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Stream: deprecated: show and tell

Topic: Quasi-categories

Thomas Read (Aug 08 2020 at 12:39):

I recently learned the definition of a quasi-category (having previously had a vague impression that it would be terribly complicated and inaccessible!), and wrote a blog post about it, starting with a description of simplicial sets and the nerve construction.

One thing I'd have liked to talk about is how to see intuitively that this is the "right" definition, beyond the fact that it generalises the nerve of a category. I'd be interested if anyone had any thoughts on that.

Christian Williams (Aug 08 2020 at 20:52):

Looks great. What software do you use to make this blog?
Ah, I see that it's https://jaspervdj.be/hakyll/

Thomas Read (Aug 08 2020 at 21:33):

Thanks! Yup - the source code is all here

philip hackney (Aug 09 2020 at 13:39):

Thomas Read said:

One thing I'd have liked to talk about is how to see intuitively that this is the "right" definition, beyond the fact that it generalises the nerve of a category. I'd be interested if anyone had any thoughts on that.

Nice post!

I'm not sure from an intuitive viewpoint that quasi-categories actually give the "right" definition. Yes, you do have n-cells for all n, but because you're modeling things with simplices, the sources and targets of these aren't just (n-1)-cells. You can see this already at the level of 2-simplices: the source doesn't look like a 1-cell, but rather as a pair of such 1-cells. I'm sure you can think of other weird anomalies, like if you start with a pair of 2-simplices, you can write down more than one inner horn of Δ[3] that will give you potential compositions.

Rune Haugseng (Aug 09 2020 at 14:40):

One motivation for the inner horn conditions is that they're the obvious way to extend both the conditions for nerves of categories (unique lifting for inner horns) and Kan complexes (non-unique liftings for all horns), so quasicategories include both categories and $\infty$ -groupoids.

Rune Haugseng (Aug 09 2020 at 14:43):

You can also prove that a simplicial set X is a quasicategory iff the restriction $X^{\Delta^2} \rightarrow X^{\Lambda^2_1}$ is a trivial fibration, which intuitively says that X is a quasicategory if every pair of composable edges has a contractible space of compositions.

Thomas Read (Aug 09 2020 at 15:37):

Rune Haugseng said:

You can also prove that a simplicial set X is a quasicategory iff the restriction $X^{\Delta^2} \rightarrow X^{\Lambda^2_1}$ is a trivial fibration, which intuitively says that X is a quasicategory if every pair of composable edges has a contractible space of compositions.

Great - I was hoping there would be a theorem roughly like this! That's enough to convince me that quasi-categories actually are a canonical way of identifying category-like simplicial sets.

philip hackney said:

Yes, you do have n-cells for all n, but because you're modeling things with simplices, the sources and targets of these aren't just (n-1)-cells. You can see this already at the level of 2-simplices: the source doesn't look like a 1-cell, but rather as a pair of such 1-cells.

Yes, this is an important point - I think I spent a while vaguely assuming that n-simplices correspond to n-morphisms, since we can certainly think of 1-simplices as 1-morphisms, but they don't correspond in general. I don't know if there's a way to think about an n-simplex as an object in its own right, in order to explain the horn condition, or whether the best we can do is to think of an n-simplex as making up part of the n-dimensional data.

Adrian Clough (Aug 10 2020 at 21:54):

Rune Haugseng said:

You can also prove that a simplicial set X is a quasicategory iff the restriction $X^{\Delta^2} \rightarrow X^{\Lambda^2_1}$ is a trivial fibration, which intuitively says that X is a quasicategory if every pair of composable edges has a contractible space of compositions.

A small remark: When I first saw this theorem it felt like cheating, as one is using the Joyal model structure to define trivial fibrations, so that one is still implicitly using inner horn filling... except that this is not really the case! All Cisinski model structures on simplicial sets have the same trivial fibrations, so somehow one is appealing to "absolute trivial fibrations" in this scenario. So e.g. the absolute trivial fibrations are the same in the Kan-Quillen and the Joyal model structures.

Adrian Clough (Aug 10 2020 at 22:01):

Thomas Read said:

I recently learned the definition of a quasi-category (having previously had a vague impression that it would be terribly complicated and inaccessible!), and wrote a blog post about it, starting with a description of simplicial sets and the nerve construction.

