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On lifting of biadjoints and lax algebras. (English) Zbl 1454.18018

E. Dubuc [Lect. Notes Math. 61, 69–91 (1968; Zbl 0172.02103)] published what is known as an “adjoint triangle theorem” which states that given functors \(R\colon A\to B\), \(U'\colon A\to C\) and \(U\colon B\to C\) such that \(UR=U'\), \(U\) and \(U'\) both have a left adjoint and \(U\) is of “descent type”, then \(R\colon A\to B\) also has a left adjoint.
The main result (Theorem 2.3) of the paper is an extension of the above theorem in the realm of 2-categories and pseudofunctors. As a direct consequence of it, one has the following result (Theorem 5.2 “Biadjoint triangle theorem”). Let \(A,C\) be 2-categories and let \(\mathcal{T}\) be a pseudomonad on \(C\). Denote by \(B\) the 2-category of lax \(\mathcal{T}\)-algebras and pseudomorphism between them. Let \(U\colon B\to C\) be the obvious forgetful 2-functor into C. Let \(U'\colon A\to C\) be a left biadjoint and let \(R\colon A\to B\) be a pseudofunctor. Let us assume that \(UR=U'\). If \(A\) has some “codescent objects” (which means that some weighted bicolimits exist), then \(R\) is a right biadjoint.

MSC:

18N10 2-categories, bicategories, double categories
18N15 2-dimensional monad theory
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)

Citations:

Zbl 0172.02103
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References:

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