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Stream: learning: questions

Topic: some sequences of categories

Tim Hosgood (Apr 01 2020 at 19:32):

A while back I came across what I thought was a nice structure, but it turns out that my example was actually flawed. I'm interested, however, in knowing if any such examples actually _do_ exist. The idea is that we have a bunch of categories $\mathcal{C}_n$ indexed by $\mathbb{N}$ along with surjective-on-objects functors $\mathcal{C}_n\to\mathcal{C}_{n+1}$ for all $n$, all such that "$\mathcal{C}_{\infty}$" is the homotopy category of $\mathcal{C}_0$ (here I'm assuming that setting $n=\infty$ makes some sort of formal sense, and that we're working with some nice type of categories that you can take homotopy categories of).

Tim Hosgood (Apr 01 2020 at 19:33):

so this data looks like a sequence of categories that somehow interpolate between a (model, say) category and its homotopy category

sarahzrf (Apr 01 2020 at 19:43):

is this really a "basic question"

Tim Hosgood (Apr 01 2020 at 19:45):

I'm not sure... if somebody answers it by saying "such a collection of data is called a so-and-so, and here's the nLab page" then I suppose it is a basic question, but I don't know what the answer will be!

sarahzrf (Apr 01 2020 at 19:46):

ah, i guess i interpreted "basic" as "elementary", and you interpreted it as "has a short answer"

Tim Hosgood (Apr 01 2020 at 19:47):

yeah, i guess so (plus it's hard to judge what "basic" means in the sense of "elementary" as somebody who isn't a category theorist)

sarahzrf (Apr 01 2020 at 19:48):

to be fair half of the stuff being discussed in this stream is stuff i couldve said the same thing to, by the standard i was applying

sarahzrf (Apr 01 2020 at 19:48):

i just saw the words "homotopy category" and snapped

Thibaut Benjamin (Apr 01 2020 at 20:53):

@Tim Hosgood So you could imagine it as a category with a "discrete process of homotopization"? I have an intuition that might do it, although I think there is a size issue hiding somewhere :
Start by taking $C_1$ to have the same objects as $C_0$, with morphisms $A\to B$ being defined as spans $A \overset{\simeq}{\rightarrow} X\leftarrow B$ in $C_0$, and then repeat this construction, $C_2$ has same objects as $C_1$, with morphisms being spans whose left leg is a weak equivalence in $C_1$, and so on... It seems intuitively that considering longer and longer zigzags, $C_\infty$ will ultimately be the homotopy category. I am a bit scared about the "going to infinity" part just by incrementing integers though, I don't master this very well, but the construction for model categories requires the small object argument and transfinite induction

Tim Hosgood (Apr 01 2020 at 21:36):

this seems similar to the original "example" that I was thinking about: you basically allow homotopies of "increasing length", with "going to infinity" meaning "of some finite but unbounded length"

Tim Hosgood (Apr 01 2020 at 21:37):

but yes, I was thinking of it as discrete homotopisation I think, that's a very good name for it

Joe Moeller (Apr 01 2020 at 23:20):

Don't you compose zigzags by concatenating? How would have category at any stage then?

Joe Moeller (Apr 01 2020 at 23:21):

(any finite stage that is)

Joe Moeller (Apr 01 2020 at 23:21):

Actually, I might be mistaken.

Tim Hosgood (Apr 01 2020 at 23:33):

yeah this is the problem that i had originally: taking "homotopies of some finite length" means that composition doesn't really work

Thibaut Benjamin (Apr 02 2020 at 07:21):

Ah yes, I didn't think about that

Amar Hadzihasanovic (Apr 02 2020 at 15:44):

@Tim Hosgood maybe not what you're looking for, but may be a somewhat similar direction:
if we have something (say $C$) of which we can take the homotopy category (say $\mathrm{Ho}(C)$), it should also be something that presents an $(\infty, 1)$-category $\tilde{C}$;
then we can see $\mathrm{Ho}(C)$ as an $(\infty, 1)$-category with trivial $k$-morphisms for $k > 1$, either as

• presented by $C$, then 1-truncating hom-spaces, or
• presented by $\mathrm{Ho}(C)$ with weak equivalences = isomorphisms.
  So, a question of which I don't know the answer: can we turn the tower of $n$-truncations of $\tilde{C}$, ending with $\mathrm{Ho}(C)$, into a tower of (model structures/cats with weak equivalences/??) presenting them, starting from ($C$, weak equivalences) and ending with ($\mathrm{Ho}(C)$, isomorphisms)?

Amar Hadzihasanovic (Apr 02 2020 at 15:46):

It would not be an $\mathbb{N}$-indexed sequence starting at $C$, but, well, it would be $\mathbb{N}$-indexed and end at the homotopy category...

