category theory

# The 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

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

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 $U R$ does.

###### Proof

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

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

and

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

where $\theta:F \to R F'$ is the mate of the equality $U R = U R$ under the adjunctions $F\dashv U$ and $F'\dashv U R$. 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 $\epsilon : F U b \to b$ 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)