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Stream: learning: questions

Topic: 3-categorical notions of universal arrow, adjunction

Patrick Nicodemus (Sep 21 2022 at 20:55):

Let $A$, $B$ be $2$-categories, and $G: B\to A$ a $2$-functor. For $x$ a $0$-cell in $A$ it appears to me that there should be notions of "strong" and "weak" universal arrow of $x$ into $G$ given by asking for either an equivalence of categories $Hom(x, Gb)\simeq Hom(F(x), b)$ or an adjunction between these categories; either way the equivalence / adjunction should be natural in $b$.

Similarly one should be able to define a "strong" or "weak" 2-adjoint to G.

I am looking for basic papers on these notions where their definition and basic properties are established. What do we know about them, etc.

Tobias Schmude (Sep 22 2022 at 06:37):

The concepts of a 2-adjunction with Hom-equivalences or Hom-adjunctions are called [[biadjunction]] and [[lax 2-adjunction]] on the nlab, there's also a bit of discussion and some sources.

Niels van der Weide (Sep 22 2022 at 07:50):

I think the main references would be:

• 'A Coherent Approach to Pseudomonads' by Stephen Lack. In this paper, pseudoadjoints are defined and it is shown that every pseudoadjunction gives rise to a pseudomonad
• 'Enriched categories, internal categories and change of base' by Dominic Verity also contains a section about adjunctions of bicategories (called 'Local adjunctions')
• 'Biequivalences in tricategories' by Nick Gurski (and also his PhD thesis). This approach defines the notion of biequivalence in arbitrary tricategories. This is more general, but it also gives more complicated diagrams.
• 'Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory' by Thomas Fiore also has some stuff about biadjoints, and here universal arrows are introduced and shown to give rise to left biadjoints.

Patrick Nicodemus (Sep 22 2022 at 09:51):

Thanks!