nLab
theory

Urs Schreiber: here is an attempt of mine to write out the very little that I think I learned of this topic from discussion elsewhere. Probably partly wrong.

A theory is…

Examples are

• essentially algebraic theory

• algebraic theory

  • commutative algebraic theory

  • finitary algbraic theory: Lawvere theory

    • Fermat theory

• geometric theory

A model? for a theory is…

Typically for a theory $T$ there exists a category $C_T$ – the syntactic category $C_T$ – such that a model for $T$ is a functor $C_T\to 𝒯$ into some topos $T$, satisfying certain conditions.

For instance the syntactic categories of Lawvere theories are precisely those categories that have finite cartesian products and in which every object is isomorphic to a finite cartesian power $x^n$ of a distinguished object $x$. A model for a Lawvere theory is precisely a finite product preserving functor $C_T\to 𝒯$.

We say a functor ${𝒯}_1\to {𝒯}_2$ of toposes (for instance a logical morphism or a geometric morphism) preserves a theory $T$ if for every model $C_T\to {𝒯}_1$ of $T$ in ${𝒯}_1$, the composite $C_T\to {𝒯}_1\to {𝒯}_2$ is a model of $T$ in ${𝒯}_2$.

For instance, every geometric morphism preserves every Lawvere theory since, being a right adjoint, it preserves limits, hence finite products.