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Stream: learning: questions

Topic: ✔ algebraic vs geometric higher categories

Ruby Khondaker (she/her) (Jul 09 2025 at 16:53):

Oh I thought you could just use, like, cuboids as your pasting diagrams

Ruby Khondaker (she/her) (Jul 09 2025 at 16:53):

so for 2D squares you'd have rectangles?

Mike Shulman (Jul 09 2025 at 16:53):

What is a cuboid?

Mike Shulman (Jul 09 2025 at 16:53):

And what do you mean by a rectangle?

Ruby Khondaker (she/her) (Jul 09 2025 at 16:54):

uh i mean in the sense of euclidean geometry, haha

Ruby Khondaker (she/her) (Jul 09 2025 at 16:54):

so a rectangle would be a $m \times n$ grid of squares?

Ruby Khondaker (she/her) (Jul 09 2025 at 16:55):

and then a cuboid a $l \times w \times h$ grid of cubes (length, width, height)

Ruby Khondaker (she/her) (Jul 09 2025 at 16:55):

the "natural" shapes you can make from squares and cubes under the composition operations, I guess?

Ruby Khondaker (she/her) (Jul 09 2025 at 16:55):

without jagged boundaries, so no tetris pieces here

Ruby Khondaker (she/her) (Jul 09 2025 at 16:56):

Mike Shulman said:

What is a cuboid?

also sorry but this is very funny to me out of context :P

Mike Shulman (Jul 09 2025 at 16:58):

Yes, $m\times n$ grids of squares are what you have in bisimplicial sets (namely $\Delta^m\Box\Delta^n$ ). A cubical set doesn't have grids, it only has individual cubes.

Ruby Khondaker (she/her) (Jul 09 2025 at 16:59):

hmmm but

Ruby Khondaker (she/her) (Jul 09 2025 at 17:00):

cubical sets are presheaves on the cube category right?

Ruby Khondaker (she/her) (Jul 09 2025 at 17:00):

ah maybe i messed something up

Ruby Khondaker (she/her) (Jul 09 2025 at 17:01):

what i thought you could do is, given a particular "shape" of cubical complex, take pushouts of the corresponding sets to find the "set of cubical complexes"

Ruby Khondaker (she/her) (Jul 09 2025 at 17:01):

something like - if you know all the squares in your set, then you know all the rectangles

Ruby Khondaker (she/her) (Jul 09 2025 at 17:01):

but maybe this doesn't work like i thought..

Mike Shulman (Jul 09 2025 at 17:02):

You can certainly construct a category of cubical pasting diagrams and consider presheaves on it, but those aren't what people mean by [[cubical set]].

Ruby Khondaker (she/her) (Jul 09 2025 at 17:03):

so if you're given a cubical set, you can't figure out what pasting diagrams there are within the cubical set?

Mike Shulman (Jul 09 2025 at 17:03):

Yes, you can.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:03):

hm now i'm even more confused lol

Mike Shulman (Jul 09 2025 at 17:03):

But to have a geometric notion of higher category, you need the "pasting diagrams" to be data rather than something defined from the available data.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:03):

what do you mean?

Mike Shulman (Jul 09 2025 at 17:04):

E.g. in a quasicategory, a 2-simplex is a pasting diagram of the form $\cdot \to\cdot\to\cdot$ .

Ruby Khondaker (she/her) (Jul 09 2025 at 17:04):

hm, right

Mike Shulman (Jul 09 2025 at 17:04):

In a directed graph you can already define the collection of 1D pasting diagrams. But you can't define a geometric notion of 1-category based on directed graphs.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:04):

why not...?

Ruby Khondaker (she/her) (Jul 09 2025 at 17:04):

maybe i'm not understanding what geometric means here

Mike Shulman (Jul 09 2025 at 17:04):

Because there's no way for a directed graph to contain information about composition.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:04):

hm...

