category theory

Idea

The adjoint triangle theorem in category theory gives conditions under which, given a pair of functors and an adjoint functor, further adjoints exist.

Depending on the specific assumptions, the theorem has several variants. The following gives the most common formulation going back to Dubuc (1968).

Statement

Theorem

Suppose that $U \colon B\to C$ is a functor which has a left adjoint $F \,\colon\, C\to B$ 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 pointwise coequalizer (i.e. $U$ is of descent type). Then for $A$ a category with coequalizers of reflexive pairs, a functor $R \colon A\to B$ 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.

Ramifications

Similarly, the adjoint lifting theorem states conditions on a square of functors in order to ensure the existence of certain adjoints. Since a triangle can be viewed as a square with ‘two sides composed’, it is possible to deduce the adjoint lifting theorem from the adjoint triangle theorem as a corollary.

It is also possible to derive the monadicity theorem from the adjoint triangle theorem Dubuc (1968).

References

Generalizations of the adjoint triangle theorem to 2-categories are considered in

Last revised on November 16, 2023 at 08:42:08. See the history of this page for a list of all contributions to it.