Beck's theorem for pseudo-monads
This paper establishes a 2-categorical analogue of Beck's theorem characterising monadic functors. The exact formulation of the main results goes as follows: Theorem 3.5. Let \((U,F,\eta, \varepsilon,s,t): {\mathcal A}\to {\mathcal C}\) be a pseudo-adjunction. The comparison functor \(K:{\mathcal A} \to{\mathcal C}^{\mathbf{T}}\) is locally an equivalence iff for each \(A\), the canonical diagram, equipped with the obvious 2-cells, \[ FU\;FU\;FU\;(A)\;\vcenter{\vtop{=.225 \line{\hss \(FU\;FU\;(\varepsilon_A)\)\hss}\vglue-4pt \line{\leaders\hrule height2.8pt depth-2.2pt\hfill\hglue-4pt\(\longrightarrow\)} \vglue-4pt \line{\leaders\hrule height2.8pt depth-2.2pt\hfill\hglue-4pt\(\;FU(\varepsilon_{FU(A)})\;\)\leaders\hrule height2.8pt depth-2.2pt\hfill\hglue-4pt\(\longrightarrow\)} \vglue-4pt \line{\leaders\hrule height2.8pt depth-2.2pt\hfill\hglue-4pt\(\longrightarrow\)} \vglue-3pt \line{\hss \(^\varepsilon FU\;FU\;(A)\)\hss}}} \;FU\;FU\;(A)\;\vcenter{\vtop{=.225 \line{\hss \(FU\;FU\;(\varepsilon_A)\)\hss} \vglue-4pt \line{\leaders\hrule height2.8pt depth-2.2pt\hfill\hglue-4pt\(\longrightarrow\)} \vglue-4pt \line{\(\longleftarrow\)\hglue-4pt\leaders\hrule height2.8pt depth-2.2pt\hfill\hglue-4pt\(\;F(\eta_{U(A)})\;\)\leaders\hrule height2.8pt depth-2.2pt\hfill} \vglue-4pt \line{\leaders\hrule height2.8pt depth-2.2pt\hfill\hglue-4pt\(\longrightarrow\)}\vglue-3pt \line{\hss \(^\varepsilon FU\;FU\;(A)\)\hss}}} \;FU\;(A)\;\overset\varepsilon{_A}\longrightarrow\;A \] is a pseudo-coequalizer in \(\mathcal A\). Theorem 3.6. Let \((U,F,\eta,\varepsilon,s,t):\mathcal A \to \mathcal C\) be a pseudo-adjunction. The comparison functor of \(K:\mathcal A \to {\mathcal C}^{\text{T}}\) is a bi-equivalence iff \(U\) reflects adjoint equivalences, \(\mathcal A\) has pseudo-coequalizers of \(U\)-absolute codescent objects and \(U\) preserves them.
scientific article; zbMATH DE number 1787496