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Stream: learning: questions

Topic: why specified isomorphisms?

Leopold Schlicht (May 31 2022 at 14:59):

A $2$-category is a class of objects $A, B, \dots$, a family of hom-categories $\hom(A,B)$, a family of composition functors $\hom(A,B)\times \hom(B,C)\to\hom(A,C)$, and a family of identity functors $1\to \hom(A,A)$, together with specified natural isomorphisms $(h\circ g)\circ f\cong h\circ (g\circ f)$, $f\circ \mathrm{id}_A\cong f$, and $\mathrm{id}_B\circ f\cong f$ (such that a few coherence diagrams commute). Often people emphasize that it is important that the natural isomorphisms $(h\circ g)\circ f\cong h\circ (g\circ f)$, $f\circ \mathrm{id}_A\cong f$, and $\mathrm{id}_B\circ f\cong f$ are really part of the data of a 2-category.

So I wonder: what goes wrong if we change the definition so that a 2-category consists only of a class of objects, a family of hom-categories, a family of composition functors, and a family of identity functors with the property that there merely exist some natural isomorphisms $(h\circ g)\circ f\cong h\circ (g\circ f)$, $f\circ \mathrm{id}_A\cong f$, and $\mathrm{id}_B\circ f\cong f$?

The same question can be asked for 2-functors (which include, say, a natural isomorphism $F(g\circ f)\cong F(g)\circ F(f)$ as part of the data), monoidal categories (which include, say, a natural isomorphism $A\otimes (B\otimes C)\cong (A\otimes B)\otimes C$ as part of the data), or monoidal functors (which include, say, a natural isomorphism $F(A)\otimes F(B)\to F(A\otimes B)$ as part of the data).

Nathanael Arkor (May 31 2022 at 15:52):

(Note that what you're calling a "2-category" is conventionally called a "bicategory", with "2-category" denoting the strict version.)

Kenji Maillard (May 31 2022 at 18:21):

I don't have an answer to the question and I would be interested understanding more precisely what goes wrong: if we only postulate the mere existence of the isomorphisms without the coherence equations, I would expect that we cannot strictify these non-coherent bicategories into an equivalent 2-category.

Mike Shulman (May 31 2022 at 18:24):

How would you define a monad in such an "incoherent bicategory"? Normally a monad is a 1-cell $t : A\to A$ with 2-cells $\mu : t\circ t \to t$ and $\eta : {\rm id} \to t$ such that some diagrams commute, one of which is that the composite $t \circ (t\circ t) \to t\circ t \to t$ is equal to the composite $t \circ (t\circ t) \cong (t\circ t) \circ t \to t\circ t \to t$. If you don't have a specified isomorphism $t \circ (t\circ t) \cong (t\circ t)\circ t$, this doesn't make sense.

Leopold Schlicht (Jun 01 2022 at 12:22):

Nathanael Arkor said:

(Note that what you're calling a "2-category" is conventionally called a "bicategory", with "2-category" denoting the strict version.)

My impression is that the modern convention is that "2-category" refers to the weak notion by default.

Leopold Schlicht (Jun 01 2022 at 12:25):

@Mike Shulman Thanks! Is there a more basic example of what goes wrong? Monads are quite advanced and I would expect that something goes wrong already for more basic concepts.

Amar Hadzihasanovic (Jun 01 2022 at 12:53):

It comes down to what a bicategory (or a monoidal category, for that matter) is “for”. I would say that it is a space in which you can interpret and compose certain pasting diagrams. The interpretation of pasting diagrams in bicategories relies on the coherence theorem.

Amar Hadzihasanovic (Jun 01 2022 at 12:55):

(Of course Mike's example of monads is a particular one of this, since a monad is defined by an equational theory of 2d pasting diagrams)

Reid Barton (Jun 01 2022 at 12:55):

Leopold Schlicht said:

Mike Shulman Thanks! Is there a more basic example of what goes wrong? Monads are quite advanced and I would expect that something goes wrong already for more basic concepts.

