nLab
adjoint triangle theorem

The adjoint triangle theorem

Statement

Theorem

Suppose that U:BCU:B\to C is a functor which has a left adjoint F:CBF:C\to B with the property that the diagram

FUFUϵFUFUϵFUϵ1 B F U F U \;\underoverset{\epsilon F U}{F U \epsilon}{\rightrightarrows}\; F U \xrightarrow{\epsilon} 1_B

is a coequalizer. Suppose that AA is a category with coequalizers of reflexive pairs; then a functor R:ABR:A\to B has a left adjoint if and only if the composite URU R does.

Proof

The direction “only if” is obvious since adjunctions compose. For “if”, let FF' be a left adjoint of URU R, and define L:BAL:B\to A to be the pointwise coequalizer of

FUFUFUϵFU F' U F U \xrightarrow{F' U \epsilon} F' U

and

FUFUFUθUFURFUϵFUFU F' U F U \xrightarrow{F' U \theta U} F' U R F' U \xrightarrow{\epsilon' F' U} F' U

where θ:FRF\theta:F \to R F' is the mate of the equality UR=URU R = U R under the adjunctions FUF\dashv U and FURF'\dashv U R. One then verifies that this works.

Remark

The hypotheses on UU are satisfied whenever it is monadic.

Remark

In fact, it suffices to assume that each counit ϵ:FUbb\epsilon : F U b \to b is a regular epimorphism, rather than it is the coequalizer of a specific given pair of maps. See (Street-Verity), Lemma 2.1.

Applications

The adjoint lifting theorem is a corollary.

References

Created on October 12, 2012 22:56:51 by Mike Shulman (192.16.204.218)