Jade Master (Apr 02 2020 at 16:49):

This kind of reminds me of a homotopy of categories. Note that the interval object in Cat is the walking arrow 0 -> 1 so a homotopy between functors F and G is a functor h:I × C -> D with h(0)=F and h(1)=G. It turns out that h is exactly the same as a natural transformation (this probably is a standard model structure on Cat and you probably know this Tim).

John Baez (Apr 02 2020 at 17:18):

There's a model structure on Cat where the interval object is the walking bidirectional arrow 0 <--> 1 (two objects and a unique isomorphism between them). For this, homotopies are natural isomorphisms. The nLab calls this the canonical model structure on Cat:

https://ncatlab.org/nlab/show/canonical+model+structure+on+Cat

even though it's not the most commonly used one.

I don't think you can have a model structure where the interval object is the walking arrow 0 --> 1, because then homotopies would be natural transformations and the relation "homotopic to" would not be symmetric! I.e. you could have F homotopic to G but G not homotopic to F.

Tim Hosgood (Apr 02 2020 at 17:59):

Amar Hadzihasanovic said:

Tim Hosgood maybe not what you're looking for, but may be a somewhat similar direction:
if we have something (say $C$) of which we can take the homotopy category (say $\mathrm{Ho}(C)$), it should also be something that presents an $(\infty, 1)$-category $\tilde{C}$;
then we can see $\mathrm{Ho}(C)$ as an $(\infty, 1)$-category with trivial $k$-morphisms for $k > 1$, either as

• presented by $C$, then 1-truncating hom-spaces, or
• presented by $\mathrm{Ho}(C)$ with weak equivalences = isomorphisms.
  So, a question of which I don't know the answer: can we turn the tower of $n$-truncations of $\tilde{C}$, ending with $\mathrm{Ho}(C)$, into a tower of (model structures/cats with weak equivalences/??) presenting them, starting from ($C$, weak equivalences) and ending with ($\mathrm{Ho}(C)$, isomorphisms)?

this seems like a more formal question that is probably more likely to have a satisfying answer!

Amar Hadzihasanovic (Apr 02 2020 at 18:05):

John Baez said:

I don't think you can have a model structure where the interval object is the walking arrow 0 --> 1, because then homotopies would be natural transformations and the relation "homotopic to" would not be symmetric! I.e. you could have F homotopic to G but G not homotopic to F.

@Alex Kavvos has some ideas on what a “directed model structure” on Cat would be, where the walking arrow is a “directed interval object”...

Alex Kavvos (Apr 02 2020 at 18:11):

Amar Hadzihasanovic said:

John Baez said:

[...]

Alex Kavvos has some ideas on what a “directed model structure” on Cat would be, where the walking arrow is a “directed interval object”...

So much is true. If some shameless plugging is allowed, I wrote a long and crazy rant on this some time ago: https://lambdabetaeta.eu/papers/meio.pdf
(Though I would like to understand more about proarrow equipments and rewrite it once more at some point.)

John Baez (Apr 02 2020 at 18:15):

I don't know if people working on directed topology have come up with a concept of "directed model structure".

Alex Kavvos (Apr 02 2020 at 18:19):

John Baez said:

I don't know if people working on directed topology have come up with a concept of "directed model structure".

There has been almost no work on the subject. Marco Grandis presents some factorisations that seem to generalise the known intuitions behind the WFSs of model categories. Paige North has independently been following some lines of thought I have laid out in that manuscript: you have to replace fibrations with two-sided fibrations, and also do some other stuff to get things working. It's unclear what this leads to. Much of it seems to be implicit in 1970s work by Ross Street.

Amar Hadzihasanovic (Apr 03 2020 at 09:50):

Tim Hosgood said:

this seems like a more formal question that is probably more likely to have a satisfying answer!

Returning to this, with $C_\infty := (C, W)$ presenting the $(\infty,1)$-category $\tilde{C}$, and $C_0 := (C[W^{-1}], \mathrm{Iso})$, in $C_n$ we should have localised strictly wrt to those weak equivalences such that “after $(n+1)$-truncation, $k$-truncation does nothing for $k \leq n$”.

Intuitively, I would suspect that those are the $(w: X \to Y) \in W$ such that, if $w^*$ is their formally adjoined inverse, the spaces of homotopies $ww^* \to \mathrm{id}_X$ and $w^*w \to \mathrm{id}_Y$ are $n$-connected in $\tilde{C}$.

Call the set of these $W_n$, in particular $W_0 = W$ and $W_\infty = \mathrm{Iso}$. So my guess would be that $C_n$ should be $C[W_n^{-1}]$ with the class of weak equivalences generated by the image of $W$ after localisation.

Things I don't know: does this give the right thing, is $W_n$ closed under 2-out-of-3, is there a characterisation of $W_n$ that does not refer to $\tilde{C}$ at least when we have more structure (eg model categories)?

Amar Hadzihasanovic (Apr 03 2020 at 09:54):

(Maybe should be $(n-1)$-connected there, I'm getting a bit confused between dimensions in the hom-spaces and dimensions in the outside category)