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Stream: learning: questions

Topic: pushouts of Theta_n-Categories and fibrant replacement

Noah Chrein (Mar 21 2025 at 21:34):

Hello, I am constructing some structures in $\Theta_nSp$ and I just want to make sure the following is cogent / ask where my intuition is flawed:

Consider a span of $\Theta_n$-Spaces $A \leftarrow M \to B$, the ordinary pushout $A+_MB$ is a presheaf $[\Theta_n^{op},sSet]$ but it is not necessarily fibrant; it does not satisfy the segal conditions. I am looking to apply fibrant replacement to this pushout $(A+_MB)^f$ to obtain a $\Theta_n$-space.

for example, gluing two representable 1-morphisms: $\Delta^1+_{\Delta^0}\Delta^1$ is not a $\Theta_1$-space, but the fibrant replacement should be like
$(\Delta^1+_{\Delta^0}\Delta^1)^f \simeq \Delta^2$

Is fibrant replacement even valid for the segal conditions? Rezk's paper defines a cartesian model category whose fibrant objects are the $\Theta_n$-spaces, so it should follow from general results on model categories, but the fibrations are built on top of a different model structure on $[\Theta_n^{op},sSet]$ namely, the injective model structure induced from kan fibrations in $sSet$. So I am a bit weary that fibrant replacement is acting with respect to the underlying model structure (i.e. adding identities) instead of with respect to the segal conditions (adding appropriate composites).

Thank you for your consideration of this question, below is a bit more info on what I am trying to do specifically.

Noah Chrein (Mar 21 2025 at 21:47):

A bit more info as to what I am trying to do:

Let $\mathbb G_k:\Theta_kSp$ the walking k-globe and $\partial \mathbb G_k:\Theta_kSp$ its boundary. We can define a "walking twisted cobordism" $T_k:\Theta_kSp$ meant to generalize the shape of the twisted arrow construction, but for iterated whiskerings of higher morphisms. Inductively $T_k$ can be defined by

$T_1 = \mathbb G_1 + \mathbb G_1 = \left \{0 \to 1\;\;\; 2\to3\right\}$ and
$T_{k+1} = \left (\;\; [1](T_k) +_{\partial\mathbb G_{k+1}} [3](\mathbb G_0,\partial \mathbb G_k,\mathbb G_0)\;\;\right )^f$

I can give more info and pictures if desired, but I just wanted to give some context for the question above.

Clémence Chanavat (Mar 21 2025 at 22:58):

Yes the fibrant replacement will "add" the Segal and completeness conditions. As you said, this model structure is built as a (left Bousfield) localisation of the injective model structure $[\Theta_n^{op}, sSet]_{inj}$, and this localisation is done at the "walking Segal conditions" (see 5.1 of the paper you linked), the "walking completeness conditions". (see 7.6 of the paper you linked), (and also at the levelwise acyclic cofibrations of the previous level $(n - 1)$)

To simplify, just consider the case $n = 1$, then the walking Segal conditions are inclusions the $j_n \colon \Delta^1 +_{\Delta^0} + \cdots +_{\Delta^0} + \Delta^1 \to \Delta^n$. The localisation of the model structure $[\Theta_1^{op}, sSet]_{inj}$ at the set of maps $J = \{ j_n \mid n \geq 1 \}$ is a new model structure on $[\Theta_1^{op}, sSet]$ with the property that an object $W$ is fibrant iff it was already fibrant in the injective model structure (i.e. $W$ is pointwise a Kan complex), and furthermore, for all map $j_n \in J$, the induced morphism of Kan complexes $j_n^* \colon \hom(\Delta^n, W) \to \hom(\Delta^1 +_{\Delta^0} + \cdots +_{\Delta^0} \Delta^1, W)$ is a weak equivalence, i.e. $W_n \to W_1 \times_{W_0} \times \cdots \times_{W_0} W_1$ is a weak equivalence, i.e. $W$ is a Segal space. Proceed similarly for completeness.

