All Questions
Tagged with homological-algebra simplicial-stuff
46 questions
7
votes
1
answer
310
views
Homotopy between posets
This is entirely a new area for me and I apologise in advance if the questions are silly.
In Quillen's paper "Homotopy properties of the posets of non-trivial $p$-subgroup of a group" (see ...
- 139
4
votes
0
answers
70
views
Explicit formula for Dold-Kan projection to normalized Moore complex
When proving the classical Dold–Kan correspondence (as e.g. in Goerss-Jardine book or here http://math.uchicago.edu/~amathew/doldkan.pdf), one associates three chain complexes to a simplicial abelian ...
- 66
2
votes
0
answers
91
views
Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms
Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....
- 245
3
votes
0
answers
167
views
Simplicial resolution for commutative group scheme
Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...
- 41
2
votes
0
answers
171
views
Motivation for working with augmented objects in homological or higher algebra
I would like to understand if there is deeper reason/motivation behind
augmentations in homological algebra. Recall classically in homology
if there is a complex of free $R$-modules ($R commutative ...
- 6,018
12
votes
4
answers
2k
views
Applications of the Dold-Kan correspondence
The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not ...
- 129
0
votes
0
answers
203
views
Double complex of simplicial resolution
In his
lectures on condensed mathematics on page 30 Peter Scholze speaks of the double complex of a simplicial resolution. How is this defined?
In the next line, he writes that if $A_\bullet$ is a ...
user473423
3
votes
0
answers
170
views
Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?
Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...
- 2,152
2
votes
1
answer
163
views
Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets
I am stuck at a critical step in my master's thesis. If someome can help me out here, I will give appropriate credit.
We know that the data of a 2-group $G$ can be given by a group $\tau_0(G)$ and a 3-...
2
votes
0
answers
140
views
$\Omega^1_{B_\bullet/A_\bullet}$ is acyclic if $A_\bullet \to B_\bullet$ is quasi-isomorphism
Let $A_\bullet \to B_\bullet$ be a quasi-isomorphism of simplicial rings in the sense of (P.62, I.3.1.7, Complexe Cotangent et Déformations I, Luc Illusie).
Then, we define the simplicial $B_\bullet$-...
- 435
2
votes
0
answers
65
views
homotopy coherent G-action on tensor product of complexes
Let $G$ be a discrete group and $k$ a field. Suppose $C_1$ and $C_2$ are complexes over $k$ with homotopy coherent actions of $G$ in the sense of Cordier (I've been reading https://arxiv.org/pdf/1801....
- 101
3
votes
0
answers
111
views
"Boundaries" in Free Simplicial Monoids
I suspect that this has been addressed somewhere already, but I cannot find anything. Let $\hat{\Delta}$ denote simplicial sets, and $Mon\hat{\Delta}$ simplicial monoids. There is a forgetful functor $...
- 713
6
votes
1
answer
385
views
Elementary proof of the exactness of Čech complex associated to a hypercovering ("Illusie's Conjecture")
Let $\mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $\mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 \to \mathcal{E} \...
- 3,312
3
votes
1
answer
193
views
Derived Category of strictly simplicial algebraic space vs. systems of objects in the derived categories
Let $X_{\bullet}^+$ be a strictly simplicial algebraic space and for a morphism $\delta:[m]\to[n]$ in $\Delta^+$, let $\delta:X_n\to X_m$ also denote the associated map (by abuse of notation). Then ...
- 443
7
votes
0
answers
178
views
Stabilisation of crossed modules?
D. Conduché has introduced a notion of a stable crossed module ("Modules croisés généralisés de longueur 2", JPAA 34 (1984) pp155–178, doi:10.1016/0022-4049(84)90034-3, Def. 3.1).
Is there a ...
- 1,407
4
votes
2
answers
389
views
Non-degenerate simplexes in a Kan complex
I have the following question on simplicial sets:
a non-constant Kan complex has a non-degenerate simplex in every sufficiently large simplicial degree?
It's Exercise 8.2.3 (p. 262) of Charles ...
- 1,906
9
votes
2
answers
991
views
Reference for homotopy colimit = total complex
I'm looking for a reference for the following fact:
take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
- 15.8k
1
vote
0
answers
72
views
Simplicial differential graded algebra and a filtration
Let $A$ be a simplicial differential algebra, i.e. for each $n \in \mathbb{N}$ a differential graded algebra $(A_n,d_n)$ and for each weakly increasing map $f \colon [n] \to [m]$ a morphism $f_* \...
- 154
1
vote
0
answers
69
views
How exactly to adapt Brown's collapse from monoids to algebras?
In The Geometry of Rewriting Systems (Springerlink behind paywall), Kenneth Brown describes a method to collapse the bar resolution of a monoid. Roughly:
Given a simplicial set $X$ equipped with a ...
1
vote
0
answers
202
views
What is the normalized complex of a simplicial set with a monoid action?
This question is a follow up to this question I posted on Math.SE. I will make this question self-contained, though.
In a certain point on the paper The Geometry of Rewriting Systems, Kenneth Brown ...
7
votes
1
answer
474
views
Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair
Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(A\otimes B)
\longrightarrow N_\ast(A)\otimes N_\ast(B)$$
and
...
- 517
12
votes
0
answers
917
views
Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?
Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...