Mike Shulman (Jul 09 2025 at 17:05):

To make a directed graph a category, you have to equip it with the structure of a composition operation, and that's an algebraic definition.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:05):

can i say what i had in mind for directed graphs? maybe that'll help resolve my confusion

Ruby Khondaker (she/her) (Jul 09 2025 at 17:05):

because i guess what i had in mind is something in between these

Mike Shulman (Jul 09 2025 at 17:05):

A quasicategory is a geometric definition because to make a simplicial set a quasicategory, you only have to assert properties, such as that for every inner horn there exists a filler.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:05):

yes yes i see what you mean

Ruby Khondaker (she/her) (Jul 09 2025 at 17:06):

in my case i have a "structure", but not quite an algebraic one?

Ruby Khondaker (she/her) (Jul 09 2025 at 17:06):

so it's somewhere in between

Mike Shulman (Jul 09 2025 at 17:06):

Sure, but maybe this should move somewhere other than Aaron's personal thread.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:06):

ah yeah :P

where would you suggest?

Notification Bot (Jul 09 2025 at 17:08):

39 messages were moved here from #community: our work > Aaron David Fairbanks by Mike Shulman.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:08):

oh nice!

Ruby Khondaker (she/her) (Jul 09 2025 at 17:09):

so, $\mathbf{Cat}$ is monadic over $\mathbf{Graph}$

Ruby Khondaker (she/her) (Jul 09 2025 at 17:09):

meaning that categories are just algebras for the "free category" monad

Ruby Khondaker (she/her) (Jul 09 2025 at 17:09):

that's what you mean by the structure of a composition operation, right?

Mike Shulman (Jul 09 2025 at 17:10):

Yes.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:10):

so what i had in mind (which turns out to be equivalent for 1-categories) is the following

Ruby Khondaker (she/her) (Jul 09 2025 at 17:11):

in higher categories, composition isn't usually a function anymore, right? it's a relation - you say that "yes, h is a composite of g o f" or "no, it's not", together with the data of some homotopy relating them

Ruby Khondaker (she/her) (Jul 09 2025 at 17:11):

if i understand correctly, this is how quasicategories work...?

Mike Shulman (Jul 09 2025 at 17:12):

Right.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:12):

so, how about, instead of making composition a function, you make it a relation!

Mike Shulman (Jul 09 2025 at 17:12):

(If by "higher categories" you mean "geometric higher categories".)

Ruby Khondaker (she/her) (Jul 09 2025 at 17:12):

mhm!

Ruby Khondaker (she/her) (Jul 09 2025 at 17:13):

so say $X$ is a directed graph, and $TX$ is the free category monad applied to the graph

Ruby Khondaker (she/her) (Jul 09 2025 at 17:13):

what you want is a relation that tells you whether or not a sequences of paths composes to a given morphism

Mike Shulman (Jul 09 2025 at 17:13):

Oh, are you back on Leinster's $L'$ ? (-:

Ruby Khondaker (she/her) (Jul 09 2025 at 17:13):

yeah, exactly!

Ruby Khondaker (she/her) (Jul 09 2025 at 17:13):

a subobject of $TX \times X$ , which i'll call $\text{Comp}$

Mike Shulman (Jul 09 2025 at 17:13):

Yes, I agree that that's something of an algebraic-geometric hybrid.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:14):

mhm mhm

Ruby Khondaker (she/her) (Jul 09 2025 at 17:14):

the axioms that Comp has to satisfy are reflexivity and transitivity

Mike Shulman (Jul 09 2025 at 17:14):

(Which, unfortunately, makes it hard to apply existing tools to understand...)

Ruby Khondaker (she/her) (Jul 09 2025 at 17:14):

as well as being a "functional" relation, i.e. for every path there's a "unique" composite

Ruby Khondaker (she/her) (Jul 09 2025 at 17:14):

oh?

