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
- …

$F(G)$ $F (G)$ , the free category generated by graph $G$ $G$
- 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 $F: \mathbf{Grph} \to \mathbf{Cat}$ $F : Grph \to Cat$
- called the free category functor

for the free category $\mathbf{C}(\Sigma)$ $C (Σ)$ generated by $\Sigma$ $Σ$
- functors $F: \mathbf{C}(\Sigma) \to \mathbf{D}$ $F : C (Σ) \to D$ from a free category are uniquely defined by their image on the signature, i.e by a morphism of signatures $F: \Sigma \to U(\mathbf{D})$ $F : Σ \to U (D)$ for $U$ $U$ 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 $\mathbf{D}$