Is there a term to denote collectively all the constructions of “theories” of some sort, without giving rise to ambiguities?

What I exactly mean is a collective term for monads, operads, clubs, Lawvere theories, and in general all categorical constructions that may have “algebras”, or which describe formal operations in some (possibly generalized) way. In my mind they all encode some sort of “theory”, but the word “theory” seems to be reserved for “algebraic theory”/”Lawvere theory”.

Which term would you use instead?

(I hope it’s clear enough what I’m asking.)

]]>It’s well known that the category of points of the presheaf topos over $Ring_{fp}^{op}$, the dual of the category of finitely presented rings, is the category of all rings (without a size or presentation restriction). In fact this holds for any algebraic theory, not only for the theory of commutative rings. One can learn about this in our entries on *Gabriel-Ulmer duality*, *flat functors*, and Moerdijk/Mac Lane.

But what if we don’t restrict the site to consist only of the compact objects? What are the points of the presheaf topos over the large category $Ring^{op}$, to the extent that the question is meaningful because of size-related issues? What are the points of the presheaf topos over $Ring_{\kappa}^{op}$, the dual of the category of rings admitting a presentation by $\lt \kappa$ many generators and relations, where $\kappa$ is a regular cardinal? (The category $Ring_{\kappa}^{op}$ is essentially small, so the question is definitely meaningful.)

The question can be rephrased in the following way: What is an explicit description of the category of *finite* limit preserving functors $F : Ring_{\kappa}^{op} \to Set$? Any such functor gives rise to a ring by considering $F(\mathbb{Z}[X])$, but unlike in the case $\kappa = \aleph_0$ such a functor is not determined by this ring.

This feels like an extremely basic question to me; it has surely been studied in the literature. I appreciate any pointers! Of course I’ll record any relevant thoughts in the lab.

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