ON 2 -CATEGORICAL ∞ -COSMOI

. Recently Riehl and Verity have introduced ∞ -cosmoi, which are certain simplicially enriched categories with additional structure. In this paper we investigate those ∞ -cosmoi which are in fact 2-categories; we shall refer to these as 2-cosmoi. We show that each 2-category with ﬂexible limits gives rise to a 2-cosmos whose distinguished class of isoﬁbrations consists of the normal isoﬁbrations. Many examples arise in this way, and we show that such 2-cosmoi are minimal as Cauchy-complete 2-cosmoi. Finally, we investigate accessible 2-cosmoi and develop a few aspects of their basic theory.


Introduction
In recent years Riehl and Verity have introduced the notion of an ∞cosmos as a framework in which to do (∞, 1)-category theory in a model independent way [15].An ∞-cosmos is a simplicially-enriched category K equipped with a class of morphisms, called isofibrations, satisfying certain properties, notably the existence of certain basic kinds of limits.Using these limits, one can develop much of the theory of ∞-categories inside an ∞-cosmos, such as the study of ∞-categories with limits and/or colimits of a given shape, or cartesian fibrations of ∞-categories, and these results then apply to various models of ∞-categories, such as quasicategories or complete Segal spaces.
The study of ∞-categories using ∞-cosmoi is very much 2-categorical in nature.In particular, each ∞-cosmos K has an associated homotopy 2-category hK and many of the definitions in an ∞-cosmos, such as equivalences and adjunctions, take place within this homotopy 2category [13].This means that ∞-categorical results such as "right adjoints preserve limits" can be given 2-categorical proofs.At the same time, much of the theory of ∞-cosmoi is guided by earlier results Date: December 21, 2023.2020 Mathematics Subject Classification.18N60, 18C35, 18D20, 18N40.The first-named author acknowledges the support of the Grant Agency of the Czech Republic under the grant 22-02964S.The second-named author acknowledges with gratitude the support of an Australian Research Council Discovery Project DP190102432.
in 2-category theory such as Street's study of fibrations in a 2-category [17] and the theory of flexible limits [1].
In the opposite direction, 2-categories can be viewed as simpliciallyenriched categories, by taking the nerves of their hom-categories, Therefore it makes sense to ask whether naturally occuring 2-categories, such as the 2-category of monoidal categories and strong monoidal functors, or that of coherent categories and coherent functors, underlie ∞-cosmoi.The main goal of this paper is to give a positive general answer to this question.In particular, in Theorem 4.4 we prove that each 2-category with flexible limits (in the sense of Bird et al [1]) forms part of an ∞-cosmos, when equipped with the class of normal isofibrations.As explained in [1], many 2-categories of categorical structures and their pseudomorphisms have flexible limits, so that this is a rather general result.
We use the term 2-cosmoi to refer to these 2-categorical examples of ∞-cosmoi -namely, 2-categories equipped with the structure of an ∞-cosmos on the associated simplicially-enriched category.For an ∞-cosmos K which arises in this way from a 2-category, the homotopy 2-category hK is just the original 2-category.Thus any ∞-cosmological notions which are defined in the homotopy 2-category, such as those discussed in [15, Chapter 2], agree with the usual 2-categorical notions in this case.
We say that a 2-cosmos is Cauchy complete if it is Cauchy complete as a 2-category, meaning that idempotents split, and if moreover the isofibrations are closed under retracts, as is the case in the key example of a flexibly complete 2-category equipped with the normal isofibrations.And in fact this notion of Cauchy complete 2-cosmos shows that the choice of the normal isofibrations is not just widely applicable but natural: we show in Theorem 4.5 that for any Cauchy complete 2-cosmos, the chosen class of isofibrations must contain the normal isofibrations.Thus the normal isofibrations are the minimal choice, at least in the Cauchy complete context.
As well as Cauchy completeness for 2-cosmoi, we also study accessibility: this is the 2-categorical version of the accessible ∞-cosmoi introduced in [4].
Let us now give an overview of the paper.In Section 2, we review the background and give an elementary characterization of 2-cosmoi, involving only 2-categorical rather than simplicially-enriched concepts.In Section 3, we investigate the different flavours of isofibration that naturally arise in 2-category theory, with a particular focus on the normal isofibrations.In Section 4 we describe the ∞-cosmos structure on a 2-category with flexible limits and provide a host of examples.Finally, in Section 5, we turn to accessible 2-cosmoi.

Basic facts about ∞-cosmoi and 2-cosmoi
Let Cat denote the cartesian closed category of small categories, and let SSet denote the cartesian closed category of simplicial sets.Central to our concerns is the adjunction whose right adjoint N takes the nerve of a small category and whose left adjoint Π takes the classifying category of a simplicial set.It is well known that the left adjoint Π, as well as the right adjoint N, preserves finite products.It follows that the adjunction lifts to a further adjunction between the categories of Cat-enriched categories (a.k.a.2-categories) and of simplicially-enriched categories.When K is an ∞-cosmos, Π * K is the homotopy 2-category hK mentioned in the introduction, and which we will have no need to discuss further.
For K a 2-category, N * K has the same objects as K and homs N * K(A, B) = N(K(A, B)) obtained by taking the nerve.We write nK as an abbreviation for N * K. Our goal is now to investigate what it means to equip nK with the structure of an ∞-cosmos.
First, let us recall from [15] that an ∞-cosmos is a simplicially enriched category K together with a class of morphisms called isofibrations, denoted A ։ B, closed under composition and containing the isomorphisms, such that ) is an isofibration of quasicategories for each isofibration p : B ։ C; (3) K has products, powers by simplicial sets, pullbacks along isofibrations, and limits of countable towers of isofibrations; (4) the class of isofibrations is stable under pullback; closed under products, limits of countable towers, and Leibniz powers by monomorphisms of simplicial sets; and contains all maps with terminal codomain.
Definition 2.1.A 2-cosmos is a 2-category K together with an ∞cosmos structure on nK.
As observed in the introduction, the homotopy 2-category of nK is then just K itself.
The following proposition describes 2-cosmoi in purely 2-categorical terms.In the statement, recall that a functor f : A → B is said to be an isofibration of categories if it has the isomorphism lifting property: namely, given an isomorphism α : Before proving this result, we will require a couple of preliminary results.The first is very well-known: Proof.By Yoneda, this is equivalent to the statement that we have a bijection natural in all variables.Now we have a natural bijection and similarly so that we must equally give a natural bijection but this follows immediately from the fact that Π preserves finite products.
Lemma 2.4.A functor f : A → B is injective on objects if and only if it is isomorphic in Cat 2 to Π(g) for some monomorphism g of simplicial sets.
Proof.Since the objects of ΠX are the 0-simplices of X, the functor Π sends monomorphisms to injective-on-objects functors, giving one direction.
For the converse, suppose that f : A → B is injective on objects.We may factorize Nf : NA → NB as where i is a monomorphism and p is a trivial fibration of simplicial sets.Furthermore, since Nf is injective on 0-simplices, we may do this in such a way that p is bijective on 0-simplices.Applying Π gives a factorization f ∼ = Π(p)Π(i), so it will suffice to show that Π(p) is invertible.
Since NB is (the nerve of) a category and p is a trivial fibration, X is a quasicategory and ΠX is the homotopy category of X, as described in [15,Lemma 1.1.12].Since p is bijective on 0-simplices and full on 1-simplices, Π(p) is bijective on objects and full; thus it will suffice to show that Π(p) is faithful.Consider then a parallel pair of morphisms in Π(X), which we may represent as homotopy classes [α] and [β], where α and β are parallel 1-simplices in X, and suppose that [pα] = [pβ].Then pα and pβ are homotopic in Y , and since p is a trivial fibration we obtain a lifted homotopy in X between α and β, and so [α] = [β], giving faithfulness of Π(p).

Proof of Proposition 2.2.
We must investigate what it means to equip nK with the structure of an ∞-cosmos.To begin with, this involves giving a class of morphisms in nK, closed under composition and containing the isomorphisms.But since K and nK have the same underlying category, this is just a class of morphisms in K satisfying the same two properties.
Condition (i) in the definition of ∞-cosmos asserts that nK is enriched in quasicategories.This is redundant since K is Cat-enriched and the nerve of a category is always a quasicategory.
Condition (ii) asserts that N(K(A, p)) is an isofibration of quasicategories for each isofibration p : B ։ C. By Observation 1.1.19(iv) of [15] a functor is an isofibration of categories just when its nerve is an isofibration of quasicategories.Thus Condition (ii) becomes Condition (a) above.
Next we show that Condition (iii) for nK is equivalent to Condition (b) for K. Condition (iii) says that three kinds of conical limits, plus powers, exist.With regards the conical limits, these are limits in the underlying category K 0 which have, moreover, the stronger universal property of being SSet-enriched limits in nK -that is, limits in K 0 preserved by each nK(A, −) : K 0 → SSet.Now since nK(A, −) is the composite N • K(A, −) : K 0 → Cat → SSet, and since N preserves and reflects limits, nK(A, −) preserves precisely those limits preserved by K(A, −) : K 0 → Cat -that is, limits in K 0 that are Cat-enriched limits.
Turning to powers, let X be a simplicial set and A ∈ K 0 .We write A X nK for the power of A by X in nK and use the corresponding notation for powers in K.We claim that we have an isomorphism A X nK ∼ = A ΠX K , either side existing if the other does.Certainly if A X nK exists, then we have the defining isomorphism Composing this with the natural isomorphism [X, N(K(B, A)] ∼ = N[ΠX, K(B, A)] of Lemma 2.3 and using fully faithfulness of the Yoneda embedding we obtain a natural isomorphism K(B, A X nK ) ∼ = [ΠX, K(B, A)], which induces -now by definition of powers in K -an isomorphism ϕ A,X : The other direction is identical.Hence nK admits powers just when K admits powers by small categories of the form ΠX. But since each small category is of this form, we conclude that nK admits powers just when K does.
Finally, we show that Condition (iv) for nK becomes Condition (c) for K.This is straightforward, bar the Leibniz conditions.For these, we note that the isomorphisms ϕ A,X : A X nK ∼ = A ΠX K constructed above are natural in A and X.Given a monomorphism i : X → Y of simplicial sets and an isofibration f : A ։ B, it therefore follows that we have a commutative square in which the vertical maps are isomorphisms and the horizontal maps are the induced ones to the pullback-product, with the upper horizontal living in nK and the lower one in K.In particular, the upper horizontal is an isofibration just when the lower one is one.Since, by Lemma 2.4, the functors of the form Π(i) for i mono are, up to isomorphism, the injective on objects functors, we conclude that isofibrations are closed under Leibniz powers by monomorphisms in nK just when they are closed under Leibniz powers by injective on objects functors in K.

Flavours of isofibration in a 2-category
In this section, we recall the various kinds of isofibrations in a 2category and the relationships between them.
We suppose throughout that K is a 2-category with pseudolimits of arrows, as in Definition/Proposition 3.2 below, and use the notation introduced there.(1) A morphism f : A → B in a 2-category K is an equivalence when there exist isomorphisms η : 1 ∼ = gf and ε : f g ∼ = 1; it is then possible to choose η and ε so that they satisfy the triangle equations.(2) If f is an equivalence and has a section, then it is possible to choose the remaining structure so that ε is an identity; we then say that f is a retract equivalence.(3) If f is an equivalence and has a retraction, then it is possible to choose the remaining structure so that η is an identity; we then say that f is an injective equivalence.
Next we recall a few basic facts about pseudolimits of arrows.Definition/Proposition 3.2.
(1) The pseudolimit L f of an arrow f : A → B is the universal diagram as on the left below.Thus for every diagram as in the centre, there is a unique arrow c : There is also a 2-dimensional aspect to the universal property.(2) In particular, there is a unique arrow (3) Thus in fact u f is a retract equivalence and d f an injective equivalence; conversely, if u f is known to be an equivalence then the 2-dimensional aspect of the universal property is automatic.(4) The pseudolimit L 1 A of an identity arrow 1 A : A → A is equally the power A I of A by the generic category I containing an isomorphism.(5) The pseudolimit L f can equally be constructed as a pullback as in the square on the right below.There is then a unique induced w f : A I → L f making the two triangles commute.
Normal isofibrations.The standard kind of isofibration in a 2category K consists of a morphism f : A → B for which the induced ) is an isofibration of categories for all C ∈ K.In elementary terms, this means that given g : X → A, h : X → B and an isomorphism α : f •g ∼ = h there exists an isomorphism α ′ : g ∼ = h ′ such that f • α ′ = α.We shall call these representable isofibrations, to distinguish them from the isofibrations that are specified as part of the structure of ∞-cosmos.
A discrete isofibration is a representable isofibration such that the lifting (α ′ , h ′ ) above is unique.
A cleavage for a representable isofibration consists of a choice, for each (g, α, h), of a pair (α ′ , h ′ ) as above such that these choices are natural in X.The cleavage is said to be normal if whenever α is an identity 2-cell so is α ′ .A normal isofibration [7] is, by definition, a representable isofibration supporting a normal cleavage.
Each discrete isofibration is a normal isofibration.Moreover in Cat each representable isofibration is normal and the corresponding statement is true in many 2-categories of categories with structure: see also Section 3.3 below.Proposition 3.3.Let f : A → B be an arrow in K, and form the pseudolimit L f of f .Then (1) be chosen with κd f an identity.
Proof.The existence of a lifting κ : Similarly, the existence of κ : x ∼ = u f satisfying the extra condition in (ii) is part of the definition of normal isofibration.For the converse, if β : b ∼ = f a is an identity, then the induced c factorizes through d f , and so κc is also an identity.
In Proposition 3.4 below we describe a weak factorization system on a suitable 2-category K whose right class consists of the normal isofibrations; this implies many of the stability properties of normal isofibrations that we will need.It is the (cofibration, trivial fibration) weak factorization system for the natural model structure [10, Section 4] on K op .The connection with normal isofibrations, as defined above, was made in [7,Remark 3.4.8],and it also featured in [6].
Proposition 3.4.Let K be a 2-category with pseudolimits of arrows.Then there is a weak factorization system on K 0 whose left class consists of the injective equivalences and whose right class consists of the normal isofibrations.
Proof.The factorization of an arrow f will be given by f = v f d f as in Definition/Proposition 3.1.For the fact that d f is an injective equivalence, and that this gives a weak factorization system consisting of the injective equivalences and some class P of arrows, see [10,Section 4].In that section, a model structure is described for any 2-category with finite limits and finite colimits.If one applies this to K op and considers the (cofibration, trivial fibration) weak factorization system, one obtains a weak factorization system (injective equivalence, P) on K.Note that the finite limits and finite colimits are assumed only because they are part of the definition of a model structure; only pseudolimits of arrows are required for this weak factorization system.The dual case of [10, Section 4], as used here, was explicity considered in [6].
We now turn to the fact that the right class P consists of the normal isofibrations.This was observed without proof in [7,Remark 3.4.8],so we now give a proof.First we show that the normal isofibrations have the right lifting property with respect to the injective equivalences; in other words, every normal isofibration is in P. To see this, consider a commutative square in which i is an injective equivalence and p is a normal isofibration.
The invertible 2-cell gη : g ∼ = gir = pf r lifts along p to give a 1cell h : B → C and invertible 2-cell θ : h ∼ = f r satisfying ph = g and pθ = gη.To prove that h is a diagonal filler, we must show that hi = f .To this end, observe that by naturality of the cleavage, hi is the domain of the lifting of gηi : gi ∼ = giri = pf ri = pf but now gηi = id and hence, by normality, hi = f .
For the converse, suppose that f : A → B is in P, and form the factorization f = v f d f .Then there exists a filler x as in and as observed in Definition/Proposition 3.2 as well as in [10, Section 4], there is an isomorphism

