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Stream: theory: category theory

Topic: lax morphism classifiers

sarahzrf (Jan 30 2021 at 02:20):

i'm a bit dismayed to find this link to a nonexistent page: image.png

i'm rather interested in the idea that we can replace lax morphisms out of something with pseudo morphisms out of some classifying object—i'd noticed a couple of instances of this, and i was wondering whether there was a general phenomenon

sarahzrf (Jan 30 2021 at 02:21):

does anyone know if this is written up somewhere? mostly for the case with stuff in Cat? preferably with explicit descriptions of what the classifier looks like

sarahzrf (Jan 30 2021 at 02:22):

for example, id like to see if theres some construction which you can feed 1 into and get out the delooping of the simplex category

Amar Hadzihasanovic (Jan 30 2021 at 09:59):

I don't know where (if anywhere) it's spelled out but I did work this out once for the case of 2-categories and strict functors, ie, the “lax morphism classifier” functor as the left adjoint to the inclusion of (2-categories, strict functors) into (2-categories, lax functors). I don't think that there is such a thing for pseudofunctors if not in a weak sense.

In a sense there's a systematic way of computing these things... namely, you compute it using the universal property on some generating set of objects and then because it's a left adjoint you extend along colimits.
The typical generating set for 2-categories would be the n-globes $O^0, O^1, O^2$ aka the classifiers for 0-cells, 1-cells, 2-cells and the classifiers $O^2 \#_0 O^2$, $O^2 \#_1 O^2$ for the horizontal and the vertical composition.
So for example you use that $\mathrm{Lax}(O^n, X)$ is $\mathrm{Str}(LO^n, X)$ to see what $LO^n$ has to be. In the case of $O^0$, which is the point/the terminal 2-cat, a lax functor from $O^0$ to $X$ is a monad in $X$, so $LO^0$ is going to be the walking monoid (aka the delooping of the augmented simplex category, as you said).
Because you will extend along colimits, this tells you that $LX$ in general will have a “walking monoid” freely attached to every 0-cell.

Amar Hadzihasanovic (Jan 30 2021 at 10:04):

Then $O^1$ is the walking arrow; looking at what a lax functor from $O^1$ to $X$ is, you get that $LO^1$ is going to be two monoids on the two 0-cells, together with a left action of the first one, and a right action of the second one on the only 1-cell... So it's a “walking bimodule”.

Amar Hadzihasanovic (Jan 30 2021 at 10:05):

And $LO^2$ is going to be the “walking bimodule homomorphism”!

Amar Hadzihasanovic (Jan 30 2021 at 10:10):

$L(O^2 \#_1 O^2)$ is not super-interesting: it's just three bimodules for the same pair of monoids and a pair of composable bimodule homomorphisms.

Amar Hadzihasanovic (Jan 30 2021 at 10:23):

Let's do $O^1 \#_0 O^1$ as a step to $O^2 \#_0 O^2$. This is the free category on a pair of arrows $a: 0 \to 1$ and $b: 1 \to 2$. You know that you'll have monoids on $0, 1, 2$ acting on $a$ and $b$ from the left or right, appropriately.

In addition $L(O^1 \#_0 O^1)$ is going to have a third 1-cell $ab: 0 \to 2$, also with actions on the left and right, that commute with it, and a 2-cell $\varphi: a \#_0 b \Rightarrow ab$, which has to commute with the various actions and also be such that (right action on $a$ followed by $\varphi$ = left action on $b$ followed by $\varphi$), it's like the condition you put on the tensor of bimodules...

Amar Hadzihasanovic (Jan 30 2021 at 10:27):

And finally $L(O^2 \#_0 O^2)$ is going to be two vertically composable instances of this, say $\varphi: a \#_0 b \Rightarrow ab$ and $\varphi': a' \#_0 b' \Rightarrow a'b'$, together with bimodule homomorphisms $\alpha: a \Rightarrow a'$, $\beta: b \Rightarrow b'$, and $\gamma: ab \Rightarrow a'b'$ satisfying

$(\alpha \#_0 \beta) \#_1 \varphi' = \varphi \#_1 \gamma$

Amar Hadzihasanovic (Jan 30 2021 at 10:33):

Now that I think of it, though, that set of objects is generating for the category of 2-categories and strict functors; it seems likely that you need some more for 2-categories and lax functors. At least $O^1 \#_0 O^1 \#_0 O^1$, which will give you an “associativity” equation between 2-cells $ab\#_0 c \Rightarrow abc$ and $a \#_0 bc \Rightarrow abc$...

