adjoint triangle theorem
The adjoint triangle theorem
Suppose that is a functor which has a left adjoint with the property that the diagram
is a coequalizer. Suppose that is a category with coequalizers of reflexive pairs; then a functor has a left adjoint if and only if the composite does.
The direction “only if” is obvious since adjunctions compose. For “if”, let be a left adjoint of , and define to be the pointwise coequalizer of
where is the mate of the equality under the adjunctions and . One then verifies that this works.
The adjoint lifting theorem is a corollary.
Created on October 12, 2012 22:56:51
by Mike Shulman