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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 10th 2014
    • (edited Aug 10th 2014)

    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?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeAug 10th 2014
    • (edited Aug 11th 2014)

    If you fix the accessibility rank then it’s quite straightforward to reduce to the case of the ordinary functor category [𝒜,𝒟][\mathcal{A}, \mathcal{D}] where 𝒟\mathcal{D} is accessible or locally presentable. More precisely, we have the following:

    1. κ\kappa-accessible functors from a κ\kappa-accessible category to any accessible category form an accessible category. (It is not so easy to say what the accessibility rank is here.)
    2. κ\kappa-accessible functors from a κ\kappa-accessible category to any locally λ\lambda-presentable category form a locally λ\lambda-presentable category.
    3. Colimit-preserving functors between locally κ\kappa-presentable categories form a locally κ\kappa-presentable category.
    4. Limit-preserving accessible functors between locally κ\kappa-presentable categories form the opposite of a locally κ\kappa-presentable category (namely, the previous one).

    Indeed, the point is this: given a κ\kappa-accessible category 𝒞Ind κ(𝒜)\mathcal{C} \simeq Ind^\kappa (\mathcal{A}) (𝒜\mathcal{A} essentially small), the category of κ\kappa-accessible functors 𝒞𝒟\mathcal{C} \to \mathcal{D} (for arbitrary 𝒟\mathcal{D}; here by “κ\kappa-accessible” I mean simply “preserves κ\kappa-filtered colimits”) is naturally equivalent to the category of all 𝒜𝒟\mathcal{A} \to \mathcal{D}. It should be well known that:

    1. If 𝒟\mathcal{D} is accessible, then so is [𝒜,𝒟][\mathcal{A}, \mathcal{D}].
    2. If 𝒟\mathcal{D} is locally λ\lambda-presentable, then so is [𝒜,𝒟][\mathcal{A}, \mathcal{D}].
    3. Colimit-preserving functors out of a locally κ\kappa-presentable category are κ\kappa-accessible.
    4. A right adjoint between locally κ\kappa-presentable categories is κ\kappa-accessible if and only if its left adjoint is strongly κ\kappa-accessible (i.e. preserves κ\kappa-presentable objects as well as κ\kappa-filtered colimits); and every limit-preserving accessible functor between locally presentable categories is a right adjoint.

    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.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 10th 2014

    Great, many thanks for such a comprehensive answer!

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeAug 11th 2014

    2 is amazingly comprehensive and clear answer!

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 11th 2014
    • (edited Aug 11th 2014)

    Too bad though that it’s not on the nLab. Somebody should add it.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 30th 2014

    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?

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeAug 30th 2014

    It’s false in general. Let 𝒞\mathcal{C} be an accessible category that is not essentially small. Consider the category 𝒜\mathcal{A} of all accessible functors 𝒞Set\mathcal{C} \to \mathbf{Set}. This is the same as the smallest full replete subcategory of [𝒞,Set][\mathcal{C}, \mathbf{Set}] containing all representable functors and closed under small colimits. In particular, 𝒜\mathcal{A} is accessible if and only if 𝒜\mathcal{A} locally presentable.

    I claim 𝒜\mathcal{A} is not accessible. Indeed, suppose 𝒜\mathcal{A} has a small generating family, say 𝒢\mathcal{G}. Then for some regular cardinal κ\kappa, every member of 𝒢\mathcal{G} is κ\kappa-accessible. So consider 𝒞(X,)\mathcal{C} (X, -) for some object XX that is not κ\kappa-presentable. (Such an XX exists because 𝒞\mathcal{C} is not essentially small.) Since 𝒢\mathcal{G} generates, there is a small diagram of κ\kappa-accessible functors whose colimit is 𝒞(X,)\mathcal{C} (X, -). But then 𝒞(X,)\mathcal{C} (X, -) is a retract of a κ\kappa-accessible functor and hence κ\kappa-accessible: a contradiction.

    That said, 𝒜\mathcal{A} is class-locally presentable.

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 31st 2014

    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.

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 31st 2014

    @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.

    • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeAug 31st 2014

    I have no objections.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2014

    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.

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 1st 2014

    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.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2014
    • (edited Sep 1st 2014)

    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.

    • CommentRowNumber14.
    • CommentAuthorZhen Lin
    • CommentTimeSep 1st 2014

    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.

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 1st 2014

    I deleted “set-theoretical” from the description.

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