Amar Hadzihasanovic (Jan 30 2021 at 11:01):

I just went forward with this answer and it's a bit of a mess, I'm sure there's a more elegant way of describing it :grimacing:

Rune Haugseng (Jan 30 2021 at 11:06):

I think this "lax envelope" L(X) of a 2-category X has an explicit description, generalizing the enveloping monoidal category of a non-symmetric operad

Rune Haugseng (Jan 30 2021 at 11:12):

The objects are the objects of X, the morphisms x -> y are sequences (f_1,..., f_n) (n \geq 0) of composable morphisms f_i : x_{i-1} -> x_i (with x_0 = x, x_n = y), with composition = concatenation, and 2-morphisms (f_1,...,f_n) -> (g_1,..,g_m) are given by a map of ordered sets h : {1,..,m} -> {1,...,n} + 2-morphisms in X from f_{h(i)} o ... o f_{h(i+1)-1} -> g_i

Rune Haugseng (Jan 30 2021 at 11:13):

(I probably got the indices wrong there, I'm trying to translate a construction in a different context and haven't had any coffee yet...)

Amar Hadzihasanovic (Jan 30 2021 at 11:18):

I think that seems correct for the classifier of strictly unital lax functors, am I right?
To have general lax functors you would also need to add a new morphism $e_x: x \to x$ for each object, so morphisms are composable sequences of morphisms of $X$ intertwined with these new endomorphisms, and the 2-morphisms are what you said, turning all the $e_x$ into the corresponding units of $X$.

Amar Hadzihasanovic (Jan 30 2021 at 11:24):

Oh I see that you already get this because the units of $X$ are different from the empty sequences. Yes then, that seems completely right.

Mike Shulman (Jan 31 2021 at 00:45):

Have a look at Blackwell-Kelly-Power "Two-dimensional monad theory". They construct classifiers for both lax and pseudo morphisms.

Mike Shulman (Jan 31 2021 at 00:46):

Also Steve Lack's "Codescent objects and coherence" is a good reference.

sarahzrf (Jan 31 2021 at 00:51):

Rune Haugseng said:

The objects are the objects of X, the morphisms x -> y are sequences (f_1,..., f_n) (n \geq 0) of composable morphisms f_i : x_{i-1} -> x_i (with x_0 = x, x_n = y), with composition = concatenation, and 2-morphisms (f_1,...,f_n) -> (g_1,..,g_m) are given by a map of ordered sets h : {1,..,m} -> {1,...,n} + 2-morphisms in X from f_{h(i)} o ... o f_{h(i+1)-1} -> g_i

hmm, im actually having trouble reading that last bit about h and the extra 2-cell data

sarahzrf (Jan 31 2021 at 00:53):

would h not go from {1, ..., n} to {1, ..., m}? and i cant tell what pattern yr getting at with the composition there

sarahzrf (Jan 31 2021 at 00:54):

i guess my intuition would be that h would go n → m and the extra 2-cells would go from the composites over each fiber to each g_i (i.e., something like "forget the 1-cell composition and horizontal 2-cell composition of X but keep the vertical 2-cell composition, then freely turn it into a strict 2-category again" nvm, youd need to remember how horizontal composition works in order to define vertical composition in L(X)...)

sarahzrf (Jan 31 2021 at 00:55):

is that, or some dualized version of that, what the notation there is meant to indicate?

Rune Haugseng (Jan 31 2021 at 01:32):

A 2-morphism from (f_1,...,f_n) to (g) should be a 2-morphism f_n o ... o f_1 -> g in X (with composition written in the usual ("backwards") order), where the empty composite means the identity if n = 0. A general 2-morphism to (g_1,...,g_m) is then a list of such, for each g_i.