and, by one of the triangle equations xδ
The following lemma is useful in recognising 2-cosmoi.Lemma 3.5.Let K be a 2-category with products and powers.Let P be a class of morphisms in K such that: • P is closed under products and composition • pullbacks of maps in P along arbitrary maps exist and are in P.
If P contains the discrete isofibrations, then it is closed under Leibniz powers by the injective on objects functors.
Proof.Let J be the class of all functors j with the property that P is closed under Leibniz powers by j.We need to show that J contains the injective on objects functors.A Leibniz power by a functor 0 → W with initial domain is just an ordinary power by W ; thus J contains such functors for W a discrete category.An arbitrary injective on objects functor j : X → Y can be factorized as j = ki as in 0 where k is bijective on objects, W is the discrete category on the set of all objects of Y not in the image of j, and the square is a pushout.Since the class J contains 0 → W , is stable under pushouts, and closed under composition, it will suffice to show that J contains the bijective on objects functors.Suppose then that j : X → Y is bijective on objects, and p : A → B is in P. In the diagram where P is a pullback, we are to show that j ⋔ p lies in P.But B j is a discrete isofibration for any bijective on objects j, thus so is its pullback g; likewise A j is a discrete isofibration.By the cancellation property for discrete isofibrations, it follows that j ⋔ p is a discrete isofibration and so in P.
Example 3.6.The representable fibrations, normal isofibrations, and discrete isofibrations all satisfy the stability properties of Lemma 3.5.Therefore each of these classes satisfies the Leibniz condition with respect to the injective on objects functors.Remark 3.7.There are other interesting classes of representable isofibrations, such as the Grothendieck fibrations.The lemma does not typically apply, since not every discrete isofibration is a Grothendieck fibration, and indeed the Leibniz condition does not typically hold.In the case K = Cat, for instance, consider the identity on objects inclusion j : 2 → 2 from the discrete category with two objects to the generic arrow {0 → 1}, and the Grothendieck fibration p : 2 → 1.The Leibniz power of p by j is just 2 j : 2 2 → 2 2 , which can be seen as the inclusion of the full subcategory of 2 2 = 2 × 2 consisting of the three objects (0, 0), (0, 1), and (1, 1).This is not a Grothendieck fibration: there is no lifting of the map (1, 0) → (1, 1).

When all isofibrations are normal.
In this section we investigate the difference between representable isofibrations and normal ones, as well as the case when this difference disappears: Definition 3.8.A 2-category K has the normal isofibration property, or NIP, if every representable isofibration in K is a normal isofibration.
We have already observed that Cat has this property.Recall the construction of w f : A I → L f as in Definition/Proposition 3.2.Proposition 3.9.For f : A → B (1) f is a representable isofibration if and only if w f is one (2) f is a normal isofibration if and only if w f is one.In either case, w f is then a retract equivalence; if moreover w f is a normal isofibration, we call it a normal retract equivalence.
Proof.Since w f is always an equivalence, it will be a retract equivalence if and only if it is a representable isofibration if and only if it has a section.
To give a lifting κ : x ∼ = u f of λ f is equivalently to give a section of w f .This proves (1).
A lifting κ : x ∼ = u f will have κd f an identity if and only if and only if the corresponding section of w f is a filler for Since d f is an injective equivalence, there will be such a filler if w f is a normal isofibration.This gives one direction of (2).
The converse follows by Lemma 3.5, since w f is the Leibniz power of f by 1 → I.
Corollary 3.10.The normal isofibration property holds just when the retract equivalences have the right lifting property with respect to the injective equivalences.
Proof.Proposition 3.9 implies that every representable isofibration is normal just when this is the case for equivalences: in other words, when every retract equivalence is a normal isofibration; or, by Proposition 3.4, when the retract equivalences have the right lifting property with respect to the injective equivalences.
We now specialize further to the case where K is the 2-category Cat(E) of internal categories in a complete category E. Then K is itself complete as a 2-category.Given an internal category A we write as usual A 0 for the object-of-objects, with a similar notation for internal functors.We often identify an object of E with the corresponding discrete internal category.
Proposition 3.11.The normal isofibration property holds in Cat(E) if and only if the split epimorphisms in E have the right lifting property with respect to the split monomorphisms.