- 721
3
votes
1
answer
335
views
Bisimplicial sets and homology
I'm not sure about the following result:
Theorem (?): Let $f_{\bullet,\bullet}: X_{\bullet,\bullet}\rightarrow Y_{\bullet,\bullet}$ a map of bisimplicial sets such that for any
natural number $n$, $...
- 1,613
5
votes
0
answers
358
views
Long exact sequence of Quillen derived functors
Let $F:\mathcal A\to \mathcal B$ be an additive right exact functor between abelian categories $\mathcal A$ and $\mathcal B$. Then for a short exact sequence
$$
0\to A\to B\to C\to 0
$$
in $\mathcal A$...
- 385
6
votes
1
answer
778
views
Commutation of homotopy groups with filtered colimits
Let $\mathcal{C}$ be a model category with a forgetful functor towards simplicial sets, and such that fibrations and trivial cofibrations are those whose underlying map on simplicial sets is a ...
user95222
3
votes
0
answers
166
views
Cubical VS Simplicial Manifold and the De Rham theorem
Let $\boldsymbol{\Delta}$ be the simplex category. Let ${\Box}$ be the cube category. I denote with $\Delta[n]$ the standard $n$-simplex and with $\Delta_{geo}[n]$ its geometric realization. On the ...
- 1,424
3
votes
1
answer
321
views
Are there Alexander-Whitney and shuffle maps for Dold-Kan for abelian categories?
I know Dold-Kan actually holds for any abelian category, i.e gives an equivalence between simplicial objects in a fixed abelian category and the connective chain complexes over it.
I've never seen an ...
- 10.5k
6
votes
1
answer
700
views
Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?
Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume $\...
- 922
37
votes
2
answers
3k
views
A more natural proof of Dold-Kan?
The Dold-Kan correspondence gives an equivalence of categories between $SAb$, the category of simplicial abelian groups, and $Ch_{\geq 0}$, the category of non-negatively graded chain complexes of ...
user42325
2
votes
0
answers
166
views
How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?
First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of $...
- 9,019
11
votes
1
answer
387
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}
$...
- 9,019
2
votes
0
answers
88
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 $M^\...
- 10.4k
6
votes
0
answers
1k
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 ...
- 461
8
votes
2
answers
2k
views
Algebraic Morse theory
In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
- 1,589
17
votes
1
answer
2k
views
Is there an algebraic "derived mapping space" construction that encompasses both Hochschild homology and loop spaces of non-simply-connected spaces?
I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a simplicial set to a ...
- 54.6k
11
votes
2
answers
2k
views
How to compute homology of symmetric products of complexes?
First, I would appreciate references on the notion of derived symmetric powers of perfect modules over various kinds of derived commutative algebras (say cdgas in characteristic zero, simplicial ...
- 661
3
votes
2
answers
261
views
When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$?
Let $A$ be a simplicial commutative ring over a field $k$ of characteristic zero (or a cdga
in non-positive degrees with differential of degree -1). Let $M$ be a perfect $A$ module. If necessary, ...
- 661
3
votes
1
answer
1k
views
Coboundary map on the cochain complex of abelian cosimplicial groups?
Maybe I'm looking at the wrong places, but I can't find a definition of
the coboundary map on the cochain complex of abelian cosimplicial groups.
What I have in mind is something similar to the "...
- 624
4
votes
1
answer
596
views
Extend Alexander-Whitney and Eilenberg-Zilber map to n-fold tensor products
See the definition of the Alexander-Whitney transformation:
http://ncatlab.org/nlab/show/Alexander-Whitney+map
and the Eilenberg-Zilber transformation:
http://ncatlab.org/nlab/show/Eilenberg-Zilber+...
- 624
5
votes
1
answer
531
views
Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets?
Recall that a (combinatorial) simplicial complex $X$ is said to be $n$-dimensional if it contains at least one face of dimension $n$ and no faces of dimension $n+1$. Further, an $n$-dimensional ...
- 19.6k
1
vote
1
answer
349
views
Is there a dual notion for the Nerve functor?
Say in the most classical case, we probe a topological space $X$ by the n-simplices $\Delta^n$ by using the nerve functor $Hom_{Top}(-,X)$. Is another functor $Hom_{Top}(X,-)$ of any use, or is there ...
- 417
9
votes
2
answers
463
views
$N$-step simplicial complexes
Recently, answering a question here, Dror Bar-Natan observed that «way too often two-step complexes have a natural extension to become many-step complexes». By such a thing I mean (and I think Dror ...
- 47.3k
8
votes
1
answer
533
views
delooping under Dold-Kan and simplicial delooping
What maps of simplicial sets exist between
the image under the Dold-Kan correspondence of a chain complex shifted up in degree
and the image under the right adjoint to simplicial looping of the DK-...
- 19.8k
4
votes
2
answers
1k
views
From chain complex to simplicial abelian group
In many places, I have seen the slogan that "simplicial abelian group = chain complexes of abelian groups". These same sources usually tell me how to go in one direction. Namely, given a simplicial ...
5
votes
1
answer
798
views
Simplicial "universal extensions", the hammock localization, and Ext
Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$
A morphism of $n$-...
- 19.6k
25
votes
4
answers
3k
views
A Peculiar Model Structure on Simplicial Sets?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
- 27.5k