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Stream: learning: questions

Topic: The universal property of Adj

fosco (Dec 19 2023 at 21:05):

I have only recently noticed that if one considers the two embeddings of $Cat$ into $Prof$, one co-contravariant and one contra-covariant (specifying the variance on 1- and 2-cells), and considers their pullback, one does not obtain something trivial at all: they get the (2-)category of adjunctions! More precisely, consider the cospan

$Cat \xrightarrow{p} Prof \xleftarrow q Cat^{op}$

where $p$ sends a functor F to hom(F,1), and inverts 2-cells, and $q$ sends F to hom(1,F), and keeps 2-cells the same way. Their (isocomma? pseudopullback?) has 1-cells the pairs $(F,G)$ such that $\hom(F,1)\cong\hom(1,G)$.

Why? What's the deep reason for this and how general it is, if at all?

Mike Shulman (Dec 19 2023 at 21:18):

If you want to get literally the 2-category whose objects are categories and whose morphisms are adjunctions, then I think you have to take this isocomma in the 2-category of bicategories, pseudofunctors, and icons. If you take it in the tricategory of bicategories, then the objects of the pullback will be triples consisting of two categories and a Morita equivalence between them; although I expect the bicategory you get would be biequivalent to the 2-category of Cauchy complete categories and adjunctions.

Mike Shulman (Dec 19 2023 at 21:19):

This should be true for any proarrow equipment. In a sense, of course, it's just the definition of "adjunction"...

Mike Shulman (Dec 19 2023 at 21:19):

(the one in terms of hom-sets)

fosco (Dec 19 2023 at 22:09):

I expected it to be something exportable to a generic equipment, probably I feel like such a neat characterization should have been mentioned elsewhere; perhaps the necessity to use icons prevented this from being stated before icons were introduced?

Mike Shulman (Dec 20 2023 at 01:09):

I would guess it's more that no one had any use for that expression of Adj as a pullback.

fosco (Dec 20 2023 at 08:52):

I see, thank you!