You can replace bicategory by monoidal category, and monad by monoid.

Leopold Schlicht (Jun 01 2022 at 13:09):

@Reid Barton Alright, but is there another example?

Reid Barton (Jun 01 2022 at 13:11):

What's wrong with this one?
I'd say the question is rather: why would you expect the notion involving an unspecified isomorphism to be useful for anything?

Leopold Schlicht (Jun 01 2022 at 13:20):

@Reid Barton

What's wrong with this one?

Nothing. I'm just asking for another one.

I'd say the question is rather: why would you expect the notion involving an unspecified isomorphism to be useful for anything?

Maybe that's your question, but I asked a different question. :P

Reid Barton (Jun 01 2022 at 13:22):

Sure. Personally, I don't find your question very interesting.

Amar Hadzihasanovic (Jun 01 2022 at 13:22):

@Leopold Schlicht is adjunctions in a bicategory to your liking? To compose $\eta L$ and $R \varepsilon$ you need to interpose a specified isomorphism between $(LR)L$ and $L(RL)$.

Reid Barton (Jun 01 2022 at 13:23):

A better version would be: Ask for a specified isomorphism, but drop the coherence condition(s).

Amar Hadzihasanovic (Jun 01 2022 at 13:23):

Here $L$ and $R$ are the adjoint 1-morphisms and $\eta, \varepsilon$ the unit and counit of the adjunction.

Amar Hadzihasanovic (Jun 01 2022 at 13:29):

Monads and adjunctions are the simplest and most common structures you can interpret in a bicategory, which involve the composition of a chain of at least three 1-morphisms (you won't need associators for less than that, although you may find “smaller” problematic examples involving only unitors.)

Leopold Schlicht (Jun 01 2022 at 13:31):

Reid Barton said:

Sure. Personally, I don't find your question very interesting.

Thanks. You are free to ignore my question if you are not interested in it. There's no need to tell me. :-)

Leopold Schlicht (Jun 01 2022 at 13:32):

@Amar Hadzihasanovic Thanks!

Nathanael Arkor (Jun 01 2022 at 13:33):

Leopold Schlicht said:

Nathanael Arkor said:

(Note that what you're calling a "2-category" is conventionally called a "bicategory", with "2-category" denoting the strict version.)

My impression is that the modern convention is that "2-category" refers to the weak notion by default.

While the nLab often uses the convention of calling bicategories "2-categories", you will rarely find this convention in the literature. You may find "weak 2-category", but in my experience it is much more common to find simply "bicategory".

Leopold Schlicht (Jun 01 2022 at 13:40):

Mhm. As another example, Lurie's Kerodon (which is a major reference in the area) uses "2-category" instead of "bicategory".

John Baez (Jun 02 2022 at 17:33):

Leopold Schlicht said:

Nathanael Arkor said:

(Note that what you're calling a "2-category" is conventionally called a "bicategory", with "2-category" denoting the strict version.)

My impression is that the modern convention is that "2-category" refers to the weak notion by default.

This is a dangerous assumption, since only a minority of researchers use this modern convention. (I used to use it, but I gave up when I realized it was confusing to many people.)

Mike Shulman (Jun 02 2022 at 17:39):

On one hand, when working at a two- or three-dimensional level, it's pretty universal that "2-category" means "strict 2-category", with "bicategory" referring to the specific classical Benabou model of weak 2-category. Similarly one has "3-category" and "tricategory".

On the other hand, when working at an $n$-categorical level, one generally says "$n$-category" to mean a weak $n$-category using some model of such, e.g. quasicategories if discussing $(n,1)$-categories, or something fancier otherwise. In that case people do sometimes specialize to $n=2$ and say "2-category" to refer to the instance of this general model in two dimensions, e.g. a quasicategory that is trivial above dimension 2. One might also do this when working or speaking "model-independently" with higher categories.

But I think it's vanishingly rare to use unadorned "2-category" to refer to the classical algebraic Benabou model traditionally called "bicategory".