One thing I'd have liked to talk about is how to see intuitively that this is the "right" definition, beyond the fact that it generalises the nerve of a category. I'd be interested if anyone had any thoughts on that.

An elementary way of seeing that the definition of quasi-categories is reasonable is the following: The horn lifting in dimension 2 is obvious. Then consider a string of 3 composable arrows, and see how far you get with only filling $\Lambda_1^2 \hookrightarrow \Delta^2$ ; you will see that if you stop one step short of getting two different compositions of all 3 arrows, you will obtain precisely one of the inclusions $\Lambda_1^3 \hookrightarrow \Delta^3$ or $\Lambda_2^3 \hookrightarrow \Delta^3$ , so these clearly need to be trivial cofibrations. Then start composing a string of 4 composable arrows with using only $\Lambda_1^2 \hookrightarrow \Delta^2, \Lambda_1^3 \hookrightarrow \Delta^3, \Lambda_2^3 \hookrightarrow \Delta^3$ ...

Thomas Read (Aug 11 2020 at 08:13):

Thanks, I like this approach. That also gives a more motivated definition of inner horn - rather than defining an inner horn as an n-simplex minus the (n-1)-simplex opposite vertex k for 0 < k < n, we can define an inner horn as an n-simplex minus an (n-1)-simplex which includes the edge <0, n>. You can really viscerally imagine holding an n-simplex, taking a knife and cutting it open from that edge...

ADITTYA CHAUDHURI (Aug 13 2020 at 06:03):

We know that if $X, Y$ are homotopy Equivalent then $\pi_{\leq 1}(X)$ and $\pi_{\leq 1 }(Y)$ are equivalent as groupoids. Since Fundamental Groupoid of a space $X$ is the Homotopy category of the Singular Simplicial Set $Sing(X)$ I am expecting an analogue of my first statement in the $\infty$ - category level.

Precisely I am expecting the following :
If $X$ and $Y$ are weakly homotopy Equivalent then $Sing(X)$ and $Sing(Y)$ are equivalent as $\infty$ -categories. Moreover since unlike fundamental groupoids, Singular simplicial sets contain the information about all Homotopy Groups of a space I am also expecting a converse of the statement. Precisely if for $X$ and $Y$ , $Sing(X)$ and $Sing(Y)$ are equivalent then $X$ and $Y$ are weakly homotopy equivalent and is in fact homotopy equivalent provided $X$ and $Y$ have the homotopy type of C.W Complex.

Are my expectations making any sense? (Apology in advance if it sounds stupid... I am a beginner)

Ulrik Buchholtz (Aug 13 2020 at 11:45):

ADITTYA CHAUDHURI said:

If $X$ and $Y$ are weakly homotopy Equivalent then $Sing(X)$ and $Sing(Y)$ are equivalent as $\infty$ -categories. Moreover since unlike fundamental groupoids, Singular simplicial sets contain the information about all Homotopy Groups of a space I am also expecting a converse of the statement. Precisely if for $X$ and $Y$ , $Sing(X)$ and $Sing(Y)$ are equivalent then $X$ and $Y$ are weakly homotopy equivalent and is in fact homotopy equivalent provided $X$ and $Y$ have the homotopy type of C.W Complex.

This is exactly right. Since you're posting under the heading of “Quasi-categories”, I take your statements to about the Joyal model structure on simplicial sets. But it doesn't really matter, since $\mathrm{Sing}(X)$ is always a Kan complex, and a map between Kan complexes is a Joyal equivalence iff it is a Quillen equivalence. (This is because the identity is a right Quillen functor from $\mathrm{sSet}_{\mathrm{Quillen}}$ to $\mathrm{sSet}_{\mathrm{Joyal}}$ , and thus preserves equivalences between fibrant objects, and as a left Quillen functor in the other direction, the identity preserves all equivalences, since all objects are cofibrant. This of course implements the model-independent fact that a functor between $\infty$ -groupoids is an equivalence iff it is an equivalence when considered as a functor between groupoidal $(\infty,1)$ -categories.)
Since $\mathrm{Sing}$ is a (right) Quillen equivalence from $\mathrm{Top}_{\mathrm{Quillen}}$ to $\mathrm{sSet}_{\mathrm{Quillen}}$ , the statements follow.