Noah Chrein (Mar 22 2025 at 18:04):

Thank you, and I understand the localization w.r.t. the set of complete/segal maps which yields the model structure on $[\Theta_n^{op},sSet]$, defining $\Theta_nSp$ as the fibrant objects. 

I am interested in gluing together $\Theta_n$ spaces via pushouts, and then "completing" them, to again be $\Theta_n$-spaces, namely $(A+_C B)^f:\Theta_nSp$

In more detail, consider the suspension functor
$[1]:\mathbb \Theta_kSp \to \mathbb \Theta_{k+1}Sp$

Is it wise to construct a sequence of $\Theta_k$-spaces by induction on k, using suspension, pushout, and fibrant replacement? I want to do something like 

$A_{k+1} = (\;\;[1]A_k\;\;\;+_{C_{k+1}} B_{k+1}\;\;\;)^f$
Where $A_k:\mathbb Cat_k$ and $A_{k+1}:\mathbb Cat_{k+1}$

In my specific case, I am consructing
$T_{k+1} = (\;\;\;[1]T_k + _{\partial \mathbb G_{k+1}} [3](\mathbb G_0, \partial \mathbb G_k, \mathbb G_0)\;\;)^f:\mathbb Cat_{k+1}$.
Where $\mathbb G_k$ is the walking k globe $[1]^k(\bullet)$ and $\partial \mathbb G_k$ is its boundary.

Clémence Chanavat (Mar 22 2025 at 19:02):

If I understand correclty, this is a picture of $T _2$ and $T_3$? pic

Clémence Chanavat (Mar 22 2025 at 19:25):

I'm not the one to say what is wise anb what is not, but I guess one problem with fibrant replacements in those kind of model structures is that it's very hard to say what they are exactly, so if you start to nest them by induction you will end up with very complicated objects.

Clémence Chanavat (Mar 22 2025 at 19:25):

If you insist that your $T_k$ is a $\Theta_k$ space, my first suggestion would be to instead define $S_1 = G_1 + G_1$, and $S_{k + 1} = [1]S_k +_{\partial G_{k + 1}} [3](G_0, \partial G_k, G_0)$, and then only later $T_k = S_k^f$ to be the fibrant replacement of walking shapes you are interested in. So that you only have one fibrant replacement to compute, and do not nest them together.

Clémence Chanavat (Mar 22 2025 at 19:25):

Then, my second, more speculative, suggestion would be to not take a fibrant replacement at all: you say that you are defining "walking shapes" and, correct me if I'm wrong, but I imagine that you will use $T_k$ to consider some collections of maps $F : T_k \to W$, for $W$ a $\Theta_k$ space. This will define the generalised twisted arrows in $W$? If so, the general philosophy is that you do not need your $T_k$'s to be fibrant in order to do that: cofibrant objects are good to model walking shapes, and fibrant objects are good to receive morphisms from those walking shapes.

Clémence Chanavat (Mar 22 2025 at 19:33):

For example, look at those walking Segal or completeness conditions whose shapes are not fibrant, and but are meant to be mapped into fibrant objects (to make them even more fibrant in some sense), or if you have a $\Theta_k$ space $W$, then for any object $K$, the "functor category" $[K, W]$ is again a $\Theta_k$ space (regardless if $K$ is fibrant or not), and $[K, W]$ is to be thought as the $\Theta_k$-spaces of $K$-shapes in $W$

James Deikun (Mar 23 2025 at 10:37):

Yes; the ability to use non-fibrant diagram shapes is one of the biggest advantages to using a model category to do higher category theory, there's no need to throw it away.

Noah Chrein (Mar 24 2025 at 17:34):

Clémence Chanavat your images:

differ from mine in the choice of
$\partial \mathbb G_{n+1} = [1](\partial \mathbb G_{n}) \overset{f}\to [3](\mathbb G_0, \partial \mathbb G_{n}, \mathbb G_0)$.
You are using the "middle inclusion" $f = [f_{1,2}](id)$,
I am using the "whiskered inclusion" $f = [f_{0,3}](!,id,!)$
where $[1] \overset{f_{i,j}}\to [3]:\Delta$ are the respective face maps.

