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This paper states: "logicians often call the category C a ‘theory’, and call the functor F:C → D a ‘model’ of this theory [...] If we think of functors as models, natural transformations are maps between models".

Could someone point me to something that allows me to understand this statement? How can a functor be a model of a theory? How is a natural transformation a map between models. Aand how in the first place, can a category be a theory?

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This is pretty loosey-goosey, but points to a very real phenomenon. For instance, a Lawvere theory is a category with finite products, and a model in sets is a finite product-preserving functor into the category of sets; a model of the Lawvere theory given by the opposite of the category of finitely generated free groups is identified with a group in the traditional sense. But this needs finite product preserving functors, and many similar categorical versions of universal algebra have similar restrictions. For a more direct example of what is described here, one can view a monoid $M$ as a one-object category $BM$, and it is then reasonable to view $BM$ as the "theory of $M$-actions." A functor from $BM$ to the category of sets is identified with an $M$-set, and this gives a way of interpreting the theory in a completely arbitrary category. Concretely, this amounts to saying that an action of $M$ on an object $x$ in some category is a monoid homomorphism from $M$ into the endomorphism monoid of $x$.

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  • $\begingroup$ We want finite products and finite product-preserving functors since most of our theories include constants or $n$-ary operations with $n>1$. There's no reason, however, that the same perspective can't be applied to arbitrary (small) categories and arbitrary functors. This would correspond to multi-sorted equational theories that only have unary function symbols with monoids being the single-sorted case of this. $\endgroup$ – Derek Elkins Jun 28 at 0:28
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    $\begingroup$ @DerekElkins Yes, but I maintain that the quote in the OP is potentially misleading: I do not think it is, in fact, at all common to refer to a plain category as a "theory", among logicians or otherwise, though it is certainly known that such a perspective is available. $\endgroup$ – Kevin Carlson Jun 28 at 0:42
  • $\begingroup$ I certainly agree that (categorical) logicians don't use the word "theory" for "category" or even the domain of a functor as a matter of course. This terminology would be reserved for certain stylized uses or perspectives. The quote is clearer about this in its full context, though it could still be even more clear. $\endgroup$ – Derek Elkins Jun 28 at 0:50

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