Ruby Khondaker (she/her) (Jul 09 2025 at 17:14):

Ruby Khondaker (she/her) said:

as well as being a "functional" relation, i.e. for every path there's a "unique" composite

but since there aren't any nontrivial 2-cells, this just means it's the same as a function

Ruby Khondaker (she/her) (Jul 09 2025 at 17:15):

so this "geometric" notion of 1-category is just the usual algebraic one

Ruby Khondaker (she/her) (Jul 09 2025 at 17:15):

but for higher categories it gives you a real weak "geometric" version, i think!

Ruby Khondaker (she/her) (Jul 09 2025 at 17:15):

or, sorry, "algebro-geometric"

Ruby Khondaker (she/her) (Jul 09 2025 at 17:16):

it's very much in the style of the definition of complete segal spaces i think

Ruby Khondaker (she/her) (Jul 09 2025 at 17:16):

actually yeah i think i got this literally from your work with riehl lol

Ruby Khondaker (she/her) (Jul 09 2025 at 17:17):

The synthetic theory of $\infty$ -categories vs the synthetic theory of $\infty$ -categories

Ruby Khondaker (she/her) (Jul 09 2025 at 17:18):

image.png
this part

Ruby Khondaker (she/her) (Jul 09 2025 at 17:19):

Ruby Khondaker (she/her) said:

Ruby Khondaker (she/her) said:

as well as being a "functional" relation, i.e. for every path there's a "unique" composite

but since there aren't any nontrivial 2-cells, this just means it's the same as a function

basically in higher categories, you can just replace "unique" with "contractible space" and it should work just fine?

Mike Shulman (Jul 09 2025 at 17:19):

If you replace composition with a relation, and then you replace associativity with a relation too, and so on all the way up, you get an honest geometric notion. What makes Leinster's $L'$ a hybrid is that although composition is a relation, composition of such relations is an algebraic operation.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:19):

hm why would you need associativity to be a relation?

Ruby Khondaker (she/her) (Jul 09 2025 at 17:20):

i thought leinster's approach combines composition and coherence into contractibility

Ruby Khondaker (she/her) (Jul 09 2025 at 17:21):

so you'd just need one relation, not many

Ruby Khondaker (she/her) (Jul 09 2025 at 17:21):

maybe i misunderstood his definition...

Mike Shulman (Jul 09 2025 at 17:23):

Saying "associativity is a relation" is somewhat handwavy. I just meant that a quasicategory, or a complete segal space, is fully geometric because there are no operations, only data and higher data and higher higher data, whereas $L'$ has data but also operations on that data.

Ruby Khondaker (she/her) (Jul 09 2025 at 17:23):

image.png
so this is the definition i'm familiar with

Ruby Khondaker (she/her) (Jul 09 2025 at 17:24):

i'm probably misunderstanding - what are the operations on the data here?

Ruby Khondaker (she/her) (Jul 09 2025 at 17:27):

this is from page 273 of Higher Operads, Higher Categories if that helps

Ruby Khondaker (she/her) (Jul 09 2025 at 17:29):

also i'll be busy for the next hour or so but will definitely reply when i get back!

Mike Shulman (Jul 09 2025 at 18:20):

The operations are the composition in a $T$ -multicategory.

Ruby Khondaker (she/her) (Jul 09 2025 at 18:41):

Mike Shulman said:

The operations are the composition in a $T$ -multicategory.

oh you mean like the things in the span?

Ruby Khondaker (she/her) (Jul 09 2025 at 18:41):

the object "over" $TX$ and $X$

Mike Shulman (Jul 09 2025 at 19:35):

Those sorts of things are the inputs and outputs of the operations.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:36):

Hm, right…

Ruby Khondaker (she/her) (Jul 09 2025 at 19:36):

I guess, a relation $R \subset TX \times X$ doesn’t feel like a collection of “operations” in the same way?

Ruby Khondaker (she/her) (Jul 09 2025 at 19:37):

It just says, giving a pasting diagram and a candidate composite, whether the result really is a composite or not

Ruby Khondaker (she/her) (Jul 09 2025 at 19:37):

So just a yes/no answer I guess

Mike Shulman (Jul 09 2025 at 19:38):

In general, the span is not just a relation.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:41):

Mike Shulman said:

In general, the span is not just a relation.

