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Stream: event: MIT Categories Seminar

Topic: April 23 - Joe Moeller's talk

Paolo Perrone (Apr 21 2020 at 20:52):

Hello all! This is the topic for Joe Moeller's talk, "The monoidal Grothendieck construction".

Paolo Perrone (Apr 21 2020 at 20:52):

Zoom meeting:
https://mit.zoom.us/j/280120646
Meeting ID: 280 120 646

Paolo Perrone (Apr 21 2020 at 20:52):

Youtube live stream:
https://youtu.be/aznXYbEmWo8

Paolo Perrone (Apr 21 2020 at 20:54):

One thing: we would like to test breakout rooms after the talk. If you are on Zoom and would like to take part, stay online after the talk is over. Otherwise, of course feel free to leave.

Paolo Perrone (Apr 21 2020 at 21:00):

Date and time: Thursday April 23rd, 12 noon EDT (Boston time).

Paolo Perrone (Apr 23 2020 at 15:58):

Hi! We start in 2 minutes.

Paolo Perrone (Apr 23 2020 at 16:00):

30 seconds

Brian Pinsky (Apr 23 2020 at 16:06):

It's not obvious to me that the output of the grothendiek construction is small in general

Brian Pinsky (Apr 23 2020 at 16:11):

I assume the $\int$ used to represent this construction is some kind of end or coend? I don't have great intuition for ends, but this seems like a nice example

Jade Master (Apr 23 2020 at 16:13):

I don't think it is an end or coend...just a coincidence of notation.

Jules Hedges (Apr 23 2020 at 16:14):

I think I remember reading that it's not quite (but it does look kinda like an end)

Paolo Perrone (Apr 23 2020 at 16:16):

Brian Pinsky said:

It's not obvious to me that the output of the grothendiek construction is small in general

You can use the fact that a locally small category with a small set of objects is small.
If you have a functor C^op -> Cat, where C is small and Cat is the category of small categories, then the Grothendieck construction is still locally small (can you see why?). It is also small, because its set of objects is a disjoint union of small sets indexed over a small set (the objects of C).

Paolo Perrone (Apr 23 2020 at 16:17):

Brian Pinsky said:

I assume the $\int$ used to represent this construction is some kind of end or coend? I don't have great intuition for ends, but this seems like a nice example

Both the Grothendieck construction and coends are written as integrals because they both look like an "area under the curve", but they are not an example of one another, as far as I've seen. Yeah, this is unfortunate.

Brian Pinsky (Apr 23 2020 at 16:17):

okay thanks

David Spivak (Apr 23 2020 at 16:34):

Brian Pinsky said:

I assume the $\int$ used to represent this construction is some kind of end or coend? I don't have great intuition for ends, but this seems like a nice example

In fact, the Grothendieck construction is a kind of coend using the coslice (or slice) construction. Given F:C-->Cat, the Grothendieck category of F is equivalent to the coend $\int^{c: C} F(c) \times c/C.$

Paolo Perrone (Apr 23 2020 at 16:41):

Oh wow, cool!

Jules Hedges (Apr 23 2020 at 17:00):

DARPA... search+rescue is search+destroy in the opposite category?

Brian Pinsky (Apr 23 2020 at 17:06):

I usually join over youtube live not zoom (it works better on my TV). Are these breakout rooms going to be a common feature in afterwards? I can connect over zoom instead.

Christian Williams (Apr 23 2020 at 17:07):

The category of elements is described as a coend in The Coend Calculus https://arxiv.org/pdf/1501.02503.pdf, 4.2.2. Then after some generalizations, in 7.3.4 they jump all the way to the $\infty$ case...
I haven't read the lax coend stuff, but I assume that for the Grothendieck construction you just need a slight weakening to pseudo-coends. Luckily it's still the same formula, and you can get the intuition from the category of elements.

Paolo Perrone (Apr 23 2020 at 17:08):

Brian Pinsky said:

I usually join over youtube live not zoom (it works better on my TV). Are these breakout rooms going to be a common feature in afterwards? I can connect over zoom instead.

It depends - let's see how the testing goes :)

Christian Williams (Apr 23 2020 at 17:15):

That's also nice because it's a case of nerve and realization -- you could call the Grothendieck construction the "slice realization". This means it has an adjoint, the "slice nerve", that turns a category $D$ into an indexed category $\mathrm{Cat}(C/-,D)$.

eric brunner (Apr 23 2020 at 17:40):

what unfortunate constructions of morphisms. there's rescue without search, and search without destroy (miss is frequent irl).

Joe Moeller (Apr 23 2020 at 17:44):

Jules Hedges said:

DARPA... search+rescue is search+destroy in the opposite category?

Unfortunately I found out later they snuck in an $^{op}$ when I wasn't looking -__-

Joe Moeller (Apr 23 2020 at 17:45):

But that's why I ended up quitting the project.

Joe Moeller (Apr 23 2020 at 17:46):

Here are my slides: https://joemathjoe.files.wordpress.com/2020/04/mit_seminar_talk.pdf

Gershom (Apr 23 2020 at 21:51):

In the spirit of "why don't they build the whole airplane out of the black box" given the equivalence between fibrations and indexed categories, and given that indexed categories are easier to present (in terms of having fewer conditions), why don't we always just do that?

Joe Moeller (Apr 23 2020 at 22:40):

That is my preference, and I think that's what is done much of the time. Dependent function types I think are basically the same as indexed categories, right?

Joe Moeller (Apr 23 2020 at 22:47):

But fibrations are sometimes the more natural thing to consider, eg graphs over sets.

Joe Moeller (Apr 23 2020 at 22:48):

Another thing, you can compose fibrations in an obvious way, I don't think it's at all easy to see how you would compose indexed categories, or even when they're composable.

Paolo Perrone (Apr 23 2020 at 22:49):

Hi all! Here's the video. https://youtu.be/DDECeQaGwYs

John Baez (Apr 24 2020 at 00:28):

Gershom said:

In the spirit of "why don't they build the whole airplane out of the black box" given the equivalence between fibrations and indexed categories, and given that indexed categories are easier to present (in terms of having fewer conditions), why don't we always just do that?

This is a fun question, so it's good to think about a decategorified version of this function. Given the equivalence between isomorphism classes of monos into $X$ and characteristic functions $\chi \colon X \to 2$, and given that characteristic functions are easier to present (in terms of having fewer conditions), why don't we always just do that?

Of course in material set theory we describe an isomorphism class of monos into $X$ as a "subset" of $X$, which simplifies things, but even without this trick I think both approaches would be useful.

Todd Trimble (Apr 24 2020 at 00:36):

Gershom said:

In the spirit of "why don't they build the whole airplane out of the black box" given the equivalence between fibrations and indexed categories, and given that indexed categories are easier to present (in terms of having fewer conditions), why don't we always just do that?

I agree, I find them easier to present that way, but anyone who wants to opine about these things had better be aware of Bénabou's notorious essay on the subject. He does make some interesting points. For example, it's easy to prove that a composite of fibrations is a fibration. He says: now try proving that using indexed categories! He also has some worthwhile remarks on the notion of definability.

Mike Shulman (Apr 24 2020 at 02:25):

As John said, I don't think it should be regarded as a choice to make between the two, but rather that the flexibility of having two presentations of the same thing is a useful tool. In between monos and fibrations we have the notion of an I-indexed family of sets that can be described as a function $I\to \mathrm{Set}$ or as a function $X\to I$, and both versions are useful.

Jules Hedges (Apr 24 2020 at 09:45):

Joe Moeller said:

But that's why I ended up quitting the project.

I'd be interested to hear your thoughts on this in detail (but possibly not right here)