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In the entry here, the fact that the functor category out of an accessible into a locally presentable category is locally presentable is referenced to Adamek & Rosicky’s book. (This edit originates from discussion in another thread).
Where in that book is this actually stated? I have trouble locating it…
Where in that book is this actually stated? I have trouble locating it…
The highlighted bullet point is Corollary 1.54. I don’t think the statement that “the category of accessible functors from an accessible category into a locally presentable category is accessible” is in the book.
Ah, thanks! I have added that pointer to the entry.
(Now I see why I didn’t find this: They spell “functor category” with a hyphen.)
But that discussion and the entry in general could do with much more polishing.
For instance, weirdly, the single reference that used to be given was the HoTT book, without commentary of what in there the reader is meant to take note of regarding functor categories. Indeed, I don’t see what that could be and have removed that reference (but let’s add it back in if we know why).
What’s the general statement regarding local presentablity of categories of enriched functors?
Do we have a statement like: If
the enriching cosmos $\mathbf{V}$ is locally presentable,
the underlying category of $\mathbf{D} \in \mathbf{V}Cat$ is small
the underlying category of $\mathbf{C} \in \mathbf{V}Cat$ is locally presentable
then $\mathbf{V}Func(\mathbf{D}, \mathbf{C})$ is locally presentable??
Re #5: Since we know VFunc(D,C) is cocomplete, it suffices to show it is accessible.
Accessibility follows since VFunc(D,C) can be constructed as a PIE-limit.
Specifically, start with the product $\prod_{d\in D}C$.
Next, for every $d,e\in D$, use an inserter to equip objects $X\in \prod_{d\in D}C$ with morphisms $D(d,e)\to C(X_d,X_e)$.
Finally, use equifiers to enforce preservation of compositions and units.
Thanks!
I admit that I don’t even know the facts that you are tacitly appealing to.
For example, what’s the statement regarding PIE-limits and accessibility? I see that at accessible weak factorization system in the proof here there is a similar allusion, where it says
…is locally presentable (being complete and a PIE limit construction…
We should add pointer there to the actual theorem being invoked.
I wonder if there are not standard accounts of these matters that we could lazily refer to. Hm, I’ll check out the references suggested at MO:q/53470: “Enriched locally presentable categories”.
For example, what’s the statement regarding PIE-limits and accessibility?
The 2-category of accessible $\mathcal{V}$-categories (for nice enough $\mathcal{V}$) is closed under small flexible limits (and thus flexible 2-limits, pseudolimits, bilimits). This is Theorem 5.5 of Lack–Tendas’s Virtual concepts in the theory of accessible categories. The same is true for locally presentable $\mathcal{V}$-categories by a result of Bird (cited in the linked paper).
[overlapped with #8 while editing]
So in
is an Example 3.4 (this page) which says that for a small $\mathcal{V}$-enriched category $\mathcal{T}$ the enriched functor category $[\mathcal{T}, \mathcal{V}]$ is locally finitely presentable in the sense of the previous Def. 3.2 (which ought to imply that the underlying category is locally enriched in the ordinary sense, if I am reading it right).
Re #7: See Chapter 5 in Makkai–Paré, Accessible Categories: The Foundations of Categorical Model Theory, in partcular, Theorem 5.1.6 and Corollary 5.1.8, which shows that PIE-limits of accessible categories are accessible. This is a very powerful tool to show accessibility (and therefore local presentability).
Another reference: Adámek–Rosický, Locally presentable and accessible categories, Section 2.H, in particular, Theorem 2.77.
Thanks varkor, thanks Dmitri! I am in the process of adding this statement to accessible category.
Just to be pedantic: What we need for the present discussion is not just that $AccCat$ has all these 2-limits, but also that they can be computed in $Cat$. Probably the inclusion of $AccCat$ into $Cat$ preserves (these) 2-limits?
Re #12: Adámek–Rosický’s formulation answers this too:
2.77 Limit Theorem. A lax limit of accessible categories is accessible. More precisely, ACC is closed under lax limits in CAT.
Thanks again! Adding this now.
By the way, Makkai & Paré say that $AccCat$ has “all limits”. The mentioning of PIE-limits I see only in Lack & Tendas. (?)
Just to be pedantic: What we need for the present discussion is not just that $AccCat$ has all these 2-limits, but also that they can be computed in $Cat$. Probably the inclusion of $AccCat$ into $Cat$ preserves (these) 2-limits?
Yes, this is also proven in the enriched setting in the paper of Lack and Tendas.
Ah, right, their theorem 5.9. Thanks, adding it now…
I have a point of (perhaps pedantic) uncertainty surrounding the notion of “the” functor category. This seems to imply that there is only one, therefore if I were to remove a particular functor from “the” functor category $D^C$ it wouldn’t be “the” functor category anymore.
Which then raises the question of what functors are in $D^C$… all possible ones? (if so what does this mean)?
I think it is unclear to me whether functors are implict or explicit constructions (as in, do declare which functors “exist”, or do they just exists automatically based on the structure of C and D).
Sorry if this is the wrong place. (Though it would be cool if this page could be tweaked/added-to so as to clear such confusion.)
Yes, I think that is a bit clearer.
One of the sticking points for me was that, since we are dealing with classes (not necessarily sets), I wasn’t sure precisely what “all functors” means (depends on formalism used for classes?), or even if that idea was well-defined.
|’ve just been thinking about it as “all functors we care about”, which seems okay based on what I’ve learned so far.
added pointer to:
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