 # On local presentability of T/A

We prove that if A is a locally λ-presentable category and T : A→A is a λ-accessible functor then T/A is locally λ-presentable.

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## 1 Introduction

Locally presentable categories form a robust class of categories which possess very nice properties, yet are general enough to encompass a large class of examples — including categories of models of algebraic theories and limit-sketches.

We refer to the standard reference  for definitions and basic properties of locally presentable categories and accessible functors. These include the following:

Proposition

[1, Prop.1.57] If is a locally -presentable category, then for each object , the slice categories and are locally -presentable.

Proposition

[1, Prop.2.43] If , and are locally -presentable categories and for , is a -accessible functor, then there exists a regular cardinal such that the comma category is locally -presentable.

Proposition

[1, Exercise 2.h] If are locally -presentable categories and for , is a -accessible functor which preserves limits, then the comma category is locally -presentable.

In this note, we add the following to the list above. (Note that denotes .)

Proposition

If is a locally -presentable category, then for each -accessible endofunctor , the comma categories and are locally -presentable.

The first claim, that is -presentable, is essentially contained already in the proof of Proposition 2.43 in . That proof begins by finding, given -accessible and , a cardinal such that and are -accessible and preserve -presentable objects. However, only the fact that preserves such objects is subsequently used. Since clearly preserves such objects, and is -accessible by hypothesis, we may take here and proceed as in  to conclude that is -accessible.

Our main contribution is thus in proving the second claim, that is -presentable as well under no additional hypothesis on beyond -accessibility.

### Assumptions

For the rest of this document, we assume the following.

• is a regular cardinal.

• is a -presentable category.

• is a set of -presentable objects generating all of under -directed colimits.

• is a -accessible endofunctor on .

## 2 Outline of the proof

This section gives a basic summary of our argument. Subsequent sections will elaborate the individual steps in this argument. Throughout, if is a poset considered as a category, we write rather than . We may also refer to “the poset ”, and denote the ordering simply by , in this case. If we write for .

By the usual arguments, (co)limits in comma categories exist, including , and are computed componentwise. is therefore cocomplete. The bulk of the argument therefore consists in exhibiting a set of -presentable objects in that generate all of under -directed colimits.

1. The set .

Let . For

 w=(A,P,Q,p,q)∈W

let , and be defined by the pushout

 Pp−−−−→TAq⏐⏐↓⏐⏐↓f(w)Qg(w)−−−−→U(w) (1)

When we need to refer to individual components of , we will write , etc. (For the above, is .)

Define

 P={(A(w),U(w),f(w))∣w∈W}

Then is a set. Indeed, for every there exists (at least one) that determines up to isomorphism. We may call such a witness that . Since is clearly a set, there are only set-many witnesses available.

2. Every element of is -presentable.

This will be proved by a direct argument in Section 3.

3. For every there is a -directed poset .

Using the fact that is locally -presentable, we first collect the following data:

• Write , where is a -directed poset, for , and for .

• For each , write , where is a -directed poset, for , and for .

• Write , where is a -directed poset, for , and for .

• For each and each , use the fact that is -presentable, , and to choose and such that

 Pi,jpi,j−−−−→TAiq(i,j)⏐⏐↓⏐⏐↓f∘TαiBk(i,j)βk(i,j)−−−−→B
###### Definition 1

For each define , where

 D (i,j,k,q)≤(i′,j′,k′,q′)

As before, we refer to individual components , and of as , and . We prove directly that is a -directed poset in Section 4.

4. There is a functor whose image consists of objects in

. For , let .

###### Definition 2

The functor is defined as follows:

• , where and , , and are as in (1).

• is defined whenever , , and . The morphism is defined by the pushout property of , as shown in either the left or the right diagram below, according as is obtained via AA or BB.

Note that, for all , is indeed in since .

5. There is a cofinal -directed sub(po)set of , and thus
 lim−→d∈D(A,B,f)Fd=lim−→d′∈D′(A,B,f)Fd′

Let be the subposet of comprising under the same ordering as on . To show that is a cofinal subposet of , suppose . Note that

 f∘Tαi∘pi,j:Pi,j⟶(→limk∈KBk)

Since is -presentable, there exists a and a morphism such that .

Since and already satisfy , by essential uniqueness of such factorizations there must exist an upper bound for such that

 βk→k′∘q=βk(i,j)→k′∘q(i,j) (2)

Then is in , and (2) confirms that via AA. (Note that .)

That is -directed is immediate from it being a confinal subposet of the -directed set .

6. Lemma

For fixed , consider the functor given by

Then the following identity holds in :

 lim−→j∈JiGi(j)=(Bf∘Tαi←−−−−TAiidTAi−−−−→TAi)

This is a straightforward computation of a colimit in a functor category. For completeness we include the proof in Section 5.

7. Lemma

For fixed , consider the functor given by

• .

• .

Then the following identity holds in :

This is also a straightforward computation of a colimit in a functor category. We omit its proof.

