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