Proof. Consider a diagram as on the left
in which i is an injective equivalence and p a retract equivalence.Then we obtain a diagram in E as on the right, with i 0 a split monomorphism and p 0 a split epimorphism.If this has a filler h 0 , then since i and p are equivalences we may extend h 0 to a functor h giving a filler in the diagram on the left.Conversely, suppose that we are given the solid part of the diagram on the right, with p 0 a split epimorphism and with i 0 a split monomorphism.If we make each object into a chaotic category with the given object-of-objects, we obtain a diagram as on the left, with i an injective equivalence and with p a retract equivalence.If this has a filler h : B → C then its action h 0 : B 0 → C 0 on objects gives a filler in the diagram on the right.Corollary 3.12.If now E is extensive, the normal isofibration property in Cat(E) is equivalent to the condition that every split monomorphism in E is a coproduct injection.
If E is a topos, the normal isofibration property holds if and only if the topos E is Boolean.
Proof.The first part follows from [16, Theorem 2.7], which says that in an extensive category, the coproduct injections and split epimorphisms form a weak factorization system.
A topos is Boolean when every monomorphism is a coproduct injection.But in a topos, every monomorphism is a pullback of a split monomorphism, and the coproduct injections are stable under pullback, so it suffices for every split monomorphism to be a coproduct injection.
This provides a large class of examples which do not satisfy the normal isofibration property.In particular, K = Cat(Set 2 ) = Cat 2 does not, since the topos Set 2 is not Boolean.Some examples of 2-categories, not of the type Cat(E), where the normal isofibration property does hold can be found in Examples 3.16 and 3.17, and in Section 5.
Proof.Any 2-functor at all preserves equivalences, injective equivalences, and retract equivalences.Preservation of representable isofibrations and normal isofibrations is an immediate consequence of Proposition 3.3.Finally the normal retract equivalences are just the retract equivalences which are also normal isofibrations.
Proposition 3.14.Let U : L → K be as in Proposition 3.13, and suppose that each U : L(A, B) → K(UA, UB) is a discrete isofibration.Then U also reflects representable isofibrations and normal isofibrations.
Proof.Let f : A → B in L have pseudolimit L f , with the usual notation.If Uf is a representable isofibration, then there is a κ : x ∼ = Uu f with Uf.κ = Uλ f .We may lift this to a unique κ : and so by the discrete isofibration property, f κ = λ f .Thus f is a representable isofibration.
If in fact Uf is a normal isofibration, then we may choose κ so that κ.Ud f = id U f .Then and so κ.d f = id f , and f is also a normal isofibration.This proves the first statement.
Since the normal retract equivalences are the retract equivalences which are also normal isofibrations, and U reflects normal isofibrations, (2) implies (3).
Suppose that U reflects normal retract equivalences, and that f : A → B is such that Uf is an equivalence.Then Uv f is a retract equivalence, thus v f is a normal retract equivalence, thus f is an equivalence.This proves that (3) implies (1).Now suppose that U reflects equivalences, and that f : A → B is such that Uf is an injective equivalence, say with retraction r : UB → UA and isomorphism Uf.r ∼ = 1.Then f is an (adjoint) equivalence, say with inverse equivalence g : B → A, unit η : 1 ∼ = gf , and counit ε : f g ∼ = 1.Then r.Uε : Ug = r.Uf.Ug ∼ = r can be lifted to γ : g ∼ = r, and now the isomorphisms γ.f : g.f ∼ = r.f and η −1 : g.f ∼ = 1 satisfy and so γ.f = η −1 and in particular r.f = 1, and f is an injective equivalence.This proves that (1) implies (4).A similar argument shows that (1) implies (2).Example 3.17.Let V be a monoidal category with finite limits, and V-Cat the 2-category of V-categories, V-functors, and V-natural transformations.Then the 2-functor U : V-Cat → Cat sending a V-category to its underlying ordinary category satisfies the assumptions of Corollary 3.15; thus V-Cat satisfies the normal isofibration property.
If in Corollary 3.15, we add the assumption that U : L → K is a fibration for the model structure on 2-Cat (of [8], but as corrected in [9]), then the condition that L has and U preserves pseudolimits of arrows is in fact redundant.This is the content of the following result.Proposition 3.18.Let K be a 2-category with pseudolimits of arrows, and U : L → K a 2-functor which (1) has the equivalence lifting property: if A ∈ L and u : L → UA is an equivalence, then there is an equivalence u : L → A with Uu = u (2) acts as a discrete isofibration L(A, B) → L(UA, UB) on homcategories.Then L has pseudolimits of arrows and U : L → K preserves them.It then follows that if K satisfies the normal isofibration property then so too does L.
Proof.Suppose that f : A → B is a morphism in L, and form the pseudolimit L in K of Uf , as displayed below on the left.Then u : L → UA is an equivalence, so we may lift it to an equivalence u : L → A. Now λ : v → Uf.u = U(f.u) is invertible, so there is a unique lifting λ : v ∼ = f.u, as below in the centre.
Now u and d are part of an adjoint equivalence with counit ud = 1 and unit δ : 1 ∼ = du.As in [9, Proposition 6], we may lift this to an adjoint equivalence involving u and some d and unit δ : 1 ∼ = du, and by the uniqueness aspect of the discrete isofibration property the counit is once again an identity.Likewise, λ d : vd ∼ = f is a lifting of the identity λd : vd ∼ = f and so is itself an identity; in particular vd = f .We shall show that u, v, and λ exhibit L as the pseudolimit of f .By Definition/Proposition 3.1(3), it will suffice to check the 1-dimensional aspect of the universal property.
Suppose then that a : Thus we have the existence of a factorization.For the uniqueness aspect, suppose that c ′ : C → L satisfies uc ′ = uc, vc ′ = vc, and λc ′ = λc.Since u is an equivalence, there is a unique isomorphism θ : c ′ ∼ = c with uθ equal to the identity.Then Uθ : c → c satisfies u.Uθ = 1 and so, since u is an equivalence, Uθ = 1.Thus θ : c ′ ∼ = c is a lifting of an identity, hence an identity, and c ′ = c.
This proves that L has the pseudolimits and U preserves them.The final sentence then follows by Corollary 3.15.

