The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
0answers
94 views

Recognition principle for 2-categories (2-groupoids)

Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...
6
votes
1answer
186 views

Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?

Consider the category C of pointed simplicial sets. The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C models the suspension and loop functors on the underlying ∞-category of C. There is another ...
8
votes
1answer
298 views

Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...
8
votes
0answers
129 views

Homotopy theory of suplattices

In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ...
10
votes
0answers
375 views

Has this chain complex associated with a simplicial complex been studied before?

I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference. Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian ...
0
votes
1answer
109 views

How to define $\Lambda^0_0$, 0-horn of a simplicial point

This is really a trivial question. The 0-horn of a simplicial point $\Delta^0$ is not defined nor remarked in the books and papers I could find. So one expect if we could make some meaningful ...
1
vote
1answer
306 views

A question about the proof of Quillen's Theorem A

(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.) Theorem (Quillen) ...
1
vote
0answers
39 views

Prove the functor $[n]\to [n]\star [n]$ preserves inner anodyne

Let $f:\Delta\to \Delta$ be the functor given by $[n]\mapsto [n]\star[n]=[2n+1]$. We can extend $f$ cocontinuously to a functor $$f_!: SSet\to SSet$$ (that is, the left adjoint of the functor $f^*$. ...
12
votes
2answers
286 views

Group cohomology without G-modules (a.k.a. what does this bar construction compute?)

Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution. For instance, let's take $G = \mathbb{Z}^2$, and "resolve": $$ 0 \to ...
5
votes
1answer
84 views

Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommutative algebras

Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where $$ C_n(A):=A^{\otimes n+1} ...
4
votes
1answer
144 views

Cocycle condition for equivariant sheaves

Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$ : $X_n := G^n \times X$, $s_0:X_0 ...
12
votes
2answers
530 views

Multisimplicial geometric realization

Does anyone know a reference or proof for the following? Let $k\geq 1$ and let $X$ be a space. There is a $k$-fold multisimplicial set whose simplices in degree $n_1,\ldots,n_k$ are the maps ...
3
votes
1answer
157 views

What is the suitable setting for supercoherence with value in a bicategory?

It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as ...
6
votes
1answer
785 views

Homotopy pullbacks of simplicial spaces, and Bousfield-Friedlander

Let $X_\bullet \longrightarrow Y_\bullet \longleftarrow Z_\bullet$ be a diagram of simplicial spaces (=bisimplicial sets, if you like). On p. 14-9 of these notes there is an example which shows that ...
9
votes
1answer
199 views

Equivariant homotopy, simplicially

It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are ...
1
vote
0answers
61 views

What is a homotopy between bisimplicial maps

I originally posted it on stackexchange, but did not get any comments or answer, so I try my luck here. I am looking for the naive notion of homotopy. For maps $f, g: A\to B$ of simplicial sets ...
1
vote
0answers
93 views

The bar construction or the quotient for monoidal category action on a category

Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category, $$ N_\bullet (M\rtimes C)= \cdots M\times ...
9
votes
1answer
294 views

What is the precise relationship between “prodsimplicial sets” and rooted trees?

In Keven Walker's answer to the question, Cubical vs. simplicial singular homology it is written: Personally, I think it is more convenient to do singular homology with the larger collection ...
8
votes
1answer
621 views

What are simplicial topological spaces intuitively?

(This is a repost of a question from MSE. I hope there is more to say.) I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial ...
0
votes
0answers
130 views

Thomason’s homotopy colimit theorem for pseudo-functor

Thomason’s homotopy colimit theorem R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Camb. Phil. Soc. (1) 85 (1979)91-109 says that for a functor $F:I^{op}\to ...
7
votes
2answers
535 views

Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds. The real question here is, do simplicial object in a ...
2
votes
3answers
221 views

How to detect if a simplicial set is the nerve of a groupoid?

I have the following question. Suppose I have a simplicial set. Is there a way to detect if it actually is isomorphic to a nerve of a groupoid? I've seen the fact that if you have a nerve ...
6
votes
0answers
135 views

Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial ...
2
votes
0answers
55 views

Relationship of height zero hypercovers to co-cartesian condition on cosimplicial modules

Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that ...
5
votes
2answers
264 views

How does one Segal-subdivide a 2-category?

Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...
1
vote
0answers
91 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
6
votes
1answer
316 views

Morita theorem for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
1
vote
0answers
100 views

Does compactly supported cohomology make sense for cosimplicial spaces?

As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only ...
4
votes
4answers
729 views

Homotopic quotients of simplicial sets as infinity-groupoids

Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of ...
3
votes
0answers
100 views

Weak notions of Kan fibrations

I have two semi-simplicial complexes $X_\bullet$ and $Y_\bullet$, along with a simplicial map $f_\bullet: X_\bullet \to Y_\bullet$. Now $f_\bullet$ is not a Kan fibration: for an $n$-simplex in ...
11
votes
2answers
515 views

Is every connected space equivalent to some B(Aut(X))?

Given a connected space $B$, is there always some space $X$ with $B \simeq \mathbf{B}(\mathrm{Aut}(X))$? Here by space I mean simplicial set, by $\mathrm{Aut}(X)$ I mean the simplicial monoid of ...
9
votes
1answer
260 views

Exponentiation in finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set ...
1
vote
2answers
241 views

Finite simplicial sets

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in ...
5
votes
1answer
125 views

On the obstruction of a sequence of simplicial spaces that is levelwise a fibration

Given a sequence of simplicial spaces (actually bisimplicial sets) $$F\to E\to B$$ that is level-wise a fibration, then the geometric realisation does not necessarily have to be a fibration. If I ...
9
votes
2answers
351 views

Do Homotopy Fully Faithful Functors Push-out?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces. Are homotopy fully faithful functors preserved under (homotopy) pushout? More ...
3
votes
1answer
364 views

Projective objects in HTT

In HTT.5.5.8.18 Lurie defines a projective object $P$ in a quasicategory $\bf C$ as an object such that its corepresented functor ${\rm Map}(P,-)$ "commutes with geometric realizations". I can catch ...
4
votes
1answer
232 views

Two definitions of modules in monoidal category

The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here ...
11
votes
2answers
1k views

What is the homotopy theory of categories?

I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of ...
4
votes
1answer
221 views

Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?

We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors: $H:S\to SS$ forgetting the ordering on simplices and ...
0
votes
1answer
101 views

What is the left adjoint for taking rows of a bisimplicial set?

Let $aSS$ / $abSS$ be the category of augmented bi/simplicial sets (one can also consider $SS$/$bSS$ be the usual bi/simplicial sets, the results should related in some reasonable way.) There is an ...
6
votes
2answers
321 views

Simplicial replacements in smoothing theory

As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...
-1
votes
1answer
69 views

Poset complex of reverse ordering [closed]

This might be too easy but I cannot proof it easily. Any reference or hint will be great. Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. ...
4
votes
0answers
164 views

What does it mean for the $\Pi_\infty$ groupoid to be fully fibrant?

The original question I had was: If I have a sequence of simplicial spaces $$A\to B\to C$$ which is degree-wise a homotopy fibration, under which conditions is the geometric ...
8
votes
0answers
102 views

Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...
2
votes
0answers
143 views

The normalised cochain complex, totalisation and cosimplicial simplicial $R$-modules

Short Version Given a cosimplicial space $X_\bullet$, what is the relationship between (co)chains on the totalisation of $X_\bullet$ and the totalisation of the cosimplicial chain complex obtained by ...
1
vote
1answer
160 views

Finding automorphism groups of simplicial complexes

Question: Given a finite simplicial complex $K$, what general techniques allow one to efficiently compute (a presentation of) the group $\text{Aut}(K)$ of $K$'s automorphisms? Since this is ...
13
votes
0answers
303 views

Asymptotics for the number of triangulations of a manifold M

In Gromov's talk at the Clay Math Research from 23:23 to 25:55 Gromov says (slightly paraphrased) I want to emphasize a problem which comes from mathematical physics which is unsolved which is ...
0
votes
1answer
200 views

What are the morphisms in the category of zig-zags?

For some reason I am having trouble locating a transparent explanation of precisely what are the morphisms in the category of zig-zags. The objects of this category are specified completely by triples ...
7
votes
1answer
261 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
5
votes
0answers
166 views

Geometric realization of simplicial spaces and finite limits

Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$. Does this geometric realization of simplicial spaces preserve finite ...