theory

:

• In mathematical logic, a theory is a formal language used to precisely axiomatize a certain classes of models
• 3 viewpoints
  • syntactic view
    • the theory itself consists of a set of formulas in the first order language $\text{Lang}(\Sigma)$ of a signature $\Sigma$
  • semantic view
    • given by the class of its models appropriate to that fragment of logic
  • categorical view
    • the logical formalism of a theory $\mathscr{T}$ can be frequently embodied in a syntactic category of term $C_{\mathscr{T}}$, so that models of a theory $\mathscr{T}$ can be identified with functors $C_{\mathscr{T}} \to \text{Set}$ that preserve some (typically property-like) structures on $C_{\mathscr{T}}$ such as certain classes of colimits or of limits, pertinent to the fragment of logic at hand.
    • In fact, the notion of model can be generalized away from Set to more general categories