The 2-cosmos structure on a 2-category with flexible limits
In this section we give one of our main results, Theorem 4.4, which shows that any 2-category with flexible limits forms a 2-cosmos in which the chosen isofibrations are the normal isofibrations.Moreover such 2cosmoi are Cauchy-complete, in a sense to be defined.
Let us begin by reminding the readers about flexible limits in 2categories.These were introduced in [1] and admit many equivalent formulations.Perhaps the most elementary description is that they are those limits constructible from products, inserters, equifiers, and splittings of idempotents [1], whilst a satisfying abstract description is that they are the cofibrantly-weighted 2-categorical limits [10].
The characterization below, proven by the first-named author as Proposition A.1 of [3], shows that that flexible limits and normal isofibrations are closely related.We include a proof since the construction of pullbacks of normal isofibrations from flexible limits is important to our concerns.Proposition 4.1.For a 2-category K, the following are equivalent (1) K has flexible limits (2) K has products, powers, splittings of idempotents, and pullbacks of normal isofibrations (3) K has products, powers, splittings of idempotents, and pullbacks of discrete isofibrations.
Proof.Since discrete isofibrations are normal isofibrations, certainly (2) implies (3).In order to prove that (3) implies (1), we use the fact that flexible limits are generated by products, inserters, equifiers, and splittings of idempotents.Thus it will suffice to show that (3) implies the existence of inserters and equifiers.The inserter of f, g : A → B, and the equifier of β, β ′ : h → k : A → B can be constructed as the pullbacks below.
Here 2 → 2 is as in Remark 3.7, whilst 2 2 → 2 is the map from the generic parallel pair of arrows {0 ⇒ 1} to 2 which identifies the parallel arrows.Since both functors are bijective on objects, it follows that the vertical restriction maps on the right of both squares are discrete isofibrations.
It remains to show that (1) implies (2), and since products, powers, and splittings of idempotents are all flexible, we need only prove the existence of pullbacks along normal isofibrations.This is the key contribution of [3,Proposition A.1].
Let f : A → C be a normal isofibration, and g : B → C arbitrary, and form the isocomma object and the lifting ξ : x ∼ = p with f ξ = ϕ (and so also f x = gq).
By the universal property of the isocomma object, there is a unique e : P → P with pe = x, qe = q, and ϕe = id f x .Then ϕee is also an identity, qee = qe, and pee = xe; while ξe : xe ∼ = pe is the lifting of the identity ϕe : gq = gq = f x which must be the identity by normality.It follows by the universal property of the isocomma object that ee = e, so that e is idempotent.By the 2-dimensional aspect of the universal property, there is a unique isomorphism ε : e ∼ = 1 with pε = ξ and qε the identity.
We claim that a splitting of e gives the desired pullback of f and g.To show this, we should show that ey = y if and only if ϕy is an identity.On the one hand, if ey = y then ϕy = ϕey which is an identity since ϕe is one.On the other hand, if ϕy is an identity then so is its lifting ξy : xy = pey ∼ = py, and so in particular pey = py.We also know that qey = qy, since qe = q; while ϕey is the identity on f pey = f py, as is ϕy.Thus ey = y by the universal property of the isocomma object.Note also that then εy is the identity.
If e splits as e = ir with ri = 1, then a straightforward calculation shows that pi and qi give the desired pullback.The two-dimensional aspect of the universal property follows easily from the fact that the splitting of e will be equivalent to P .
We now turn to the (new) concept of Cauchy-completeness in the context of ∞-cosmoi and 2-cosmoi.Definition 4.2.An ∞-cosmos K is Cauchy-complete if K has split idempotents and any retract of an isofibration is an isofibration.A 2cosmos K is Cauchy-complete if the ∞-cosmos nK is Cauchy complete.
Since the Cauchy-completeness condition for an ∞-cosmos does not involve the simplicial enrichment, clearly we have

Proposition 4.3. A 2-cosmos K is Cauchy complete just when it has split idempotents and a retract of an isofibration is an isofibration.
With this, we are in a position to prove: Theorem 4.4.Let K be a 2-category.Then K has flexible limits if and only if the normal isofibrations equip K with the structure of a Cauchy complete 2-cosmos.
Proof.One direction follows immediately from Proposition 4.1.
For the converse, suppose that K admits flexible limits.We shall verify the conditions of Proposition 2.2.Since each normal isofibration is a representable isofibration, Condition (a) holds.
Since K has flexible limits, it has products, powers, splittings of idempotents and pullbacks of normal isofibrations by Proposition 4.1 again.As far as the existence of limits in Condition (b) goes, it remains to construct limits of towers of normal isofibrations.
Consider a tower of normal isofibrations in K, with objects A n for n < ω, and morphisms f m n : A m → A n for n ≤ m < ω.Form the pseudolimit P with projections p n : P → A n and structure isomorphisms π m n : Step 1. Define inductively a cone over the tower with vertex P and morphisms q n : P → A n , and with isomorphisms α n : q n ∼ = p n , compatible in the sense that is equal to f m n α m for all n < m.Start by setting q 0 = p 0 and α 0 = 1.For a successor ordinal n + 1, we have p n+1 : P → A n+1 , and isomorphisms and so may use the normal isofibration structure to lift to an isomorphism α n+1 : q n+1 → p n+1 with f n+1 n q n+1 = q n and f n+1 n Step 2. Next we show that a morphism x : X → P has α n x : q n x → p n x an identity for all n < ω if and only if π m n x : p n x → f m n p m x is an identity for all n < m < ω.