Mhm mhm, but what if you just have a relation?

Ruby Khondaker (she/her) (Jul 09 2025 at 19:41):

Like, define an omega category to be a relation that’s reflexive and transitive and “contractible” in an appropriate sense

Ruby Khondaker (she/her) (Jul 09 2025 at 19:41):

I must’ve misunderstood L’ because I thought that’s what it was saying

Ruby Khondaker (she/her) (Jul 09 2025 at 19:42):

Any “algebraic” omega category would embed into this by just taking the span to be TX with the identity to TX and the algebra map to X

Ruby Khondaker (she/her) (Jul 09 2025 at 19:43):

Would this somehow not be general enough…?

Ruby Khondaker (she/her) (Jul 09 2025 at 19:43):

Again apologies if I’m missing something obvious

Mike Shulman (Jul 09 2025 at 19:44):

Leinster doesn't say anywhere that the span $X_1 \to T X_0 \times X_0$ should be monic.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:44):

Ahhhh ok

Ruby Khondaker (she/her) (Jul 09 2025 at 19:44):

If you require it to be monic does that work?

Mike Shulman (Jul 09 2025 at 19:44):

Well, as far as I know $L'$ has not been proven equivalent to any other definition of $\omega$ -category, so no one knows whether it "works" at all anyway....

Ruby Khondaker (she/her) (Jul 09 2025 at 19:45):

Wait really??

Ruby Khondaker (she/her) (Jul 09 2025 at 19:45):

What’s the issue

Mike Shulman (Jul 09 2025 at 19:45):

I don't know if anyone has even really tried.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:45):

Huh…

Ruby Khondaker (she/her) (Jul 09 2025 at 19:45):

L’ felt so natural to me, why don’t people like it

John Baez (Jul 09 2025 at 19:45):

Ruby Khondaker (she/her) said:

What’s the issue

Not many people work on this stuff.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:46):

But Amar was sending me all those awesome papers

Ruby Khondaker (she/her) (Jul 09 2025 at 19:46):

Seems like a lot of people are working on this…?

Mike Shulman (Jul 09 2025 at 19:46):

There are a lot of meanings of "this".

Ruby Khondaker (she/her) (Jul 09 2025 at 19:47):

True lol

Ruby Khondaker (she/her) (Jul 09 2025 at 19:47):

Hm maybe I could try showing L’ is equivalent to some other def?

Mike Shulman (Jul 09 2025 at 19:47):

A lot of people work on geometric definitions of higher category because they've been widely proven to be useful.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:47):

Mhm, I guess L’ feels “geometric” enough to me

Ruby Khondaker (she/her) (Jul 09 2025 at 19:48):

But it’s not quite fully geometric as you say

Ruby Khondaker (she/her) (Jul 09 2025 at 19:48):

Since the composition isn’t witnessed by the shapes themselves

Ruby Khondaker (she/her) (Jul 09 2025 at 19:48):

It’s done externally through the relation

Mike Shulman (Jul 09 2025 at 19:49):

Remember Amar's advice in the other thread?

Amar Hadzihasanovic said:

I would recommend that the workflow should be

Find an open problem that you want to solve,

See if it can be solved with existing models/techniques,

Only if 2. fails start thinking about new models/techniques

As far as I know, no one has yet encountered a problem that seems to require $L'$ for its solution.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:51):

Yeah I guess this is where I don’t know enough open problems :(

John Baez (Jul 09 2025 at 19:51):

Mike Shulman said:

A lot of people work on geometric definitions of higher category because they've been widely proven to be useful.

