signature

• In logic, a signature is a collection of data which prescribes the ‘alphabet’ of non-logical symbols of a logical theory, saying which operations and predicates are taken to be primitive.
• The signature of a structure is the union of the collection of constant symbols, function symbols, and relation symbols in the language of said structure.

from DisCoPy

• a simple signature $\Sigma$ is given by a pair of sets and a pair of functions
  • $\Sigma_{0}$: the set of generating objects
  • $\Sigma_{1}$: the set of generating morphisms
  • $\text{dom}: \Sigma_{1} \to \Sigma_{0}$
  • $\text{cod}: \Sigma_{1} \to \Sigma_{0}$
  • ↳ a graph
• A morphism of signatures $F: \Sigma \to \Sigma'$ is a pair of functions
  • $F_{0}: \Sigma_{0} \to \Sigma'_{0}$
  • $F_{1}: \Sigma_{1} \to \Sigma'_{1}$
  • which commute with $\text{dom}$ and $\text{cod}$
    • $\forall$ morphism $a \in \Sigma_{1}$
    • $F_{0}(dom(a)) = \text{dom}(F_{1}(a))$
    • $F_{0}(cod(a)) = \text{cod}(F_{1}(a))$
    • ↳ my guess
  • graph homomorphism?

free category generated by signature

• given a simple signature $\Sigma$, the free category $\mathbf{C}(\Sigma)$ generated by $\Sigma$ is defined as
  • objects: given by $\Sigma_{0}$
  • morphisms: all possible combinations of arrows in $\Sigma_{1}$ that connects
    • identity is the empty list of path
    • composition is given by list concatenation