nLab
commutative diagram

Commutative diagrams

Idea

In category theory, a commutative diagram is a free diagram in which all parallel morphisms obtained by composing morphisms in the diagram agree.

For example that a square diagram of the form

X f Z g g Y f W \array{& X & \overset{f}\rightarrow & Z & \\ g & \downarrow &&\downarrow & g'\\ &Y & \underset{f'}\rightarrow& W & \\ }

commutes is to say that gf=fgg' \circ f = f' \circ g (see also at commuting square).

Definitions

Slick definition

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

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

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 posets, and prosets with certain categories in the usual ways.

Elementary definition

Recall that a free category 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 free 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.

References

Revised on May 5, 2017 14:20:14 by Urs Schreiber (92.218.150.85)