψ푐 , ̃ ′ ′ ̃−1 so Ψ푐′ 퐹 (푓)퐺 (푓)Ψ푐 is the identity. Thus the diagram

퐹′(푓) 퐹′(푐) 퐹′(푐′)

훿푐 훿푐′ 퐺(푐) 퐺(푐′)

휇푐 휇푐′ 퐺′(푐) 퐺′(푐′) 퐺′(푓) commutes. The naturality of 휇 implies that the diagram (3.1.1) commutes, as desired. Then ′ since 훿푐′ 퐹 (푓) ∘ 휀푐 = 퐺(푓)훿푐 ∘ 휀푐 and the enhanced factorization system on 푲 distinguishes ′ ′ between parallel pairs, we see that 퐺(푓)훿푐 = 훿푐′ 퐹 (푓), so the 1-cells 훿푐 ∶ 퐹 (푐) 퐺(푐) assemble into a unique 2-natural transformation 훿. ̃ Now let us show that the invertible 2-cells Ψ푐 in 푲 assemble into an invertible modifica- ′ ̃ ′ tion. Suppose that 푓∶ 푐 푐 is a morphism in 퐶. A priori it is not clear that Ψ푐′ 퐹 (푓) = ′ ̃ 퐺 (푓)Ψ푐; to see this, we exploit the fact that E consists of representable epimorphisms. Pre- ̃ ′ composing Ψ푐′ 퐹 (푓) with 휀푐 we see that ̃ ′ ̃ ′ Ψ푐퐹 (푓)휀푐 = Ψ푐′ 휀푐′ 퐹(푓) = Ψ푐′ 퐹(푓) = 퐺 (푓)Ψ푐 , ̃ by the naturality of 휀 and the fact that Ψ is a modification. Moreover, Ψ푐 has the property that ′ ̃ ′ ̃ ′ 퐺 (푓)Ψ푐휀푐 = 퐺 (푓)Ψ푐. Since 휀 is a representable epimorphism this implies that Ψ푐′ 퐹 (푓) = ′ ̃ ̃ ̃ ′ 퐺 (푓)Ψ푐. Hence the 2-cells Ψ푐 assemble into an invertible modification Ψ∶ 훼 ⟹ 휇훿, which ̃ □ is unique because the Ψ푐 are. 3.2. Lemma. Suppose that 푲 is a 2-category with an enhanced factorization system (E, M) and that 퐶 is a small category. If (E, M) separates parallel pairs and E consists of representable epimorphisms, then the pointwise factorization (E퐶, M 퐶) satisfies the last condition1.2 ( .c) of Definition 1.2 to define an enhanced factorization system on 푲퐶. 10 PETER J. HAINE

Proof. Suppose that we are in the situation indicated in the last condition (1.2.c) of Defini- tion 1.2 (since it is lengthy, we will not spell it out again here.) For each 푐 ∈ 퐶, write 훿1,푐 and 훿1,푐 for the components of 훿1 at 푐 and 훿2 at 푐, respectively. Since (E, M) is an enhanced factorization system on 푲, for each 푐 ∈ 퐶, there exists a unique invertible 2-cell Δ푐 ∶ 훿1,푐 ⟹ 훿2,푐 ′ in 푲 so that Δ푐휀푐 = Ψ푐 and 휇푐Δ푐 = 훷푐. To see that the 2-cells Δ푐 for 푐 ∈ 퐶 assemble into an ′ invertible modification Δ∶ 훿1 ⟹ 훿2, notice that for all morphisms 푓∶ 푐 푐 in 퐶, since Ψ is a modification we have

퐺(푓)Δ푐휀푐 = 퐺(푓)Ψ푐 = Ψ푐′ 퐹(푓) . Similarly, ′ Δ푐′ 퐹 (푓)휀푐 = Δ푐′ 휀푐′ 퐹(푓) = Ψ푐′ 퐹(푓) ′ hence 퐺(푓)Δ푐휀푐 = Δ푐′ 퐹 (푓)휀푐. Then since E consists of representable epimorphisms, in particular 휀푐 is a representable epimorphism, so we see that ′ 퐺(푓)Δ푐 = Δ푐′ 퐹 (푓) .

Hence the invertible 2-cells Δ푐 for 푐 ∈ 퐶 satisfy the necessary conditions to define a modification. □

Combining Proposition 1.8 and Lemmas 3.1 and 3.2 proves Proposition 2.7.

4. Conditions for Rigidity In this section we analyze conditions on the enhanced factorization system on 푲 which yield a rigid enhanced factorization system on 푲퐶, for any small category 퐶. 4.1. Lemma. Suppose that 푲 is a 2-category with an enhanced factorization system (E, M). If post-composition with 1-cells in M creates invertible 2-cells, then the enhanced factorization system on 푲 is rigid. Proof. Suppose that we have 1-cells 휇∶ 퐹 퐺 ∶ 훼, where 휇 ∈ M, and we have an invertible 2-cell Ψ∶ 휇훼 ⟹ id퐺. Then the 2-cell Ψ휇∶ 휇훼휇 ⟹ 휇 is invertible. For notational simplic- ity, write Φ ≔ Ψ휇. Then since post-composition with 휇 creates invertible 2-cells, there is a unique invertible 2-cell ̂ Φ∶ 훼휇 ⟹ id퐹 ̂ □ so that 휇Φ = Ψ휇. In particular, 훼휇 ≅ id퐹.

4.2. Lemma. Suppose that 푲 is a 2-category with an enhanced factorization system (E, M) and that 퐶 is a small category. If M consists of representable monomorphisms and post-composition with 1-cells in M creates invertible 2-cells, then post-composition with 1-cells in M 퐶 creates invertible 2-cells in 푲퐶. Proof. Suppose that we are given 1-cells 휇∶ 퐹 퐺 ∶ 훼, where 휇 ∈ M 퐶, and we have an invertible 2-cell Ψ∶ 휇훼 ⟹ id퐺. Then for each 푐 ∈ 퐶, there exists an invertible 2-cell Ψ푐 ∶ 휇푐훼푐 ⟹ id퐺(푐) in 푲. As shown in the proof of Lemma 4.1, there exists a unique invertible 2-cell ̂ Φ푐 ∶ 훼푐휇푐 ⟹ id퐹(푐) ̂ ̂ so that 휇푐Φ푐 = Ψ푐휇푐. We want to show that the Φ푐 assemble into the components of an invertible modification. LIFTING ENHANCED FACTORIZATION SYSTEMS TO FUNCTOR 2-CATEGORIES (PRELIMINARY VERSION) 11

To do this, suppose that 푓∶ 푐 푐′ is a morphism in 퐶. Writing Φ ≔ Ψ휇, since Φ is a modification, we have

휇 ′ 훼 ′ 휇 ′ 휇푐훼푐휇푐 푐 푐 푐

⟹ ⟹ 퐺(푓) ′ 퐹(푓) ′ ′ 퐹(푐) Φ푐 퐺(푐) 퐺(푐 ) = 퐹(푐) 퐹(푐 ) Φ푐′ 퐺(푐 ) .

휇푐 휇푐′ Hence we see that ̂ 휇푐′ Φ푐′ 퐹(푓) = Φ푐′ 퐹(푓) = 퐺(푓)Φ푐 . Hy the naturality of 휇 and the fact that 휇Φ̂ = Φ we see that ̂ ̂ 휇푐′ 퐹(푓)Φ푐 = 퐺(푓)휇푐Φ푐 = 퐺(푓)Φ푐 . Then since ̂ ̂ 휇푐′ Φ푐′ 퐹(푓) = 휇푐′ 퐹(푓)Φ푐 ̂ ̂ and 휇푐′ is a representable monomorphism, we have Φ푐′ 퐹(푓) = 퐹(푓)Φ푐, hence the compo- ̂ ̂ □ nents Φ푐 assemble into an invertible modification Φ. Hence Proposition 2.7 and Lemmas 4.1 and 4.2 together prove the main result Theo- rem 2.8.

References 1. G. M. Kelly, Enhanced factorization systems, 13 Jan. 1988. Lecture to the Australian Category Seminar. 2. G. M. Kelly and Ross Street, Review of the elements of 2-categories, Category seminar: Proceedings Sydney category seminar 1972/1973, 1974, pp. 75–103. 3. Stephen Lack, Codescent objects and coherence, Journal of Pure and Applied Algebra 175 (2002), 223–241. 4. A. J. Power, A general coherence result, Journal of Pure and Applied Algebra 57 (1989), 165–173.

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