Here is how you do this gluing, note my usage of independent labels 1,2,1',2',1'',2'' to indicate gluing:
T

Re-arranged, these cobordisms look like classical cobordisms between boundaries of globes:
cob.

I would like to glue composible s-length chains together to get something like
$T_{n,s} = T_n+_{\partial \mathbb G_n}...+T_n$, and $T_{n,0} = \partial \mathbb G_n$

Further, I would like to define a simplicial object
$K_{\partial \mathbb G_n}:\Delta^{op} \to sSet$
by $K_{\partial \mathbb G_n}([s]) = Map(T_{n,s},K)$
and I want $K_{\partial \mathbb G_n}$ to be a $\Theta_1$-space.

I am wondering, to both Clémence and James Deikun's point, if regardless of the fibrancy of the $T_{n,s}$ and due to the fibrancy of $K$, whether $K_{\partial \mathbb G_n}$ is automatically fibrant (i.e. is a $\Theta_1$-space). I need $K_{\partial \mathbb G_n}$ to behave like a category because it will induce composable fibered actions on $Map(\mathbb G_n, K)$

Thank you for your interest and consideration!

Noah Chrein (Mar 24 2025 at 17:59):

Btw the difference between the fibrancy points that both of you brought up, that [T,K] is automatically fibrant, and the property that I am looking for is that $[T_n,K]$ is not the exact structure I am looking to work with, it has the wrong objects and morphisms. I am constructing $K_{\partial \mathbb G_n}$ by hand.

Perhaps there is a general notion that, given a cosimplicial object $\Delta\overset{T}\to\Theta_nSp$, the resulting simplicial object $Map(T(-), K):\Delta^{op} \to sSet$ is a $\Theta_1$-space?

Noah Chrein (Mar 24 2025 at 18:32):

This conjecture could take on the assumption that $T:\Delta \to \Theta_nSp$ is inductively built from colimits of suspensions of representables. Or, even more, if T is built from fibrant replacements of colimits of suspensions of representables

Clémence Chanavat (Mar 24 2025 at 21:34):

Ok I see better what you want to do (I was confused as to why you would called it generalised twited arrow, that makes more sense to me now hehe)

Clémence Chanavat (Mar 24 2025 at 21:34):

Concerning your definition of $K_{\partial G_n }$, I do not know how to make a simplicial object at all, since I do not see any way to construct the inner faces $d_i \colon T_{n, s + 1} \to T_{n, s}$ for $0 < i < s + 1$. Like you would want them to be a composite of the $i$ th and $(i + 1)$ th copy of the "tube" $T_n$, but those composite do not exist in $T_{n, s + 1}$, so I guess let's take a fibrant replacement: but even then, if you somehow manage to explicit all your simplicial structure for $s \mapsto T_{n, s}$ the best you can hope for is that your simplicial identities hold only up to some homotopy.

Clémence Chanavat (Mar 24 2025 at 21:34):

Maybe a not too bad alternative would be to construct your shapes $T_{n, s}$ in strict $\omega$-category: since everything will be a polygraph it should not be too hard there to manage the composites and exhibit the simplicial structure for $s \mapsto T_{n, s}$, (or even better, you can do this very easily with maps and comaps of regular directed complexes, but I don't want to do no proselytism) and then you use a nerve to send your construction to $\Theta_n$-spaces (I assume there is such nerve, but wouldn't know where to find it in the literature), so you will end up indeed with a cosimplicial object $T_n \colon \Delta \to \Theta_n Sp$

Clémence Chanavat (Mar 24 2025 at 21:49):

As for your last question, I don't think this is the case in general, but yeah it feels it should be true in your case: like the inclusion $i \colon T_{n ,1} +_{T_{n, 0}} T_{n, 1} \to T_{n, 2}$ wants to be a weak equivalence, in fact even a fibrant replacement! But this inclusion $i$ is nothing else than $T(\Lambda^1_2 \to \Delta_2)$, thus by adjunction $Map(T-, K)$ has the right lifting property against $\Lambda^1_2 \to \Delta_2$, and similarly for all other inner horns: saying this should not be too far from saying that $Map(T-, K)$ is a $\Theta_1$ space

Clémence Chanavat (Mar 25 2025 at 07:42):

Actually, I find your last question very instructive, at least for me: simplify it to the maximum so consider your basic twisted arrow shape to be only the arrow $\Delta_1$, and works with quasicategories $\cong \Theta_1$ spaces.