I wonder what "a lot of people" means here? 15? 50? 50 seems high.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:51):

I just like L’ because it feels conceptually cleanest to me

Ruby Khondaker (she/her) (Jul 09 2025 at 19:53):

I might stick with it as my “internal” idea of what an $\omega$ -category is, and see how far that gets me

Ruby Khondaker (she/her) (Jul 09 2025 at 19:53):

Either I’ll get stuck and see why other definitions are better, or I’ll be fine

Ruby Khondaker (she/her) (Jul 09 2025 at 19:53):

Though to be fair comical sets are also pretty cool

Ruby Khondaker (she/her) (Jul 09 2025 at 19:55):

For some reason I thought even defining the $\omega$ -category of $\omega$ -categories was open, but clearly not lol

Amar Hadzihasanovic (Jul 09 2025 at 19:55):

One reason that geometric definitions are more used is that often they fit into model categories where all objects are cofibrant, whereas typically in algebraic definitions the cofibrants should be the free (computad-like) objects.
It just so happens that a lot of model category theory has been developed in the special case of “all objects are cofibrant” because, well, geometric definitions are very natural for the homotopical/homological purposes that model category theory was created for :)

Ruby Khondaker (she/her) (Jul 09 2025 at 19:56):

Hm what about this mixed algebro-geometric def?

Ruby Khondaker (she/her) (Jul 09 2025 at 19:56):

(Also this is where my lack of homotopy theory knowledge holds me back, I barely understand fibrations and cofibrations)

Amar Hadzihasanovic (Jul 09 2025 at 19:56):

So when people had to build the “homotopy theory of higher categories”, there simply was a lot more machinery available to do it for geometric definitions.

Amar Hadzihasanovic (Jul 09 2025 at 19:57):

Ruby Khondaker (she/her) said:

Hm what about this mixed algebro-geometric def?

As soon as there's any possibility of “non-trivial equations” you're outside of the “everything is cofibrant” scenario.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:57):

Is there a laywoman’s way of saying what the “homotopy theory of higher categories” is here?

Ruby Khondaker (she/her) (Jul 09 2025 at 19:57):

Hm what do you mean by “nontrivial equations”?

Mike Shulman (Jul 09 2025 at 19:58):

Amar Hadzihasanovic said:

It just so happens that a lot of model category theory has been developed in the special case of “all objects are cofibrant” because, well, geometric definitions are very natural for the homotopical/homological purposes that model category theory was created for :)

I think that's a bit misleading. To first approximation, model category theory is self-dual: the opposite of a model category in which all objects are cofibrant is a model category in which all objects are fibrant, and conversely.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:58):

ooh ok, neat

Mike Shulman (Jul 09 2025 at 19:58):

And of the ur-examples of model categories, topological spaces are surely way up there, and in that model category every object is fibrant.

Mike Shulman (Jul 09 2025 at 19:59):

I think I kind of get what you're trying to say though.

Ruby Khondaker (she/her) (Jul 09 2025 at 19:59):

Hm maybe I should make a thread to ask why model categories are so useful

Ruby Khondaker (she/her) (Jul 09 2025 at 19:59):

I barely understand their definition

Mike Shulman (Jul 09 2025 at 20:00):

Maybe I would say that some of the important tools of model category theory, notably left Bousfield localization, tend to take you out of the world where all objects are fibrant, but don't take you out of the world where all objects are cofibrant.

Mike Shulman (Jul 09 2025 at 20:00):

Of course right Bousfield localization behaves dually; the real difference is that left localizations tend to exist more often because more naturally-occurring categories are locally presentable than co-locally-presentable.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:01):

that sounds like nonlocally-presentable, haha

Mike Shulman (Jul 09 2025 at 20:02):

(Sorry for going over your head, Ruby -- I'm writing to Amar here.)

Amar Hadzihasanovic (Jul 09 2025 at 20:02):

Yes, sorry for being handwavy. You are giving a better explanation of what I was aiming for.

Mike Shulman (Jul 09 2025 at 20:02):

But I'm not sure that even that gets at the point, because at least a priori it's not clear that a model category of algebraic objects would need a Bousfield localization to be constructed -- you only need it in the geometric case because of how you usually build such a model category, by localizing a presheaf model category.