Since the diagram
commutes, if each α n x is an identity then so is each π m n x.Suppose conversely that each π m n x is an identity.We show by induction that each α n x is an identity.
First of all α 0 is itself an identity hence so is α 0 x.For a successor ordinal n + 1, we obtain α n+1 x : q n+1 x → p n+1 x as a lift of x so it will suffice, using normality, to show that this is an identity.But α n x is an identity by inductive hypothesis, and π n+1 n x by assumption.
Step 3. Next we define an idempotent e : P → P ; later, we shall see that splitting e gives the desired limit.By the universal property of the pseudolimit, there is a unique induced e : P → P with p n e = q n for all n < ω, and equal to the identity for all n < m < ω.By Step 2 it follows that α n e : q n e → p n e is an identity for all n < ω; and now p n ee = q n e = p n e for all n, while each π m n ee and π m n e are both the (same) identity, whence e 2 = e by the universal property of the pseudolimit P .
Step 4: Next we obtain a (strict) cone over the original tower.
Let the idempotent e : P → P split as and observe that for all n < m, and so, cancelling the split epimorphism r, we deduce that f m n p m i = p n i.Thus the p n i : L → A n define a strict cone.
Step 5: This cone is a limit.If the maps x n : X → A n define another cone, then there is a unique x : X → P with p n x = x n for all n and π m n x an identity for all n < m.By Step 2, it follows that α n x : q n x → p n x is an identity for all n < ω, and so in particular that q n x = p n x.Now p n ex = q n x = p n x for all n, while π m n ex and π m n x are both the (same) identity, and so ex = x by the universal property of the pseudolimit P .It follows that p n irx = p n ex = p n x = x n for all n < ω, and so rx is a factorization.
For the uniqueness, suppose that y is any map for which p n iy = x n for all n < ω.Since π m n i = π m n iri = π m n ei is an identity for all n < m < ω, so too is π m n iy, and thus iy = x by the universal property of the pseudolimit.Then y = riy = rx giving the uniqueness.
We leave the 2-dimensional aspect of the universal property as an exercise for the reader.
Finally, we turn to the properties in Condition (c).Clearly, each map p : A → 1 is a normal isofibration: indeed, since the only isomorphism pu ∼ = v ∈ K(A, 1) is the identity, defining its lifting to be 1 : u ∼ = u gives a normal cleavage.
Stability of normal isofibrations under products, pullbacks and limits of towers follows from the fact, established in Proposition 3.4, that they are the right class of a weak factorisation system on K. Since the normal isofibrations are closed under composition and contain the discrete isofibrations, they satisfy the Leibniz property by Lemma 3.5.
The following result shows that normal isofibrations are the minimal choice of isofibration in the Cauchy-complete setting.Theorem 4.5.In a 2-cosmos, each normal isofibration is a retract of a chosen isofibration.In particular, in a Cauchy-complete 2-cosmos, each normal isofibration is chosen.
Proof.Let f : A → B be arbitrary.We may form the pseudolimit of f as the pullback of the chosen isofibration B I → B × B, and so u v is a chosen isofibration; but so too are the projections of A × B, hence so too are u and v.
If now f is a normal isofibration, then the filler w in Earlier we studied 2-categories in which every representable isofibration is normal, and we shall return to this topic in Section 5.In relation to such 2-categories we have the following observation: Corollary 4.6.If K is a 2-category with flexible limits in which every representable isofibration is normal, then there is a unique Cauchycomplete 2-cosmos structure on K.
Proof.By definition every chosen isofibration is a representable isofibation.By the theorem, every normal isofibration is a chosen isofibration.Thus we have implications normal =⇒ chosen =⇒ representable between the classes of isofibrations.If all representable isofibrations are normal, the result follows.
Examples 4.7.As explained in [1], many important 2-categories of categorical structures and their pseudomorphisms admit flexible limits.For instance, the 2-categories Lex, Reg, Ex, Coh, and Pretop, of small finitely complete categories, regular categories, Barr-exact categories, coherent categories, and pretopoi all admit flexible limits, as does the 2-category MonCat of monoidal categories and strong monoidal functors.
Therefore, by Theorem 4.4, each of these gives an example of a 2cosmos with isofibrations the normal isofibrations.In all of the above examples, the forgetful 2-functor to Cat satisfies the conditions of Proposition 3.18, and so the normal isofibrations and representable isofibrations coincide, and are just the morphisms whose underlying functor is an isofibration of categories.
In Proposition 2.6 of [14], Riehl and Wattal show that the 2-category Icon of bicategories, normal pseudofunctors and icons is a 2-cosmos with chosen isofibrations the representable ones, equally the normal ones.This is another instance of the canonical 2-cosmos structure on a 2-category with flexible limits.As pointed out in the proof of Proposition 2.6 of [14], the NIP holds in Icon.
Similarly the 2-category of bicategories, (not necessarily normal) pseudofunctors, and icons is a 2-cosmos.Indeed, following Section 6.3 of [4] it has flexible limits 1 , and so by Theorem 4.4 is a 2-cosmos with isofibrations the normal ones -once again, the NIP is satisfied.
Example 4.8.We end this section with a non-example.Consider the 2-category sMonCat of strict monoidal categories, strong monoidal functors, and monoidal natural transformations.By Proposition 3.14, both the isofibrations and the normal isofibrations are those strong monoidal functors whose underlying functors are (necessarily normal) isofibrations of categories.We shall show that sMonCat does not have pullbacks of these (normal) isofibrations, and so that choosing them as isofibrations does not make sMonCat into a 2-cosmos.
By [3, Section 6.2], idempotents do not split in sMonCat.To see this, let C be a small monoidal category.Its strictification A ∈ sMonCat has as objects the words in C, while its morphisms are such that the mapping q : A → C which evaluates words using the left bracketing is fully faithful.Then q : A → C is a retract equivalence in MonCat, with section r : C → A sending each object to the corresponding word of length 1, and so can be made into an adjoint equivalence with unit α : 1 ∼ = rq.In particular, e = rq is then an idempotent in sMonCat and the triangle equations yield eα = id = αe.If e splits in sMonCat, then it follows that C is isomorphic to the splitting, a strict monoidal category.But not every monoidal C is isomorphic to a strict monoidal category: as explained in loc.cit., an example is given by the skeletal category of at-most-countable sets.
So suppose that e : A → A as above does not split, and let M be the chaotic category with two objects 0 and 1, made into a strict monoidal category under addition mod 2. There is a unique functor χ : A → M sending a ∈ A to 0 if ea = a and to 1 otherwise.It admits a unique structure of strong monoidal functor and is, moreover, an isofibration: on the one hand, if χa = 1 then a ∼ = ea and χea = 0; on the other, if χa = 0 then a ∼ = a.i,where i is the monoidal unit, and χ(a.i) = 1 since a.i is a not a singleton.
Suppose that there were a pullback in sMonCat as below left 1 Pie limits were discussed earlier in [11].
where 0 : 1 → M is the (strict) monoidal functor picking out the unit object 0. Since e is idempotent, the square above right commutes, so that we obtain a unique morphism k : A → B to the pullback such that f k = e.We claim that kf = 1 B .Since 0 : 1 → M has terminal domain, it is monic, whence so is its pullback f .Therefore it suffices to show that f kf = f or, since f k = e, that ef = f .Indeed α f b : f b ∼ = ef b has identity component for each b ∈ B since f b, being sent by χ to 0, is a word of length 1, and so in the image of e. Hence f = ef .Therefore B splits e, a contradiction.

