## Franklin's Notes

### Category of models

If $\mathscr{L}$ is a first-order language including constant, function, and relation symbols, and $\mathrm{T}$ is a theory over this language, then there is a category $\mathsf{Model}_\mathrm{T}$ whose objects are structures/models satisfying $\mathrm{T}$, and whose morphisms are isomorphisms. Riehl lists several special cases of such categories which are well-known by other names, such as:

• $\mathsf{Group}$ is the category which has groups as its objects and group homomorphisms as its morphisms.

• $\mathsf{Graph}$ is the category whose objects are graphs and whose morphisms are incidence-preserving functions.

• $\mathsf{Poset}$ is the category with objects given by partially ordered sets and morphisms given by order-preserving (increasing) functions.