nLab
adjoint triangle theorem

Statement

Theorem

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

$$FUFU\underset{ϵFU}{\overset{FUϵ}{\rightrightarrows }}FU\stackrel{ϵ}{\to }{1}_B$$F U F U \;\underoverset{\epsilon F U}{F U \epsilon}{\rightrightarrows}\; F U \xrightarrow{\epsilon} 1_B

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

Proof

The direction “only if” is obvious since adjunctions compose. For “if”, let $F\prime$ be a left adjoint of $UR$, and define $L:B\to A$ to be the pointwise coequalizer of

$$F\prime UFU\stackrel{F\prime Uϵ}{\to }F\prime U$$F' U F U \xrightarrow{F' U \epsilon} F' U

and

$$F\prime UFU\stackrel{F\prime U\theta U}{\to }F\prime URF\prime U\stackrel{ϵ\prime F\prime U}{\to }F\prime U$$F' U F U \xrightarrow{F' U \theta U} F' U R F' U \xrightarrow{\epsilon' F' U} F' U

where $\theta :F\to RF\prime$ is the mate of the equality $UR=UR$ under the adjunctions $F⊣U$ and $F\prime ⊣UR$. One then verifies that this works.

Remark

The hypotheses on $U$ are satisfied whenever it is monadic.

Remark

In fact, it suffices to assume that each counit $ϵ:FUb\to b$ 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

• Eduardo Dubuc, “Adjoint triangles”, Lecture Notes in Mathematics 61

• Ross Street and Dominic Verity, “The comprehensive factorization and torsors”, 2010 TAC.