Accessible 2-cosmoi
In this section, we investigate the notion of accessible 2-cosmoi, the 2-categorical version of the accessible ∞-cosmoi introduced in [4].In Theorem 5.5, we show that the condition concerning bicolimits in an accessible 2-cosmos can be replaced by a simpler condition concerning colimits of pointwise equivalences being equivalences.Combining this with the existing literature, we are able to give many examples of accessible 2-cosmoi including all of the examples of Cauchy complete ∞-cosmoi in Example 4.7.
To get started, let us recall the definition of accessible ∞-cosmos, before breaking down what it means.Definition 5.1.An ∞-cosmos K is said to be accessible if (1) K is accessible as a simplicially-enriched category; (2) the chosen isofibrations are accessible; (3) there exists a regular cardinal λ such that λ-filtered colimits exist in K and are homotopy colimits.
First, by Proposition 2.4 of [4], K is accessible as a simpliciallyenriched category just when K 0 is accessible and, for each simplicial set X, the powering functor (X ⋔ −) 0 : K 0 → K 0 is accessible.
Since K 0 is accessible, so is the category of arrows K 2 0 .We call a class of morphisms J in K accessible just when the corresponding full subcategory of K 2 0 is accessible and accessibly embedded in K 2 0 .In particular, in Condition (2) we require that the class of chosen isofibrations be accessible.
With regards Condition (3), the colimit of a diagram D : J → K 0 is said to be a homotopy colimit if the induced morphism is an equivalence of quasicategories, where p : Q → ∆1 ∈ [J op , SSet] is a cofibrant replacement in the projective Joyal model structure. 2efinition 5.2.A 2-cosmos K is accessible if the ∞-cosmos nK is accessible.
The following proposition refers to bicolimits rather than homotopy colimits.Bicolimits can be defined in the same way as homotopy colimits above, but with p : (2) the chosen isofibrations are accessible; (3) there exists a regular cardinal λ such that λ-filtered colimits exist in K and are bicolimits.
Proof.As remarked above, nK is accessible as a simplicially-enriched category just when K 0 is accessible and, for each simplicial set X, the powering functor (X ⋔ −) 0 : K 0 → K 0 is accessible.Since we have a natural isomorphism (X ⋔ A) nK ∼ = (ΠX ⋔ A) K and each small category is of the form ΠX for some X, this is equally to say that the powering functor (X ⋔ −) 0 : K 0 → K 0 is accessible for each small category X.But combined with accessibility of K 0 , this is exactly to say that K is accessible as a Cat-enriched category.Thus Condition (1) in Definition 5.1 corresponds to (1) above.There is no difference between the two Condition (2)s: they involve only the underlying ordinary category.
It remains to prove the equivalence of Condition (3) with Definition 5.1.(3).For this, we will prove that a (conical) colimit colim D in K is a bicolimit just when it is a homotopy colimit in nK.
To do this, first observe that the Quillen adjunction Π ⊣ N gives rise to a Quillen adjunction Π • − ⊣ N • − between the projective model structures on [J op , Cat] and on [J op , SSet].Given a cofibrant replacement p : Q → ∆1 ∈ [J op , SSet], we will show that Πp : ΠQ → Π1 = ∆1 is a cofibrant replacement in [J op , Cat].Indeed ΠQ is cofibrant since Π • − is left Quillen.Moreover, at X ∈ J , p X is a weak equivalence of cofibrant objects (all simplicial sets being cofibrant); hence the left Quillen Π sends it to a weak equivalence Π(p X ) so that, in particular, Π(p) is a weak equivalence in [J op , Cat], as required.
Therefore colim D is a bicolimit just when is an equivalence of categories or, equivalently, just when the induced map N(Πp * ) on nerves is an equivalence of quasicategories.But it follows easily from Lemma 2.3 that this is isomorphic to and hence an equivalence of quasicategories just when colim D is a homotopy colimit in nK.
Condition (3) about bicolimits in the above result can be replaced by an apparently simpler condition, as in the following result.

Proposition 5.4. A 2-cosmos K is accessible if and only if it is satisfies Conditions (1) and (2) of Proposition 5.3 above and
(3) there exists a regular cardinal λ such that λ-filtered colimits of pointwise equivalences in K are equivalences.
Proof.First note that a 2-cosmos satisfying Conditions (1) and ( 2) above is Cauchy-complete, and hence has flexible limits by Theorem 4.4.
In order to prove the claim, it will therefore suffice to show that given an accessible 2-category with flexible limits, Condition (3) above is equivalent to λ-filtered colimits in K being bicolimits.We defer the proof of this non-trivial result to the appendix, where it appears in more general form as Proposition A.1.
Building on Proposition 5.4, we can give several different formulations of accessible 2-cosmoi, each of which avoids any mention of bicolimits.This result is closely connected to [3, Proposition 8.1], although here the isofibrations are not necessarily the representable ones.
Theorem 5.5.Let K be a 2-cosmos for which K is accessible as a Cat-enriched category.The following are equivalent.
(1) K is an accessible 2-cosmos; (2) The chosen isofibrations and equivalences are accessible; (3) K is a Cauchy complete 2-cosmos and the trivial fibrations are accessible.
Proof.By Proposition 5.4, we have (2 =⇒ 1).For (1 =⇒ 3), we must show that the trivial fibrations are accessible.This is established in Theorem 6.3 of [4].Assuming (3), the proof that the equivalences are accessible is exactly as in the proof of Proposition 6.4 of loc.cit.
To prove that the isofibrations are accessible, we prove that there is a pullback where S is accessible.The result then follows from the Makkai-Paré limit theorem [12] for accessible categories -see Proposition 2.3 of [4] for a description of the relevant special case of pullbacks.
Here S : K 2 0 → K 2 0 is the functor sending f to w f , as in Definition/Proposition 3.2.This is accessible, since it is constructed using finite limits.
It remains to prove that f is an isofibration if and only if w f : A I → L is a trivial fibration; but w f is always an equivalence, so this says that f is an isofibration if and only if w f is one.Since w f is the Leibniz power of f by the injective-on-objects 1 : 1 → I, one direction is clear.Suppose conversely that w f is an isofibration.The composite of p f : L → B I and dom : B I → B is the isofibration appearing in the Brown factorization of f [15, Lemma 1.2.19].Thus the composite dom .pf .wf = dom .fI = f.dom : A I → B is also an isofibration; but f is a retract of this, hence it too is an isofibration.
In order to give examples of accessible 2-cosmoi, we will use the following result.
Corollary 5.6.Let K be a 2-category with flexible limits in which all representable isofibrations are normal.Then the normal 2-cosmos structure on K of Theorem 4.4 is accessible just when K is an accessible 2-category and the retract equivalences in K are accessible.
Proof.Recall that the normal 2-cosmos structure is Cauchy-complete and has as chosen isofibrations the normal ones.By our additional assumption, these are just the representable isofibrations.Hence a morphism is a trivial fibration just when it is an equivalence and a representable isofibration, which amounts to being a retract equivalence.The result then follows directly from Theorem 5.5.
Examples 5.7.In [3], the first-named author described a class LP M of 2-categories which are closely related to the above.A 2-category K belongs to LP M just when it is accessible, has filtered colimits and flexible limits, finite flexible limits commute with filtered colimits in K, and retract equivalences in K are accessible.Now if K belongs to LP M and all representable isofibrations are normal then, by Corollary 5.6, it follows that K is an accessible 2-cosmos.
As shown in [3], many 2-categories of categorical structures and their pseudo-morphisms belong to LP M .For instance, each of the following from Examples 4.7 belongs to LP M : • For T a 2-monad on a 2-category K, let w ∈ {s, p, l, c} stand for "strict/pseudo/lax/colax", and let w-T -Alg denote the 2-category of w-T-algebras and pseudomorphisms, and U : w-T -Alg → K the forgetful 2-functor.If T is a filtered colimit preserving 2-monad on K ∈ LP M , and w = s, then by Proposition 7.1 of [3] w-T -Alg also belongs to LP M .The corresponding result where w = s holds only under further assumptions, such as when T is flexible -see Theorem 7.2 of [3] and the discussion below it.In all of these cases, Corollary 3.15 applies to U, so that if K satisfies the NIP so does w-T -Alg which thereby forms an accessible 2-cosmos.For A a small 2-category, we can consider the 2-monad T on Cat obA with the simple formula T X(a) = Σ b∈A A(b, a) × Xb -see Section 6.6 of [2] for a complete description.Then the above setting specialises to capture each of the 2-categories of lax/pseudo/oplax functors A → Cat and pseudonatural transformations between them, so that each of these form accessible 2-cosmoi.As proven in Theorem 5.8 of [3], the 2-category of 2-functors and pseudonatural transformations A → Cat belongs to LP M if A is cellular, and again it follows that it is an accessible 2-cosmos.A key special case is when A is the cellular 2-category 2 -then the corresponding accessible 2-cosmos is the 2-category Ps(2, Cat) of arrows and pseudo-commutative squares.
On the other hand, we have the following non-example, making clear that the accessibility of K alone is not enough: Proposition 5.8.There is no accessible ∞-cosmos structure on Cat 2 .
Proof.If there were, then by Theorem 4.5 every normal isofibration would be a chosen isofibration, and so in particular a representable isofibration.By Theorem 5.5, there would be a regular cardinal α for which the trivial fibrations are closed under α-filtered colimits.Since every normal retract equivalence is a trivial fibration, and every trivial fibration is a retract equivalence, this would imply that every α-filtered colimit of normal retract equivalences is a retract equivalence.We shall show that this is impossible.
Let α be an arbitrary regular cardinal.For any set Y , let S Y be the set of all subsets of Y of cardinality strictly less than α, seen as a chaotic category, so that the functor S Y → 1 is a retract equivalence.Let P Y be the category whose objects are pairs (U ∈ S Y , x ∈ U), made into a category in such a way that the projection π Y : P Y → Y is a retract equivalence.Write f Y for the resulting morphism Since Y and 1 are discrete, there are no non-trivial liftings, and f Y is a normal isofibration.By construction the maps P Y → Y and S Y → 1 are retract equivalences; the map f Y will itself be a retract equivalence if and only if it has a section.
If Y has cardinality less than α, then f Y does have a section, with 1 → S Y picking out Y ∈ S Y and with Y → P Y sending y to (Y, y).On the other hand if Y has cardinality α, then f Y does not have a section: there is no way to choose U ⊆ Y with |U| < α containing every element of Y .Finally the fact that f Y is an α-filtered colimit of the f X with |X| < α gives the desired contradiction.
Perhaps the way to think about this is that if one wants to model arrows in Cat in an ∞-cosmological way, the "correct" thing to do is not use Cat 2 itself but rather the full subcategory Cat Our goal is to prove the following proposition.
Proposition A.1.Let K be an accessible 2-category with flexible limits, and suppose further that it admits J -colimits for some small category J .Then J -colimits are bicolimits in K if and only if J -colimits of pointwise equivalences in K are equivalences.
We will break the proof of the non-trivial direction into a few steps.We fix a small category J and an accessible 2-category K with flexible limits and J -colimits.We further fix a trivial fibration t : K → ∆1 with cofibrant domain in the projective model structure on [J op , Cat].
For any functor S : J → K and object A ∈ K, there is an induced functor t * : [J op , Cat](∆1, K(S, A)) → [J op , Cat](K, K(S, A)) and we are to show that this is an equivalence.Proposition A.2.There is an enriched monad R on [J , K] sending an object T : J → K to RT : J → K, where RT J = KJ ⋔ T J.The unit r : T → RT is induced by t : K → ∆1.
Proof.The 2-category [J op , Cat] is monoidal with respect to the product, and there is an enriched action Φ : [J op , Cat] op × [J , K] → [J , K] given by Φ(F, T )J = F J ⋔ T J.
Every object of [J op , Cat] has a unique comonoid structure; in particular, K does so, and thus Φ(K, −) : [J , K] → [J , K] has the structure of a monad.Proposition A.3.For each S : J → K there exists a QS : J → K and η : S → RQS such that the induced map is a retract equivalence.
Proof.If K is accessible and has flexible limits, then the same is true of [J , K], and moreover R : [J , K] → [J , K] will be accessible and preserve these limits.Thus it will have a weak left adjoint by [5,Theorem 8.10].This means that that for each S : J → K there exist a QS and an η : S → RQS such that the displayed map is a shrinking morphism in Cat; that is, a retract equivalence.
with cofibrant codomain, and so has a section in [J op , Cat], and is therefore an equivalence in the 2-category [J op , Cat].This completes the difficult direction of the proof.
On the other hand, if J -colimits are bicolimits in K, then their universal property with respect to pseudo-natural transformations implies that they they take pseudo-natural equivalences to equivalences in K.But since each pointwise equivalence is a pseudo-natural equivalence, this implies the result.
part of the definition of representable isofibration.Conversely, given κ : x ∼ = u f and a lifting problem β : b ∼ = f a, by the universal property of L f there is an induced c with u f c = a, v f c = b, and λ f c = β and now κc : xc ∼ = u f c = a gives the required cleavage.

