free category (path category)

Idea

: the free category on a directed graph (a set of arrows) is the category, with
- objects: vertices of the graph
- morphisms: are tuples of composable edges
  - morphisms are freely generated from directed edges
  - 5.1.2.33 Category Theory for the Sciences:
free categories are also called path categories
- morphisms can be thought of paths in the directed graph
formally, there is a forgetful functor
- from 1-category \(\text{Cat}_{\text{small}}^{\text{strict}}\)of small strict categories
- to that of directed graphs
- ...

\(F(G)\), the free category generated by graph \(G\)
- for any two vertices \(v\) and \(v'\), the hom-set \(\text{Hom}_{F(G)}(v, v')\) is the set of paths in \(G\) from \(v\) to \(v'\)
- identity elements are given by the trivial paths
- composition formula is given by concatenation of paths
a graph homomorphism \(f: G \to G'\) induces a functor \(F(f): F(G) \to F(G')\)
we can say that we have a functor \(F: \mathbf{Grph} \to \mathbf{Cat}\)
- called the free category functor

for the free category \(\mathbf{C}(\Sigma)\) generated by \(\Sigma\)
- functors \(F: \mathbf{C}(\Sigma) \to \mathbf{D}\) from a free category are uniquely defined by their image on the signature, i.e by a morphism of signatures \(F: \Sigma \to U(\mathbf{D})\) for \(U\) the forgetful functor
  - since \(\mathbf{C}\) is "free", it does not really have "structure" on its morphisms. the morphism composition is generated. so the mapping of signature (constituents) is enough to tell us the what maps to things in \(\mathbf{D}\)

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