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Recall that the microcosm principle in category theory asserts that algebraic structures live most naturally inside categories equipped with categorified versions of the algebraic structures in question.

I am curious about a related 'macrocosm principle':

Say that a theory $T$ satisfies the macrocosm principle iff the collection $\mathcal{T}$ of all set-sized models of $T$ also carries a canonical $T$-model structure.

An obvious example is the theory of categories, since ${\bf Cat}$ can be viewed as a $1$-category. Are there any others?

Edit: As pointed out by varkor and Maxime in the comments, the word 'canonical' is doing pretty much all of the heavy lifting in the above "definition" (as it sometimes tends to do).

Accordingly it is not really a definition of the principle I'm interested in, rather a heuristic; a more appropriate first question is thusly

What is the correct formal definition of the macrocosm principle?

Once this is settled, examples besides the category of of categories would be very interesting to me.

Alec Rhea
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  • Any reason for the downvote? – Alec Rhea Oct 31 '22 at 22:50
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    The theory of categories is not algebraic. Furthermore, if "collection" means set/class, then the collection of categories may be equipped with trivial categorical structure, so it's not a particularly interesting example. As such, I don't see that there's a good motivating example for this question. (However, I haven't downvoted the question.) – varkor Oct 31 '22 at 23:21
  • @varkor Thank you; I will remove reference to algebraic theories. Avoiding trivial structures was why I included the word ‘canonical’, as there is a canonical nontrivial category structure on the collection of all set-sized categories, but if this question doesn’t strike others as interesting I am fine removing it. – Alec Rhea Oct 31 '22 at 23:28
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    I like this question but I think a more appropriate question would be : what is a precise statement of the macrocosm principle ? This, I think no one knows (and if someone knows, they should write something !!) – Maxime Ramzi Nov 01 '22 at 08:35
  • @MaximeRamzi I see! I’ll edit accordingly today, thank you. – Alec Rhea Nov 01 '22 at 13:12
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    As I expressed above, I do not know the answer to your main question. However, you ask about examples : there are many examples beyond the category of categories, that motivate the idea that there should be a "macrocosm principle". Not least is the notion of semi-additivity : the category of semiadditive categories is itself semi-additive. Or maybe, to phrase this in terms of structures as you do : the category of commutative monoids is itself a (highly coherent, up-to-homotopy) commutative monoid. – Maxime Ramzi Nov 01 '22 at 17:56
  • @MaximeRamzi It was the ‘up to homotopy’ part that made me hesitate on monoids, but the semi-additive example is a good ‘strict’ one. – Alec Rhea Nov 01 '22 at 18:08
  • Any general enough macrocosm principle will have to allow things up to (coherent) homotopy, because moving from sets to categories introduces higher categorical structures. Another key example of the macrocosm principle lies in the theory of operads: to define an $O$-algebra in $C$, one needs an $O$-monoidal structue on $C$. But the $O$-monoidal structure on $C$ only makes $C$ an $O$-algebra in an up-to-coherent-homotopy sense – Maxime Ramzi Nov 01 '22 at 18:20
  • @MaximeRamzi Very cool — naively I had wondered if the differing levels of strictification might serve to classify different theories in a certain sense, so some theories satisfy a ‘strict macrocosm principle’ and we have a hierarchy of ‘$n$-lax macrocosm principles’ where we allow for $n$ levels of ‘up to coherent isomorphism’. – Alec Rhea Nov 01 '22 at 19:52
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    I am really not convinced by the premise of this question, or at least that it holds beyond vanilla structures: the category of fields looks nothing like a field. However, I will offer a positive example: categories of domains (in computer science) enjoy a "cofiltered limit / filtered colimit coincidence" that makes them look like domains. This example is what came to mind when I first heard of Voevodsky's Univalence Principle. – Paul Taylor Nov 01 '22 at 20:48
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    @PaulTaylor I can respect that, this is just something that occurred to me while making pizza dough (maybe I was considering making many tiny pizzas vs one big pizza — who knows). I appreciate the good-faith example! – Alec Rhea Nov 01 '22 at 22:07

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