Colimits of accessible categories

Accessible categories are closed under constructions of "a limit type". More precisely, the 2-category of accessible categories and accessible functors has all limits appropriate for 2-categories calculated in the 2category of categories and functors (see [5]). The situation is much less satisfactory for colimits. The only general result is that lax colimits of strong diagrams of accessible categories and accessible functors exist and are calculated as the idempotent completion of the lax colimit of categories (see [5], Theorem 5.4.7). In this paper we show that any directed colimit of accessible categories and accessible full embeddings is accessible and, assuming the existence of a proper class of strongly compact cardinals, accessible categories are closed under directed colimits of embeddings. We do not know whether set theory is really necessary for the second result. We also do not know anything about general directed colimits. We will start with an example of a colimit of accessible categories which is not accessible (but has split idempotents).

All undefined concepts concerning accessible categories can be found in [1] or [5]. Recall that a functor F : K → L is called λ-accessible if K and L are λ-accessible categories and F preserves λ-directed colimits. F will be called strongly λ-accessible if, in addition, it preserves λpresentable objects. Any λ-accessible functor is strongly µ-accessible for some regular cardinal µ. F is (strongly) accessible if it is (strongly) λ-accessible for some regular cardinal λ. CAT will denote the (nonlegitimate) category of categories and functors while ACC is the (nonlegitimate) category of accessible categories and accessible functors.

Example 1.1. Let K be a combinatorial model category and W its class of weak equivalences. Then W is an accessible category and its embedding G : W → K → into the category of morphisms of K is accessible (see [4] A.2.6.6 or [6] 4.1). Let dom, cod : W → K be the functors assigning to each f ∈ W its domain or codomain. These functors are accessible and let ϕ : dom → cod the natural transformation such that ϕ f = f . Then the coinverter of ϕ is the homotopy category Ho K = K[W -1 ] of K. The homotopy category has very often split idempotents (for instance if K is stable) and is almost never accessible; e.g., if K is the model category of spectra then Ho K has split idempotents and is not accessible.

Theorem 2.1. Let F ij : K i → K j , i ≤ j ∈ I be a directed diagram of accessible categories and accessible full embeddings. Then its colimit in CAT is accessible as are the colimit injections, and is in fact the colimit in ACC.

Proof. We can assume the F ij are full inclusions. Then we want to show that K = i∈I K i is accessible. Let κ be such that each K i is κ-accessible, each inclusion K i ⊆ K j is strongly κ-accessible , and κ > |I|.

Let

We claim that there is an i 0 ∈ I and a cofinal subset M 0 ⊆ M such that K m ∈ K i 0 for all m ∈ M 0 . Otherwise there would be, for every i,

Then as M is κ-directed and κ > |I|, there is one p ≥ m i for all m i , and the corresponding K p is not in any K i , a contradiction. Now M 0 is κ-directed as it is cofinal in M so colim m∈M 0 K m exists in K i 0 . Now for any cocone k m : K m → L m∈M in K, L will be in some K i and if we take i 1 ≥ i, i 0 then we get a cocone k m : K m → L m∈M 0 in K i 1 . As K i 0 ⊆ K i 1 preserves κ-directed colimits, this cocone factors uniquely through colim m∈M 0 K m , so colim m∈M 0 K m is the colimit in K as well.

If our diagram lies entirely in one K i to start with we can take i 0 = i and M 0 = M, so the inclusion K i ⊆ K preserves κ-directed colimits.

If K is κ-presentable in K i , and K m m∈M is a κ-directed diagram in K, then we can choose the i 0 above so that it is also ≥ i. Then

the inclusions K i ⊆ K are strongly κ-accessible.

Every object of K is a κ-directed colimit of κ-presentables in some K i so also in K. Thus K is κ-accessible.

Finally, K is the colimit of K i in ACC, for if G i : K i → L i∈L is a compatible family of κ i -accessible functors, we can choose the κ in the above argument to be larger than all κ i , and then the extension G : K → L will preserve κ-directed colimits.

Example 2.2. Let n be the ordered set {1 < 2 < • • • < n} and consider the chain of accessible embeddings

where the transition for n to n + 1 extends a path of length n to one of length n + 1 by adding an identity at the end. The colimit can be identified with the category of infinite paths A 1 → A 2 → A 3 → • • • which are eventually constant, i.e. there is an N such that A n → A n+1 is an identity for all n ≥ N . It is ω 1 -accessible but not ω-accessible.

Remark 2.3. Theorem 2.1 can be extended to directed colimits of embeddings F ij such that for each commutative triangle

In fact, we can choose m 0 ∈ M 0 and repeat the argument above to get i 1 > i 0 and a cofinal subset

Using this colimit in the proof above instead of colim m∈M 0 K m , we get the extension of

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