Clémence Chanavat (Mar 25 2025 at 07:42):

Now the question becomes: is there a functor $S \colon \Delta \to sSet$ such that for all $n \geq 1$,

• $S_n$ is a quasicategory, and is somewhat a fibrant replacement of $\Delta_1 +_{\Delta_0} \cdots +_{\Delta_0} \Delta_1$.
• The simplicial set $sSet(S-, K)$ is a quasicategory, for all quasicategory $K$ ?

Clémence Chanavat (Mar 25 2025 at 07:43):

well yes there is such thing, it's none other than the Yoneda embedding! Indeed, for all $n$, and $S_n = \Delta_n$ is a quasicategory, since it is the nerve of the category $[n] = \{ 0 \to 1 \to \cdots \to n\}$, and the inclusion $\Delta_1 +_{\Delta_0} \cdots +_{\Delta_0} \Delta_1 \to \Delta_n$, also known as the spine, is an acyclic cofibration for quasicategories! And $sSet(S-, K)$ is a quasicategory, since it is isomorphic to $K$ by the Yoneda Lemma

Clémence Chanavat (Mar 25 2025 at 07:45):

Now, the same idea will also work to define your first twisted arrow shape $T_1$, since $T_{1, n} = S_n + S_n$ is also the nerve of the coproduct $[n] + [n]$, and then maybe it reinforces the idea of building your cosimplicial object $T_n$ first in polygraphs then passing to the nerve to get a $\Theta_n$ space, since this is what this does for the base case $T_1$ anyway

Noah Chrein (Mar 27 2025 at 22:55):

It would be nice if the $T_{n,-}:\Delta \to \Theta_nSp$ is actually a theta space. But to the above points, I am using them as probes so for now we seek $T_{n,-}:\Delta \to [\Theta_n^{op},sSet]$ and leave fibrant replacement out of the picture.

Let $A:\Theta_n$, we have the map $[3]([0],A,[0]) \to [1](A)$ to be viewed as collapsing the left and right 1-morphisms represented by the $[0]$ into identities.

In particular we have maps:
$[3](\mathbb G_0,\partial \mathbb G_n, \mathbb G_0) \overset{d_{n+1}}\to [1](\partial \mathbb G_n) \cong \partial \mathbb G_{n+1}$

we also have
$T_{1} = y\mathbb G_1+y\mathbb G_1 \overset{\delta_1}\to y\mathbb G_0 + y\mathbb G_0 \cong y\partial \mathbb G_1$

Assume for induction $T_n \overset{\delta_n} \to \partial \mathbb G_n$ and build
$\delta_{n+1} = [1]\delta_n +_{\partial \mathbb G_{n+1}} yd_{n+1}$.

There are a few details missing here because I haven't posted the full details of the $T_n$ construction (I can but I would like to keep things intuitive). Essentially the $\delta_n$ contracts all whiskering higher morphisms to identities, so $[1](\delta_n)$ contracts all whiskering $k\geq 2$ morphisms, and $d_{n+1}$ handles the 1-morphisms.

Anyway, we are left with a reflexive graph
$\Delta|^1 \to [\Theta_n^{op}, sSet]$
$T_n \rightrightarrows \partial \mathbb G_n$ and $T_n \overset{\delta_n}\leftarrow \partial \mathbb G_n$
so we can generate the face and degeneracy maps of $T_{n,s}$ by pushout

Clémence Chanavat this and essentially everything you said below that is what I am trying to do. I think at this point, the easiest thing to do is forget the fibrancy requirements on $T$ and rearrange the pushouts in the definition of $T_n$ to be amenable to the segal condition on $[T_{n,s}, K]$.

Thank you for the help!