By Barr M.

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Ds1 = Dtm ◦ Dtm−1 ◦ . . 7) should happen to be 0? If, say, m = 0, then we interpret the above equation to be meaningful only if the nodes i and j are the same (you go nowhere on an empty path) and the meaning in this case is that Dsn ◦ Dsn−1 ◦ . . ◦ Ds1 = idDi (you do nothing on an empty path). In particular, a diagram D based on the graph ❘ e i commutes if and only if D(e) is the identity arrow from D(i) to D(i). 1 Diagrams 39 have models that one might think to represent by the diagram ❘ f A but the diagram based on (a) commutes if and only if f = idA , while the diagram based on (b) commutes automatically (no two nodes have more than one path between them so the commutativity condition is vacuous).

F1 ) to be (φ1 (fn ), φ1 (fn−1 ), . . , φ1 (f1 )) and know we get a path in F (H ). That F preserves composition of paths is also clear. 2 as the deﬁning property of freeness. We will describe the property for free categories since we use it later. 2. Let G be a graph and F (G ) the free category generated by G . There is a graph homomorphism with the special name ηG : G − → U (F (G )) which includes a graph G into U (F (G )), the underlying graph of the free category F (G ). The map (ηG )0 is the identity, since the objects of F (G ) are the nodes of G .

For an arrow f of G , (ηG )1 (f ) is the path (f ) of length one. 14, since f and (f ) are really two distinct entities. 16 Proposition Let G be a graph and C a category. Then for every graph homomorphism h : G − → U (C ), there is a unique functor h : F (G ) − → C with the property that U (h) ◦ ηG = h. Proof. If () is the empty path at an object a, we set h() = ida . For an object a of F (G ) (that is, node of G ), deﬁne h(a) = h(a). And for a path (an , an−1 , . . , a1 ), h is ‘map h’: h(an , an−1 , .