Amar Hadzihasanovic (Jul 09 2025 at 20:03):

There is in practice a certain bias towards e.g. cofibrant generation which probably comes from the bias of naturally-occurring categories towards local presentability.

Mike Shulman (Jul 09 2025 at 20:03):

Maybe it's that localizing a presheaf model category is such an easy way to build model categories, and we don't have similarly powerful tools for the algebraic case?

Ruby Khondaker (she/her) (Jul 09 2025 at 20:03):

Oh no apologies needed, it’s just a skill issue on my part :P

Mike Shulman (Jul 09 2025 at 20:07):

Or maybe another way of writing the same thing is that we expect the $(\infty,1)$ -category of higher categories to be locally presentable, i.e. to be an $\infty$ -categorical localization of a presheaf $\infty$ -category, and when we import that construction into model category theory we get a Bousfield localization of a presheaf model category, i.e. a nonalgebraic definition.

Amar Hadzihasanovic (Jul 09 2025 at 20:09):

Yes, that's a convincing way of seeing it.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:14):

Oh ok i was looking this up on nlab, is locally presentable related to essentially algebraic theory?

Mike Shulman (Jul 09 2025 at 20:25):

Yes, locally finitely presentable categories are basically the same as models of essentially algebraic theories.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:26):

hm ok so i'm going off this screenshot
image.png

Ruby Khondaker (she/her) (Jul 09 2025 at 20:26):

particularly the last line

Ruby Khondaker (she/her) (Jul 09 2025 at 20:27):

for $\mathbf{L}'$ , the composite of a pasting diagram isn't literally "unique", but the space of such composites is contractible

Ruby Khondaker (she/her) (Jul 09 2025 at 20:27):

would that be "unique enough" to be essentially algebraic...?

Ruby Khondaker (she/her) (Jul 09 2025 at 20:28):

again i'm going off that def of complete segal spaces, where every pair of arrows has a "unique" binary composite in the HoTT sense

Mike Shulman (Jul 09 2025 at 20:28):

No.

Mike Shulman (Jul 09 2025 at 20:29):

Not in the usual 1-categorical sense of "essentially algebraic theory".

Ruby Khondaker (she/her) (Jul 09 2025 at 20:29):

oh, is there a higher-categorical sense where this would work?

Mike Shulman (Jul 09 2025 at 20:30):

Yes, the $\infty$ -category of $\omega$ -categories should be locally finitely presentable. But that's a "model-independent" statement, it doesn't say anything about particular 1-categorical presentations such as $L'$ .

Ruby Khondaker (she/her) (Jul 09 2025 at 20:31):

i guess what i'm going off of is:

$(\infty, 1)$ -category of higher categories should be locally presentable
This is the same as saying they're models of essentially algebraic theories
Rezk spaces are good models of higher categories
Rezk spaces are models of essentially algebraic theories...?

Ruby Khondaker (she/her) (Jul 09 2025 at 20:31):

i don't know enough to tell why this is wrong

Mike Shulman (Jul 09 2025 at 20:37):

Yes, Rezk spaces are a model of an essentially algebraic $\infty$ -theory.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:38):

Mike Shulman said:

Yes, Rezk spaces are a model of an essentially algebraic $\infty$ -theory.

hm, how is that possible if they're "geometric" in nature? is it cause the space of composites is contractible?

Mike Shulman (Jul 09 2025 at 20:38):

(For any reasonable meaning of the latter -- there may or may not exist a "syntactic" one in the literature.)

Ruby Khondaker (she/her) (Jul 09 2025 at 20:38):

oh ok!

Mike Shulman (Jul 09 2025 at 20:38):

Geometric vs algebraic is a property of a 1-categorical presentation, not of the resulting $\infty$ -categorical object.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:38):

hm, i don't quite understand what you mean

Ruby Khondaker (she/her) (Jul 09 2025 at 20:40):

would i be wrong in saying that $\mathbf{L}'$ categories are an essentially algebraic $\infty$ -theory?

