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Stream: theory: category theory

Topic: 2-universal morphisms/biadjoints for cheap

Max New (Feb 13 2023 at 22:37):

The nice thing about the definition of an adjunction via universal morphisms is that you get a lot of structure for free: If you already have G, you can define F by specifying the action on objects, the unit and its UMP and you get the action of F on morphisms and the naturality of the unit for free (relying on Yoneda lemma and some other facts about natural transformations).

What is the version of this for bi-adjoints? I.e., if I have G : D -> C then a left bi-adjoint should be a functor F : C -> D with a natural equivalence D(F A, B) =~ C(A, G B) but this is even more data than a 1-adjunction, so what is the minimalistic universal morphism version of a bi-adjoint? You have an action on objects and a unit C(A, G F A) such that composition with the unit (G - o \eta) : D(F A, B) -> C(A, G B) is an equivalence of categories. Can it be simplified any further?

Max New (Feb 13 2023 at 23:08):

I guess the most minimalistic then is to prove (G - o \eta) is an equivalence by showing it's fully faithful and eso

Kobe Wullaert (Feb 13 2023 at 23:20):

In the case of (1-)categories you have that functor F:C->D has a left adjoint whenever you have the following:

An assignment on objects G:ob D -> ob C
for every object a, a morphism from a to GFA
The canonical function from D(F a, b) to C(A,G b) is a natural isomorphism of sets.
The bicategorical statement is the same but you now require an equivalence of categories.
In particular, you, also do not need that G is a functor.
So in what sense is this not minimalistic compared to the 1-dimensional one?

Max New (Feb 14 2023 at 01:11):

Just in that an equivalence of categories is more data than a bijection. But then we can use a definition like ff eso to minimize the data needed to define that

Jonas Frey (Feb 14 2023 at 04:47):

If by universal morphism you mean an initial object in $a/U$ , that ~~doesn't work~~ is more subtle for 2-functors.

More precisely, TFAE for a functor $U:B\to A$ between 1-categories:

$U$ has a left adjoint $F$
the functor $\hom(a,U-):B\to Set$ is representable for all $a\in A$
the comma category $a/U$ has an initial object for all $a\in A$

Conditions 2. and 3. are equivalent because $a/U$ is the category of elements of $\hom(a,U-)$ , and a set-valued functor is representable iff its category of elements has a an initial object.

However, the latter equivalence fails in the context of $2$ -categories and $Cat$ -valued functors: (bi)representability of a 2-functor/pseudofunctor $F:X\to Cat$ (where $X$ is a 2-category) cannot be rephrased in terms of a condition on any category of elements alone -- we have to remember the cocartesian arrows.

Jonas Frey (Feb 14 2023 at 05:24):

Specifically, a pseudofunctor $\Phi:X\to Cat$ is bi-representable iff the full sub-2-category of $Elts(\Phi)$ on cocartesian arrows has a bi-initial object $J$ , and for all $A\in Elts(\Phi)$ , the essentially unique cartesian arrow $j:J\to A$ is terminal in $\hom(J,A)$ , where the latter $\hom$ -category also comprises non-cartesian arrows.

Jonas Frey (Feb 14 2023 at 05:25):

In the case of presheaves $\hom(a,-)$ this amounts to looking at lax comma-categories $a//U$ while remembering which of the $1$ -cells are triangles that commute up to iso. (those are the cocartesian arrows)

Jonas Frey (Feb 14 2023 at 05:32):

In the end, it's still a simplification I think in the sense that we can check existence of a biadjoint by checking a bunch of universal properties without having to verify any coherence laws.

Max New (Feb 14 2023 at 18:36):

It will take me some time to stew on this answer. Do you have a reference that covers bi-representability?

Niels van der Weide (Feb 14 2023 at 22:57):

Bi-universal arrows have been studied. See for example Theorem 9.17 in https://arxiv.org/pdf/math/0408298.pdf. The same definition is also used in https://arxiv.org/pdf/1904.06538.pdf (Definition II.4). It seems like this particular definition goes back to "Formal category theory: adjointness for 2-categories" by Gray, but I did not check that source. Note that the definition of biuniversal arrows in these sources are the same as the one given by Kobe. So, one does not need to require G to be a pseudofunctor, and it suffices to prove that a certain functor is an adjoint equivalence of categories (for which it is often convenient to use that eso+ff implies adjoint equivalence).

Jonas Frey (Feb 15 2023 at 01:11):

Max New said:

It will take me some time to stew on this answer. Do you have a reference that covers bi-representability?

I don't have a good reference, unfortunately.

I call a pseudofunctor $F:A\to Cat$ out of a bicategory birepresentable if it is equivalent to a hom-pseudofunctor $Y(a)=A(a,-)$ in the $2$ -category $[A,Cat]$ of pseudofunctors, pseudonatural transformations, and modifications.

Since we have $\hom(Y(a),F)\simeq F(a)$ by the bicategorical Yoneda lemma, any equivalence $Y(a)\simeq F$ corresponds to an object $x\in Fa$ , which we could call a biuniversal element.

In particular, if $F$ is of the form $B(b,U-)$ then we could call a biuniversal element a "biuniversal arrow". This is the notion of biuniversal arrow in @Niels van der Weide 's reference (Def 9.4 here).

Jonas Frey (Feb 15 2023 at 01:13):

So in this sense, the answer to your original question is: Yes, there is a well behaved notion analogous to universal arrows to characterize and exhibit biadjoints.

Jonas Frey (Feb 15 2023 at 01:20):

So let me rephrase what I wrote before:

Biadjoints can be characterized and exhibited in terms of birepresentability of Cat-valued functors, which is great and far more efficient than defining the biadoint, two pseudonatural transformations, and two modifications by hand, and verifying a lot of axioms.
However, birepresentability of Cat-valued pseudofunctors can not be reformulated in terms of (bi)initiality/(bi)terminality.

In general, while in 1-cats basically all universal properties can be reformulated in terms of initiality/terminality, this is not the case for 2-categories. Here, (bi)representability is the fundamental notion.

Philip Saville (Feb 28 2023 at 09:23):

Niels van der Weide said:

Bi-universal arrows have been studied. See for example Theorem 9.17 in https://arxiv.org/pdf/math/0408298.pdf. The same definition is also used in https://arxiv.org/pdf/1904.06538.pdf (Definition II.4). It seems like this particular definition goes back to "Formal category theory: adjointness for 2-categories" by Gray, but I did not check that source.

a bit late to the party, but: in my thesis, which included the work in the second link, I explored Tom Fiore's biuniversal arrows a little further because I wanted to be able to define (preservation of) cartesian closed structure purely in biuniversal-arrow terms. There's nothing there that's surprising from the categorical setting, though.

also, some of the issues Jonas is talking about are discussed for various kinds of limits in this paper and this followup. (Incidentally I believe these show Remark 2.2.5 in my thesis is not right)