path category

There are several concepts often called a path category.


Free category on a directed graph

There is a forgetful functor from small strict categories to quivers. This forgetful functor has a left adjoint, giving the free category or path category of a quiver, whose objects are the vertices of the quiver. The morphisms from aa to bb in this free category are not merely the arrows from aa to bb in the quiver but instead are lists of the form (a n,f n,a n1,,a 2,f 1,a 0)(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0) where n0n \geq 0 is a natural number, a 0,a 1,,a na_0,a_1,\ldots,a_n are vertices of the graph, a=a 0a = a_0, b=a nb = a_n, and for all 0<in0 \lt i \leq n, f i:a i1a if_i\colon a_{i-1} \to a_i is an edge from a i1a_{i-1} to a ia_i. The composition is given by the concatenation

(a n,f n,a n1,,a 2,f 1,a 0)(b m,g m,a m1,,b 2,g 1,b 0):=(a n,f n,a n1,,a 2,f 1,a 0=b m,g m,a m1,,b 2,g 1,b 0)(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0)\circ (b_m,g_m,a_{m-1},\ldots,b_{2},g_1,b_0) := (a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0= b_m,g_m,a_{m-1},\ldots,b_{2},g_1,b_0)

whenever a 0=b ma_0 = b_m, and the target and source maps are given by s(a n,f n,a n1,,a 2,f 1,a 0)=a 0s(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0)=a_0 and t(a n,f n,a n1,,a 2,f 1,a 0)=a nt(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0) = a_n. One informally writes ff for the morphism (b,f,a):ab(b,f,a)\colon a \to b in the free category and the identities of the free category are id a=(a,a)id_a = (a,a); thus f nf n1f 1=(t(f n),f n,t(f n1),,t(f 1),f 1,s(f 1))f_n \circ f_{n-1} \circ \ldots \circ f_1 = (t(f_n),f_n,t(f_{n-1}),\ldots,t(f_1),f_1,s(f_1)). The standard reference is Gabriel–Zisman.

Path category of a space

Given a topological space XX (or some other kind of space with a notion of maps from intervals into it), there are various ways to obtain a small strict category whose objects are the points of XX and whose morphisms are paths in XX. This is also often called a path category.

  • In particular, for every topological space there is its fundamental groupoid whose morphisms are homotopy classes of paths in XX.

  • If XX is a directed space there is a notion of path category called the fundamental category of XX.

  • When XX is a smooth space, there is a notion of path category where less than homotopy of paths is divided out: just thin homotopy. This yields a notion of path groupoid.

  • If parameterized paths are used, there is a way to get a category of paths without dividing out any equivalence relation: this is the Moore path category.

Arrow category

Given a category XX, the functor category [I,X][I,X] for II the interval category might be called a “directed path category of XX” (similar to path space). However, this functor category is referred to instead as the arrow category of XX and sometimes even denoted Arr(X)Arr(X).

Revised on October 9, 2012 01:06:34 by Urs Schreiber (