Mike Shulman (Jan 31 2021 at 02:11):

What does a reaction of :eyes: mean?

John Baez (Jan 31 2021 at 02:14):

You said "have a look", so I assume it means "I'm looking".

John Baez (Jan 31 2021 at 02:15):

Sometimes it means "staring at something fascinating", so maybe you're in luck and it actually meant that.

Mike Shulman (Jan 31 2021 at 16:13):

I look forward to the day when the Internet progresses from ideograms to the invention of a true writing system.

John Baez (Jan 31 2021 at 17:42):

The main thing is, Mike, you don't need to care about the emoticons. 99% of the time here it's just someone having fun. Nobody is gonna say "no, you're wrong - that's not a Kan extension" using emoticons.

Jules Hedges (Jan 31 2021 at 17:52):

Maybe someone will take that as a challenge

Nathanael Arkor (Jan 31 2021 at 18:34):

It's really difficult with the size constraints, but I gave it a shot.

Oscar Cunningham (Jan 31 2021 at 18:54):

A kan't extension

Fabrizio Genovese (Jan 31 2021 at 23:41):

Mike Shulman said:

I look forward to the day when the Internet progresses from ideograms to the invention of a true writing system.

I'd argue that ideograms are a true writing system. Personally I much rather prefer languages based on ideograms than languages based on alphabets. :smile:

John Baez (Feb 01 2021 at 16:21):

Nathanael Arkor said:

It's really difficult with the size constraints, but I gave it a shot.

Cute! I wanted to try it with pre-existing emoticons. I was looking for a "can" emoticon, but the best I can do is this:

:not_allowed: :candy:

Christian Williams (Jul 14 2022 at 20:06):

Thanks all; I'm happy to find this thread again. I think lax functors are fundamental to category theory, but they are more complex, and this "relaxation" of a 2-category provides a way to encode 2Cat_lax into 2Cat_ps.

Christian Williams (Jul 14 2022 at 20:07):

I want to use this to construct the higher coend calculus of Equ, the universe of equipments.

Christian Williams (Jul 14 2022 at 20:08):

At first I was constructing the universe of equipments and lax functors, but I began to see that "lax ends" add too much complexity to the language.

Christian Williams (Jul 14 2022 at 20:10):

So, I need to check that this encoding is "nice", i.e. there should be an equivalence between 2Cat_lax and some other 3-category. But which one? This thread answers the question: because "relaxation" is a left adjoint, it defines a comonad on 2Cat_str. Then we should have 2Cat_lax is equivalent to the coKleisli of "relaxation".

Christian Williams (Jul 14 2022 at 20:15):

It is certainly surprising that this stuff is (apparently) folklore. I imagine when certain experts "saw" how relaxation works, it was so simple that nobody bothered writing it out explicitly. As was said earlier in this thread: the construction takes a 2-category, forms lists of 1-cells and 2-cells, and then formally adjoins 2-cells for composition and units.

Christian Williams (Jul 14 2022 at 20:17):

I was initially confused because this sounds just like the "free monoid" construction, which we normally understand as a monad on Set. But here, 2Cat_str is the "algebraic" side, so it gets the comonad.

Christian Williams (Jul 14 2022 at 20:46):

So 2Cat has two "comodalities", Lax and Colax. We can restrict to 2-functors, and get a well-behaved language of 2-categories.

Mike Shulman (Jul 14 2022 at 21:21):

These modalities don't live on the 3-category 2Cat, though, because lax functors don't interact well with natural transformations. Some places they do live are:

• The 1-category $2\rm Cat_1$ of 2-categories and 2-functors
• The 2-category $2\rm Cat_2$ of 2-categories, 2-functors, and icons
• The 2-category $\rm Dbl$ of double categories, double functors, and vertical transformations (where the laxness is in the horizontal direction)

Christian Williams (Jul 14 2022 at 21:59):

Ah yes, thanks. The case that I care about is like $\mathrm{Dbl}$.

Mike Shulman (Jul 14 2022 at 21:59):

I guessed that might be the case... (-: