Timeline for Why is $\rm{Cat}$ a Cartesian-closed category?
Current License: CC BY-SA 4.0
22 events
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| Jan 2 at 19:04 | comment | added | varkor | I believe this is essentially a duplicate of this question. | |
| May 28, 2023 at 20:50 | comment | added | Arshak Aivazian | I mean patterns among essentially algebraic theories such as commutativity. Yes, I also wanted to suggest delta-generated spaces when you wrote about compactly generated ones. Only they are not a finitary theory, in this sense $\mathrm{Cat}$ seems more interesting/fundamental. | |
| May 28, 2023 at 20:16 | comment | added | Tim Campion | If the goal is to find deep patterns, then it may be a bit artificial to restrict to the presentable case. At any rate, $\Delta$-generated spaces are locally presentable and cartesian closed, and perhaps just as mysterious as $Cat$ in that regard. | |
| May 28, 2023 at 20:09 | comment | added | Arshak Aivazian | This explains many examples of closed monoidal categories that we encounter in mathematics, but not $\mathrm{Cat}$. I wonder what deep pattern (class of theories) is behind this particular example. | |
| May 28, 2023 at 20:09 | comment | added | Arshak Aivazian | Compactly generated spaces are not locally presentable categories, are they? So they don't apply to my question, if that's what you're talking about. I position Grothendieck topoi here as initial contexts for doing mathematics. Then we can interpret certain theories in them. It turns out that there is a common pattern: the category of algebras of a commutative algebraic theory (in a Grothendieck topos) has a natural closed monoidal structure. | |
| May 28, 2023 at 20:02 | comment | added | Tim Campion | Is the cartesian closedness of $Cat$ more mysterious than the cartesian closedness of compactly-generated spaces, or less mysterious? | |
| May 24, 2023 at 14:53 | comment | added | Philippe Gaucher | The category of small semicategories is not Cartesian-closed. So the identity maps play an important role. | |
| May 24, 2023 at 14:17 | comment | added | Jonas Frey | 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 | |
| May 24, 2023 at 12:36 | comment | added | Zhen Lin | I think there's a joke answer one can give here. It is not hard to check that the reflector of any reflective exponential ideal – such as $\textbf{Cat}$ in $\textbf{sSet}$ – preserves cartesian products. But a reflector that preserves cartesian monoids is a commutative monad (with respect to the cartesian monoidal structure, of course). Therefore it is covered by your "super example". | |
| May 24, 2023 at 8:30 | comment | added | Fernando Muro | It fits into the class of locally presentable categories whose Cartesian product has a right adjoint. | |
| May 24, 2023 at 7:16 | comment | added | Peter LeFanu Lumsdaine | 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). | |
| May 24, 2023 at 0:36 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 24, 2023 at 0:33 | comment | added | Arshak Aivazian | Thank you! I realized that I was interested in the explanation from a syntactic point of view (as in my examples before). I somewhat reflected that clarification in the question now, I apologize. From this point of view, $\rm{Cat}$ is still just lucky to be embedded that way in simplicial sets (or rather, I don't know of a class of limit theories that have that property). | |
| May 24, 2023 at 0:21 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 24, 2023 at 0:19 | comment | added | Tim Campion | One natural way to answer the question would be to show that $Cat$ has some sort of universal property qua cartesian closed category. For instance, it might be enlightening to try to understand the category $Fun^{L,\times}(Cat, \mathcal K)$ of functors $Cat \to \mathcal K$ which preserve colimits and finite products, where $\mathcal K$ is an arbitrary cartesian closed locally presentable category. It may also be relevant to mention that there are exactly two monoidal biclosed structures on $Cat$. | |
| May 24, 2023 at 0:19 | comment | added | Todd Trimble | 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.) | |
| May 24, 2023 at 0:17 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 24, 2023 at 0:11 | comment | added | Arshak Aivazian | @ToddTrimble Thanks for your attention! Perhaps I didn't articulate well what I mean: I would like to put $\rm{Cat}$ into a natural family of locally presentable categories that are Cartesian-closed for some general reason. I'll add this clarification to the text of the question. | |
| May 24, 2023 at 0:09 | comment | added | Todd Trimble | Pos can piggyback on Cat because (like the Segal conditions) what it means for a category to be a preorder or poset can be described by certain limit conditions. One could say that Cat is an "exponential ideal" of s-sets (if C is a category and X is an s-set, then the s-set C^X is a category), and likewise, Pos is an exponential ideal of Cat. | |
| May 24, 2023 at 0:05 | comment | added | Todd Trimble | 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. | |
| May 23, 2023 at 23:47 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 23, 2023 at 23:33 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |