5
$\begingroup$

I am interested in naturally occurring symmetric closed monoidal structures on locally presentable categories.

Two general examples:

  1. Grothendieck topos with Cartesian structure. Here, for example, $\mathrm{Set}, \mathrm{sSet}$, smooth sets, etc.
  2. Category algebras over a commutative algebraic theory. Here, for example, $R\text{-}\mathrm{Mod}, \mathrm{Set}_*$.

I think, it's easy to see that they can be combined into one super example: the category of algebras over a commutative monad on a Grothendieck topos (in the case of a trivial monad, we have the topos itself; we can also consider pointed objects in any topos, in particular we get some sorts of pointed spaces).

Separate from this business is the category $\mathrm{Cat}$.

Is there a natural family of limit theories, including $\rm{Cat}$ theory, which are Cartesian closed for some general reason? And $\mathrm{Pos}$ (category of posets)?

Improve this question
$\endgroup$
17
  • 4
    $\begingroup$ It's hard to know what sort of answer would satisfy you. I'm going to guess that you could grind through a proof if you had to, so that what you want is some sort of conceptual reason. Here's one attempt: via the nerve functor N, Cat embeds fully faithfully into simplicial sets, and are characterized as s-sets satisfying "Segal conditions". Meanwhile, s-sets form a cartesian closed category (since, e.g., it's a topos). It then suffices to check that for a category C and simplicial set X, that NC^X is also the nerve of a category, by showing it too satisfies the Segal conditions. $\endgroup$
    – Todd Trimble
    Commented May 24, 2023 at 0:05
  • 4
    $\begingroup$ And I'm trying to sketch a general reason: roughly speaking, if you start with a cartesian closed category, and if the objects of certain full subcategory are definable as those objects c that satisfy certain limit properties (like Segal conditions), then exponentials c^x will also satisfy those conditions since (-)^x preserves limits. (I guess you also need that the full embedding preserves products. This is true for the nerve functor, for instance.) $\endgroup$
    – Todd Trimble
    Commented May 24, 2023 at 0:19
  • 3
    $\begingroup$ I like this question. I feel like @ToddTrimble’s comments gives a good conceptual reason why Cat is CCC, for a particular presentation of Cat (the simplicial one) — and the comments would make a nice answer — but like @ Aivazian, I would also be interested to know if there’s a way to explain it in terms of the traditional algebraic presentation of Cat (or perhaps I should say, one of the several traditional such presentations). $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented May 24, 2023 at 7:16
  • 3
    $\begingroup$ Not an answer, but Joyal proposed the name "paratopos" for reflective subcats of presheaf cats where the reflector only preserves finite products (rather than finite limits as for toposes). As pointed out above, this preservation of finite products by the reflector is equivalent to the subcategory being an exponential ideal. Here's a video where Joyal makes this suggestion: youtube.com/watch?v=qmCh9KwrQq8 $\endgroup$
    – Jonas Frey
    Commented May 24, 2023 at 14:17
  • 6
    $\begingroup$ The category of small semicategories is not Cartesian-closed. So the identity maps play an important role. $\endgroup$
    – Philippe Gaucher
    Commented May 24, 2023 at 14:53

0

Reset to default

You must log in to answer this question.