8. Every object in is a -directed colimit of elements of .

We have that is the colimit of . Indeed, we have

 lim−→d∈D(A,B,f)Fd =lim−→d∈D′(A,B,f)Fd =lim−→(i,j,k,q)∈D′(A,B,f)F(i,j,k,q) =lim−→i∈Ilim−→j∈Jilim−→k∈k(i,j)↑F(i,j,k,βk(i,j)→k∘q(i,j)) =lim−→i∈Ilim−→j∈Jilim−→k∈k(i,j)↑Pushout(Bkβk(i,j)→k∘q(i,j)←−−−−−−−−−Pi,jpi,j−−−−→TAi) ={since pushouts commute with colimits (% see Section ???)} lim−→i∈Ilim−→j∈JiPushoutlim−→k∈k(i,j)↑(Bkβk(i,j)→k∘q(i,j)←−−−−−−−−−Pi,jpi,j−−−−→TAi) ={by Lemma in point 7} ={since pushouts commute with colimits} ={by Lemma in point 6} lim−→i∈IPushout(Bf∘Tαi←−−−−TAiidTAi−−−−→TAi) ={pushing out by identity is identity} lim−→i∈I(f∘Tαi) ={by computation of colimits in comma % categories} f

This completes the proof of the theorem.

## 3 Elements of P are λ-presentable

Let be a -directed poset and let in . By computation of colimits in comma categories, we have that

• in , with structure morphisms and .

• in , with structure morphisms and .

• The structure morphisms for in are .

• Since is -accessible, in .

Now let be determined by via the pushout

 Pp−−−−→TAq⏐⏐↓⏐⏐↓fQg−−−−→B

and suppose in . Then

 f∗∘Tα=β∘f (3)

We want to show that there exists a such that factors essentially uniquely through , as in the characterization of -presentable objects in Definitions 1.13 and 1.1 of Adámek and Rosický .

Since , is -presentable, and is a colimit, there exists , and such that

 α=αd∘α∘ (4)

Similarly, since , is -presentable, and is a colimit, there exists and such that

 β∘g=βd′∘g′ (5)

Without loss of generality we may assume , so that

 αd=αd′∘αd→d′ (6)

Next, observe that

 βd′∘fd′∘Tαd→d′∘Tα∘∘p =f∗∘Tαd′∘Tαd→d′∘Tα∘∘p by point 3 above =f∗∘Tαd∘Tα∘∘p by Equation 6 =f∗∘Tα∘p by Equation 4 =β∘f∘p by Equation 3 =β∘g∘q by definition of (A,B,f) =βd′∘g′∘q by Equation 5

This exhibits two factorizations of the same morphism from the -presentable object to the colimit via . By the essential uniqueness of such factorizations, there exists a , such that

 βd′→d0∘fd′∘Tαd→d′∘Tα∘∘p =βd′→d0∘g′∘q (7)

By the pushout property of , there is therefore a unique such that

 β′∘f =βd′→d0∘fd′∘Tαd→d′∘Tα∘ =fd0∘Tαd′→d0∘Tαd→d′∘Tα∘ =fd0∘Tαd→d0∘Tα∘ (8) β′∘g =βd→d0∘g′ (9)

Letting , Equation (8) states that is a morphism in . That the first component of this morphism composes with to is obvious:

 αd0∘α′=αd∘α∘=α

The first equality is by Equation 6 and the second is by Equation 4. To see that the second component of this morphism composes with to , we use uniqueness property of the morphism from the pushout to . That is, we show that

 βd0∘β′∘f=β∘f βd0∘β′∘g=β∘g

to conclude that . To that end, observe that

 βd0∘β′∘f =βd0∘fd0∘Tαd→d0∘Tα∘ by Equation 8 =f∗∘Tαd0∘Tαd→d0∘Tα∘ by point 3 above =f∗∘Tαd∘Tα∘ by point 1 above =f∗∘Tα by Equation 4 =β∘f by Equation 3 βd0∘β′∘g =βd0∘βd′→d0∘g′ by Equation 9 =βd′∘g′ by point 2 above =β∘g by Equation 5

So and thus in .

To see that this factorization is essentially unique, suppose . We must show that there exists an such that . Since and are morphisms in , we have

 β′∘f=fd0∘Tα′ (10) β′′∘f=fd0∘Tα′′ (11)

Moreover, since is -presentable and , there exists a such that

 αd0→d1∘α′=αd0→d1∘α′′ (12)

Let and . Then

 βd1∘γ =βd1∘βd0→d1∘β′∘q by definition of γ =βd0∘β′∘q =β∘q by hypothesis =βd0∘β′′∘q by hypothesis =βd1∘βd0→d1∘β′′∘q =βd1∘γ′ by definition of γ′

That is, we have two factorizations and of the same morphism from the -presentable object to . There must therefore exist an such that

 βd1→l∘γ=βd1→l∘γ′ (13)

We want to show that

 (αd0→l,βd0→l)∘(α′,β′)=(αd0→l,βd0→l)∘(α′′,β′′)

For the first components we have

 αd0→l∘α′ =αd1→l∘αd0→d1∘α′ =αd1→l∘αd0→d1∘α′′ by Equation~{}??? =αd0→l∘α′′

For the second components we first observe that

 βd0→l∘β′∘f =βd0→l∘fd0∘Tα′ by Equation 10 =fl∘Tαd0→l∘Tα′ (αd0→l,βd0→l):(Ad0,Bd0,fd0)→(Al,Bl,fl) =fl∘Tαd0→l∘Tα′′ by the calculation for the first components =βd0→l∘fd0∘Tα′′ (αd0→l,βd0→l):(Ad0,Bd0,fd0)→(Al,Bl,fl) =βd0→l∘β′′∘f by Equation 11