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I am interested in conditions that guarantee that the category of functors between two given categories is locally presentable or accessible.
Some (conjectural, possibly wrong) statements that I have in mind:
1) Accessible functors between accessible categories form an accessible category.
2) Accessible functors from an accessible category to a locally presentable category form a locally presentable category.
3) Cocontinuous functors between locally presentable categories form a locally presentable category.
4) Continuous accessible functors between locally presentable categories form a locally presentable category.
Which of these have a chance of being true? Is there any place in the literature that considers such statements?
If you fix the accessibility rank then it’s quite straightforward to reduce to the case of the ordinary functor category $[\mathcal{A}, \mathcal{D}]$ where $\mathcal{D}$ is accessible or locally presentable. More precisely, we have the following:
Indeed, the point is this: given a $\kappa$-accessible category $\mathcal{C} \simeq Ind^\kappa (\mathcal{A})$ ($\mathcal{A}$ essentially small), the category of $\kappa$-accessible functors $\mathcal{C} \to \mathcal{D}$ (for arbitrary $\mathcal{D}$; here by “$\kappa$-accessible” I mean simply “preserves $\kappa$-filtered colimits”) is naturally equivalent to the category of all $\mathcal{A} \to \mathcal{D}$. It should be well known that:
Statements 1 and 2 are proved in [Adamek and Rosický, Locally presentable and accessible categories], statement 3 is obvious, and statement 4 is a straightforward exercise. Thus the claims follow.
Great, many thanks for such a comprehensive answer!
2 is amazingly comprehensive and clear answer!
Too bad though that it’s not on the nLab. Somebody should add it.
What happens if we don’t restrict the accessibility rank in (1) and (2), i.e., take all accessible functors and not just κ-accessible ones for some κ? Is there any chance that the statements still hold?
It’s false in general. Let $\mathcal{C}$ be an accessible category that is not essentially small. Consider the category $\mathcal{A}$ of all accessible functors $\mathcal{C} \to \mathbf{Set}$. This is the same as the smallest full replete subcategory of $[\mathcal{C}, \mathbf{Set}]$ containing all representable functors and closed under small colimits. In particular, $\mathcal{A}$ is accessible if and only if $\mathcal{A}$ locally presentable.
I claim $\mathcal{A}$ is not accessible. Indeed, suppose $\mathcal{A}$ has a small generating family, say $\mathcal{G}$. Then for some regular cardinal $\kappa$, every member of $\mathcal{G}$ is $\kappa$-accessible. So consider $\mathcal{C} (X, -)$ for some object $X$ that is not $\kappa$-presentable. (Such an $X$ exists because $\mathcal{C}$ is not essentially small.) Since $\mathcal{G}$ generates, there is a small diagram of $\kappa$-accessible functors whose colimit is $\mathcal{C} (X, -)$. But then $\mathcal{C} (X, -)$ is a retract of a $\kappa$-accessible functor and hence $\kappa$-accessible: a contradiction.
That said, $\mathcal{A}$ is class-locally presentable.
Thanks a lot, the concept of class-locally presentable categories does clarify things quite a bit. And apparently there are also class-combinatorial model categories, of which there are several interesting examples.
@Urs: It seems to me that the above writeup by Zhen Lin could be incorporated (by me, for example) in one of the nLab pages without much difficulty, as long as the original author does not object to it. The only question is in which article this material should go, one possible choice is http://ncatlab.org/nlab/show/functor+category#properties.
I have no objections.
The only question is in which article this material should go, one possible choice is http://ncatlab.org/nlab/show/functor+category#properties.
That sounds good.
I added the material in this thread to http://ncatlab.org/nlab/show/functor+category#accessibility_and_local_presentability.
While editing, I formulated some statements that could describe what happens for the case of a class-accessible category and class-locally presentable category, though I am hesitant to say more in the article because my knowledge of class-accessible and class-locally presentable categories is too shallow, so I place them here instead.
Accessible functors between accessible categories form an class-accessible category.
(Strongly?) class-accessible functors between class-accessible categories form an class-accessible category.
Accessible functors from an accessible category to a locally presentable category form a class-locally presentable category.
(Strongly?) class-accessible functors from a class-accessible category to a class-locally presentable category form a class-locally presentable category.
Thanks. I have added some more hyperlinks to keywords (minimum at class-locally presentable category) and have made it a subsection of the Properties-section.
I cannot comment on class-accessibility.
In general, accessible functors between accessible categories do not form an accessible category due to set-theoretical size issues.
I am not inclined to call these issues “set-theoretical”. That seems to suggest that adding set-theoretic axioms (e.g. Vopěnka’s principle) would solve the problem, but this is very different.
I deleted “set-theoretical” from the description.
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