commutative diagram

Commutative diagrams


A commutative diagram is a diagram in which composition is path-independent?.


Slick definition

For our purposes, a diagram DD in a category CC consists of a quiver JJ and a functor to CC from the free category on JJ:

JDC,Ja quiver. J \overset{D}\to C ,\; J\;\text{a quiver}.

Then this diagram DD commutes if this functor DD factors (up to natural isomorphism) through a poset PP:

JPCJC,Pa poset; J \to P \to C \;\cong\; J \to C ,\; P\;\text{a poset} ;

or equivalently (treating CC as a strict category) if the functor factors up to equality through a proset QQ:

JQCJC,Qa proset. J \to Q \to C \;\cong\; J \to C ,\; Q\;\text{a proset} .

In the above, we are identifying quivers, posets, and prosets with certain categories in the usual ways.

Elementary definition

Recall that a quiver JJ consists of a set VV of vertices, a set EE of edges, and two functions s,t:EVs,t\colon E \to V. Given a category CC, a diagram DD of shape JJ in a category CC is consists of a map from VV to the objects of CC and a map from EE to the morphisms of CC, both denoted FF, such that F(s(e))=S(F(e))F(s(e)) = S(F(e)) and F(t(e))=T(F(e))F(t(e)) = T(F(e)) for each edge ee, where S,TS,T are the source and target maps in CC.

Recall that a path pp in JJ consists of a list (v 0,v 1,,v n)(v_0,v_1,\ldots,v_n) of vertices and a list (e 1,,e n)(e_1,\ldots,e_n) of edges such that s(e i)=v i1s(e_i) = v_{i-1} and t(e i)=v it(e_i) = v_{i} for each ii, where nn is any natural number (possibly zero). We say that v 0v_0 is the source of the path and that v nv_n is its target. Given a path pp and a diagram DD, the composite of pp under DD is the composite F(e 1);;F(e n):F(v 0)F(v n)F(e_1);\ldots;F(e_n)\colon F(v_0) \to F(v_n) in CC.

A diagram DD commutes if, given any two vertices x,yx,y in JJ and any two paths p,pp,p' with source xx and target yy, the composites of pp and pp' under DD are equal in CC.

Revised on February 13, 2011 19:26:57 by Toby Bartels (