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