ADITTYA CHAUDHURI (Aug 13 2020 at 15:41):

@Ulrik Buchholtz Thank you!! Yes, I assumed the Joyal Model structure on Simplicial Sets.

ADITTYA CHAUDHURI (Aug 15 2020 at 14:28):

@Ulrik Buchholtz Sorry for being late. I have a little confusion which I am describing below. (Thanks in advance)

I asked
If $X$ and $Y$ are weakly homotopy Equivalent then $Sing(X)$ and $Sing(Y)$ are equivalent as $\infty$ -categories. Moreover since unlike fundamental groupoids, Singular simplicial sets contain the information about all Homotopy Groups of a space I am also expecting a converse of the statement. Precisely if for $X$ and $Y$ , $Sing(X)$ and $Sing(Y)$ are equivalent then $X$ and $Y$ are weakly homotopy equivalent and is in fact homotopy equivalent provided $X$ and $Y$ have the homotopy type of C.W Complex. Are my expectations making any sense?

My understanding:
There exist a Quillen Equivalence between sSet and Top where Singular Simplicial Set Sing is the right adjoint and the Geometric Realization $|-|$ is the left adjoint. Now with the standard model structure on Top weak homotopy equivalences are the Weak Equivalences .

(1) Now if $f: X \rightarrow Y$ be a Weak Homotopy Equivalence, since Sing preserves Weak Equivalences it is evident $Sing(f): Sing(X) \rightarrow Sing(Y)$ is a Weak Equivalence which is precisely an equivalence of Quasicategories.(Equivalence of Infinity categories which I was mentioning).

(2) Now if it is mentioned that $X$ and $Y$ are Weak homotopy equivalent that is by definition (https://ncatlab.org/nlab/show/weak+homotopy+equivalence#RelationToHomotopyTypes Remark 3.6) there exists a ZigZag of Weak Homotopy Equivalences between $X$ and $Y$ then also it is evident that $Sing(X)$ is Weakly Equivalent to $Sing(Y)$ as Sing preserves Weak Equivalences.

(3) Converse Direction:
Let for spaces $X$ and $Y$ , $Sing(X)$ and $Sing(Y)$ are Weakly Equivalent in sSets. So by definition there exists a ZigZag of Weak Equivalence between $Sing(X)$ and $Sing(Y)$ . We know that the Geometric Realization $| - |$ preserves Weak Equivalences. Hence $|Sing(X)|$ is Weakly homotopy Equivalent(Hence Homotopy Equivalent) to $|Sing(Y)|$ .
My Claims were:
(A) $X$ and $Y$ are Weakly Homotopy Equivalent.
(B) If $X$ and $Y$ has the Homotopy type of C.W Complexes then $X$ and $Y$ are Homotopy Equivalent.
Part(A)
Now to prove (A) it is sufficient to show that for a given space $M$ the Geometric Realization of the corresponding Singular Simplicial Set $|Sing(M)|$ is weakly homotopy equivalent to $M$ .

Now we know that $| - |$ and $Sing$ are adjoint pairs . So if we can prove that the counit of the adjunction $\epsilon_X: |Sing(X)| \rightarrow X$ is a Weak Equivalence in Top we will be done. But I am not able to prove it immediately.( I am a beginner)

Can you please suggest any reference where I can find the proof of the fact that for a given space $M$ the Geometric Realization of the corresponding Singular Simplicial Set $|Sing(M)|$ is weakly homotopy equivalent to $M$ ? I am guessing that my claim is true from the following question in Math Overflow https://mathoverflow.net/questions/25541/is-geometric-realization-of-the-total-singular-complex-of-a-space-homotopy-equiv?noredirect=1&lq=1

Rune Haugseng (Aug 15 2020 at 16:26):

That the counit is a weak equivalence is part of the statement that you have a Quillen equivalence between topological spaces and simplicial sets, so any reference for that should prove this.

ADITTYA CHAUDHURI (Aug 15 2020 at 16:32):

@Rune Haugseng Thanks

ADITTYA CHAUDHURI (Aug 15 2020 at 18:00):

@Rune Haugseng @Ulrik Buchholtz Thanks!! and also Sorry for asking a stupid question. I realised that the Answer of my Part (A) is directly following from the definition of Quillen Equivalence.