Mike Shulman (Jul 09 2025 at 20:40):

Sorry, when I said this

Mike Shulman said:

Yes, Rezk spaces are a model of an essentially algebraic $\infty$ -theory.

I was referring to the $\infty$ -categorical notion of Rezk space. If you mean the 1-categorical presentation thereof, then no, they are not models of an essentially algebraic 1-theory.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:40):

hm lemme pull up the definition of Rezk I mean

Ruby Khondaker (she/her) (Jul 09 2025 at 20:42):

image.png
this is from riehl's talk

Ruby Khondaker (she/her) (Jul 09 2025 at 20:43):

image.png
and this is the def of segal type given

Mike Shulman (Jul 09 2025 at 20:44):

That definition is written inside simplicial homotopy type theory, which is a totally different context from any of the usual definitions of higher category that are usually written in set theory. So it's not really comparable.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:44):

hm...

Ruby Khondaker (she/her) (Jul 09 2025 at 20:44):

so, for these Rezk spaces, are they models of an essentially algebraic $\infty$ -theory?

Ruby Khondaker (she/her) (Jul 09 2025 at 20:45):

or is that a concept that doesn't even apply here

Mike Shulman (Jul 09 2025 at 20:45):

I would say it doesn't apply.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:46):

ah ok! thanks for the clarification :)

Ruby Khondaker (she/her) (Jul 09 2025 at 20:46):

if you converted the def into bare set theory, would it apply then?

Ruby Khondaker (she/her) (Jul 09 2025 at 20:46):

i just remember riehl saying during the talk that a lot of the $\infty$ -categories that come up in "practice" are naturally complete segal spaces

Ruby Khondaker (she/her) (Jul 09 2025 at 20:47):

so i assumed that complete segal spaces would be pretty nice

Ruby Khondaker (she/her) (Jul 09 2025 at 20:47):

which meant i thought they'd be locally presentable, which i thought meant models of an essentially algebraic theory

Ruby Khondaker (she/her) (Jul 09 2025 at 20:47):

clearly there's still something i'm misunderstanding..

Mike Shulman (Jul 09 2025 at 20:49):

There is no straightforward way to "convert" a definition in sHoTT into bare set theory, but you can interpret it in the simplicial spaces model of sHoTT defined in set theory. In that case, what you get from this definition is (up to equivalence) the same as Rezk's original definition of complete Segal space.

Mike Shulman (Jul 09 2025 at 20:49):

There are many ways that something can be "nice". Locally presentable categories are nice (in one way), but not everything that's nice is a locally presentable category.

Ruby Khondaker (she/her) (Jul 09 2025 at 20:50):

hm, right

Ruby Khondaker (she/her) (Jul 09 2025 at 20:50):

Mike Shulman said:

Or maybe another way of writing the same thing is that we expect the $(\infty,1)$ -category of higher categories to be locally presentable, i.e. to be an $\infty$ -categorical localization of a presheaf $\infty$ -category, and when we import that construction into model category theory we get a Bousfield localization of a presheaf model category, i.e. a nonalgebraic definition.

i guess i was just going off you said here - i interpreted this as saying "any $(\infty, 1)$ -category of higher categories must be locally presentable"

Ruby Khondaker (she/her) (Jul 09 2025 at 20:51):

but clearly i misunderstood!

Mike Shulman (Jul 09 2025 at 20:52):

That's what I said, but the key part of the statement is the $(\infty,1)$ .

Ruby Khondaker (she/her) (Jul 09 2025 at 20:53):

oh, so the $(\infty, \infty)$ -category of higher categories may not be locally presentable?

Mike Shulman (Jul 09 2025 at 20:53):

No, it's the 1-category of them that may not be.

Mike Shulman (Jul 09 2025 at 20:53):

The $(\infty,1)$ -category of $(\infty,1)$ -categories is locally presentable. That is a model-independent statement, so it's true about the $(\infty,1)$ -category of quasicategories, and the $(\infty,1)$ -category of complete Segal spaces, and so on, because they're all equivalent.