Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: a certain groupoid

Tim Hosgood (Oct 25 2020 at 11:41):

I recently came across the following groupoid in my research, and it seems like something that probably has a much slicker description and that has been studied before, so any comments or observations would be lovely :blush:

The objects of $\mathcal{C}$ are partitions of a finite set into two subsets, or, equivalently, just a subset of a finite set, e.g. $\{1,3,4\}\subset\{1,2,3,4,5\}$. There is exactly one morphism between any two objects if the finite sets that they partition are of the same size, and there are no (non-identity) morphisms otherwise.

Nathanael Arkor (Oct 25 2020 at 12:06):

This is equivalently the set of pairs $(\alpha \leq \beta)$ of cardinal numbers; the groupoidal structure doesn't seem interesting.

Tim Hosgood (Oct 25 2020 at 12:24):

I don’t quite see how that’s true. For example, $\{1,3,4\}\subset\{1,2,3,4,5\}$ and $\{1,2,4\}\subset\{1,2,3,4,5\}$ are different objects , but both correspond to $3\leqslant5$

Tim Hosgood (Oct 25 2020 at 12:25):

maybe I should have replaced “finite sets” with “finite sets of the form $\{1,\ldots,n\}$”

Nathanael Arkor (Oct 25 2020 at 12:25):

But there will be an invertible morphism between them, because they have the same cardinality, no?

Nathanael Arkor (Oct 25 2020 at 12:25):

Perhaps I misunderstood the description.

Tim Hosgood (Oct 25 2020 at 12:26):

no, you’re right, but I don’t want to think of isomorphic objects as being equal

Tim Hosgood (Oct 25 2020 at 12:27):

which I guess is “evil” maybe

Reid Barton (Oct 25 2020 at 12:33):

You might as well call $\mathcal{C}$ a set equipped with an equivalence relation, then

Reid Barton (Oct 25 2020 at 12:46):

I guess examples where something like this comes up are the $A_\infty$ and $E_\infty$ operads (nonsymmetric in the case of $A_\infty$) valued in categories, which encode the theory of monoidal/symmetric monoidal categories. By design, the $n$ th category of $A_\infty$ is a contractible groupoid for each $n$, but they can't all just be the terminal category because $A_\infty$ needs to be cofibrant.
There's a similar gadget involving cubes which appears in the formula for the simplicial nerve, which could be related to your example because vertices of an $n$-cube can be identified with subsets of $\{1, 2, \ldots, n\}$.

sarahzrf (Oct 25 2020 at 23:53):

Tim Hosgood said:

no, you’re right, but I don’t want to think of isomorphic objects as being equal

how come? is it because you make use of which things are elements of the sets?

sarahzrf (Oct 25 2020 at 23:53):

if so, then you should probably be thinking of this as a groupoid equipped with certain functors

Tim Hosgood (Oct 26 2020 at 10:43):

yeah, i use the numbers to label some things, so the actual elements are important in this sense

Tim Hosgood (Oct 26 2020 at 10:43):

what do you mean by certain functors?

sarahzrf (Oct 26 2020 at 14:26):

umm, maybe not "certain functors" exactly, i havent thought about it really, but

sarahzrf (Oct 26 2020 at 14:28):

i just mean that it's common for issues of "the objects im working with have a naturally finer notion of comparison than the natural categorical one" to be a symptom of failing to make certain data an explicit part of the objects

sarahzrf (Oct 26 2020 at 14:28):

the archetypal example i have in mind is like

sarahzrf (Oct 26 2020 at 14:28):

for a group G, you can have two subgroups of G that are isomorphic as groups, but induce non-isomorphic quotients—which seems problematic at first glance...

sarahzrf (Oct 26 2020 at 14:29):

but the issue is that a "subgroup of G" is not an object of Grp, categorically speaking—it's an object of Grp/G

sarahzrf (Oct 26 2020 at 14:29):

you need the map into G as part of the data

Reid Barton (Oct 26 2020 at 15:15):

The category $\mathcal{C}$ is equivalent to the set $\mathbb{N}$ viewed as a discrete category. More specifically, the functor sending any $S \subset \{1, \ldots, n\}$ to $n$ is an equivalence of categories. We could think of $\mathcal{C}$ as a version of $\mathbb{N}$ where each $n \in \mathbb{N}$ has been "blown up" into a contractible groupoid.

Clearly, there must be some reason that you need this "blown up" $\mathcal{C}$, otherwise you'd just use $\mathbb{N}$ instead. A common reason in homotopy theory/higher category theory is that you need to take a cofibrant replacement in order to make some 1-categorical construction be homotopy/equivalence-invariant. For example, consider the associator in a monoidal category. We want to say that in the expression $x \otimes y \otimes z$, it's "the same" whether you consider it as $(x \otimes y) \otimes z$ or as $x \otimes (y \otimes z)$. But to ask for a literal equality of objects would be "evil"--not equivalence-invariant. So, instead of having a single operation $\{(12)3 = 1(23)\}$, we "blow up" the possible ways to evaluate $x \otimes y \otimes z$ to the equivalent groupoid $\{(12)3 \xleftrightarrow{\sim} 1(23)\}$. The interpretation of this isomorphism is as the associator $\alpha_{x,y,z} : (x \otimes y) \otimes z \xleftrightarrow{\sim} x \otimes (y \otimes z)$. In general, for each $n$, we have a contractible groupoid of ways to interpret the expression $x_1 \otimes \cdots \otimes x_n$, and what forces us to use these fancy blown-up contractible groupoids is the operad structure which relates these operations for varying $n$.

Reid Barton (Oct 26 2020 at 15:20):

So, it's possible there is something similar in your situation--some setting in which the specific category $\mathcal{C}$ (up to isomorphism) is a cofibrant replacement of the category $\mathbb{N}$.

Tim Hosgood (Oct 26 2020 at 17:54):

all very helpful comments, thank you!

John Baez (Oct 26 2020 at 21:01):

Quite generally, Tim, you'll find category theorists puzzled if you declare yourself unwilling to think of equivalent categories as "the same". You can always take a category C and find an equivalent category called a "skeleton" which has one object for each isomorphism class of objects in C. In this skeleton isomorphic objects are equal. So, when you say something like

I don’t want to think of isomorphic objects as being equal

then category theorists start wondering what you're up to.

John Baez (Oct 26 2020 at 21:02):

If you want, you can tell them you're working with the 1-category Cat, not the 2-category Cat. That may assuage them - or they may ask you why.

sarahzrf (Oct 26 2020 at 21:12):

or he could just say that he's a homotopy theorist

Dan Doel (Oct 26 2020 at 21:19):

If that's what homotopy theorists do, why are they trying to build a formal system where isomorphic things are considered equal?

Reid Barton (Oct 26 2020 at 21:27):

because it's "equal" that should change, not "isomorphic"!

sarahzrf (Oct 26 2020 at 21:55):

Dan Doel said:

If that's what homotopy theorists do, why are they trying to build a formal system where isomorphic things are considered equal?

it's pl theorists who are building hott, not homotopy theorists :upside_down:

sarahzrf (Oct 26 2020 at 21:55):

tongue in cheek, tongue in cheek

Dan Doel (Oct 26 2020 at 21:55):

Voevodsky was a PL theorist?

sarahzrf (Oct 26 2020 at 21:56):

i did say i was being tongue in cheek :p

sarahzrf (Oct 26 2020 at 21:57):

but i mean, more seriously, isnt half of the field of homotopy theory all about dealing with strict representatives of weak objects

John Baez (Oct 26 2020 at 23:53):

Indeed, homotopy theory is a really different subject than homotopy type theory, with a different crew of people doing different things.