adjoint triangle theorem

The adjoint triangle theorem


The adjoint triangle theorem is a useful item in the categorist’s toolbox as it gives conditions under which, given a pair of functors and an adjoint, 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).



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 pointwise coequalizer (i.e. UU is of descent type). 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.


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


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.


The hypotheses on UU are satisfied whenever it is monadic.


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.


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.


  • Michael Barr, Charles Wells, Toposes, Triples and Theories , Springer Heidelberg 1985. (Reprinted as TAC reprint no.12 (2005); section 3.7, pp.131ff)

  • Eduardo Dubuc, Adjoint triangles, pp.69-81 in LNM 61 Springer Heidelberg 1968.

  • I. B. Im, G. M. Kelly, Adjoint-Triangle Theorems for Conservative Functors , Bull. Austral. Math. Soc. 36 (1987) pp.133-136.

  • John Power, A unified approach to the lifting of adjoints , Cah. Top. Géom. Diff. Cat. XXIX no.1 (1988) pp.67-77. (numdam)

  • Ross Street, Dominic Verity, The comprehensive factorization and torsors , TAC 23 no.3 (2010) pp.42-75. (abstract)

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

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

  • Fernando Lucatelli Nunes, On biadjoint triangles, TAC

  • Fernando Lucatelli Nunes, On lifting of biadjoints and lax algebras, arXiv

Revised on June 26, 2017 12:21:16 by Mike Shulman (