Corollary 3 . 15 .
Suppose that the 2-category L has pseudolimits of arrows, and the 2-functor U : L → K preserves them; and further that U induces discrete isofibrations on the hom-categories.If K satisfies the normal isofibration property then so too does L. Example 3.16.For a 2-monad T on K, let T -Alg be the 2-category of strict T -algebras and pseudomorphisms, and U : T -Alg → K be the forgetful 2-functor.If K has pseudolimits of arrows, then U : T -Alg → K satisfies the assumptions of Corollary 3.15.Thus T -Alg satisfies the NIP if K does so.
C → A, b : C → B and β : b ∼ = f a, as displayed above on the right.Apply U to these data and use the universal property of L to obtain a unique map c : UC → L with u.c = Ua, v.c = Ub, and λ.c = Uβ.Now δc : c ∼ = d.u.c = d.Ua = U(d.a)has a unique lifting γ : c ∼ = da.We shall show that c provides the required factorization through L. Now • uγ : u c ∼ = uda = a is a lifting of an identity, thus an identity • λc : v c ∼ = f u c = f a is a lifting of Uβ, thus equal to β and so u c = a, v c = b, and λc = β.

։
of isofibrations.This is accessible, by [4, Proposition 4.1] and the fact that Cat is itself an accessible ∞-cosmos.Alternatively, in the 2-categorical setting one can use the accessible 2-cosmos Ps(2, Cat) of arrows and pseudocommutative squares, as discussed in Examples 5.7.Appendix A.
cosmos is a 2-category K together with a class of morphisms called isofibrations, denoted A ։ B, closed under composition and containing the isomorphisms, such that (a) the morphism K(A, p) : K(A, B) ։ K(A, C) is an isofibration of categories for each isofibration p : B ։ C; (b) K has products, powers by small categories, pullbacks along isofibrations, and limits of countable towers of isofibrations; (c) the class of isofibrations is stable under pullback; closed under products, limits of countable towers, and Leibniz powers by injective on objects functors; and contains all maps with terminal codomain.
Cat] a cofibrant replacement now in the projective model structure on [J op , Cat].Proposition 5.3.A 2-cosmos K is accessible if and only if (1) K is accessible as a Cat-enriched category; Lex by [3, Section 6.4] • Reg, Ex by [3, Section 6.5] • Coh, Pretop by adapting the techniques of [3, Section 6.5] • MonCat, Icon by [3, Section 7].As observed in Examples 4.7, in each case the normal isofibrations are just the representable isofibrations.It follows that each of these examples is an accessible 2-cosmos.