nLab adjoint triangle theorem

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) and Huq (1970).

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).

Related entries

• adjoint functor theorem

• adjoint lifting theorem

References

• Michael Barr, Charles Wells, section 3.7, pp.131 in: Toposes, Triples, and Theories, Springer (1985), Reprints in Theories and Applications of Categories 12 (2005) 1-287 [tac:tr12]

• Eduardo Dubuc, Adjoint triangles, pp.69-81 in LNM 61 Springer Heidelberg 1968. [doi:10.1007/BFb0077118]

• S. A. Huq, An interpolation theorem for adjoint functors, Proc. AMS 25 (1970) pp.880-883. (pdf)

• Geun Bin Im, Gregory Maxwell Kelly, Some remarks on conservative functors with left adjoints, J. Korean Math. Soc. 23 (1986), no. 1, 19–33, MR87i:18002b, pdf

• I. B. Im, G. M. Kelly, Adjoint-Triangle Theorems for Conservative Functors, Bulletin of the Australian Mathematical Society 36 1 (1987) pp.133-136. [doi:10.1017/S000497270002637X]

• John Power, A unified approach to the lifting of adjoints, Cahiers de Topologie et Géométrie Différentielle Catégoriques 29 1 (1988) 67-77. (numdam)

• Ross Street, Dominic Verity, The comprehensive factorization and torsors, Theory and Applications of Categories 23 3 (2010) 42-75. (TAC)

• Walter Tholen, Adjungierte Dreiecke, Colimites und Kan-Erweiterungen, Mathematische Annalen 217 (1975) pp.121-129. (gdz)

See also Proposition 27 and Corollary 28 of:

• Ross Street, Wood fusion for magmal comonads, arXiv:2311.07088 (2023)

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

• Fernando Lucatelli Nunes, On biadjoint triangles, Theory and Applications of Categories 31 9 (2016) 217-256. TAC

• Fernando Lucatelli Nunes, On lifting of biadjoints and lax algebras, General Algebraic Structures with Applications 9 1 (2018) 29-58. [doi:10.29252/CGASA.9.1.29, arXiv:1607.03087]