nLab
model

Idea

In model theory, a model of a theory is a realization of the types, operations, relations, and axioms of that theory. In ordinary model theory one usually studies mainly models in sets, but in categorical logic we study models in other categories, especially in topoi.

The term structure is often used to mean a realization of types, operations, and relations in some signature, but not satisfying any particular axioms. This is of course the same as a model for the “empty theory” in that signature, which has the same types, operations, and relations, but no axioms at all. One then talks about whether a given structure is, or is not, a model of a given theory in a given signature.

Definition

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For Lawvere theories

For $\mathrm{Syn}(T)$ the syntactic category of a Lawvere theory, and for $C$ any category with finite limits, a model for $T$ in $C$ is a product-preserving functor

$$N:\mathrm{Syn}(T)\to C.$$N : Syn(T) \to C \,.

The category of models in this case is hence the full subcategory of the functor category $[\mathrm{Syn}(T),C]$ on product-preserving functors.