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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. (Even more interesting would be if this somehow singles out the theory of categories.)

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.

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?

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. (Even more interesting would be if this somehow singles out the theory of categories.)

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 an algebraic 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?

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?