# How can we see a functor $F:C\to D$ as a “model” of the “theory” $C$?

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?

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 $$x$$.
• 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. – Derek Elkins Jun 28 at 0:28