On local presentability of T/A

07/06/2018 ∙ by Patricia Johann, et al. ∙ 0

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 [1] 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 [1]. 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 [1] 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

let , and be defined by the pushout

(1)

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

Define

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

Definition 1

For each define , where

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

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

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

(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 :

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

This completes the proof of the theorem.

3 Elements of are -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

and suppose in . Then

(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 [1994].

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

(4)

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

(5)

Without loss of generality we may assume , so that

(6)

Next, observe that

by point 3 above
by Equation 6
by Equation 4
by Equation 3
by definition of
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

(7)

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

(8)
(9)

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

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

to conclude that . To that end, observe that

by Equation 8
by point 3 above
by point 1 above
by Equation 4
by Equation 3
by Equation 9
by point 2 above
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

(10)
(11)

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

(12)

Let and . Then

by definition of
by hypothesis
by hypothesis
by definition of

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

(13)

We want to show that

For the first components we have

For the second components we first observe that

by Equation 10
by the calculation for the first components
by Equation 11

In addition, we have