Questions tagged [homotopy-limits]
The homotopy-limits tag has no usage guidance.
24
questions
2
votes
0
answers
92
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Alternative construction for the loop space (?)
There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
7
votes
1
answer
406
views
Different definitions of homotopy colimits
I was reading about the definition of homotopy limits and colimits, and I have seen two different approaches in "Homotopy theories and model categories" by Dwyer and Spalinski, and in "...
0
votes
0
answers
108
views
Locally constant (homotopy) pre-factorization algebras
In my thesis, I'm using the theory of (homotopy) factorization algebras and particularly locally constant ones. While reading an article that I can't find again I read that already a locally constant ...
1
vote
1
answer
90
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Find a functorial zig-zag of spaces
This is a rather broad question. Suppose you have an ordinary category $C$ (for example, $\Delta$), and two diagrams $X_{\bullet}, Y_{\bullet} : C \to \textrm{Top}$. Suppose also that $X_c$ is ...
4
votes
1
answer
167
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Homotopy totalization and chains - reference
Simple case. Take $X_{\bullet}$ a cosimplicial space. Is it true that the chain complex of its homotopy totalization is quasi-isomorphic to the homotopy totalization of its chain complex? Because of ...
3
votes
1
answer
390
views
Homotopy colimit commutes with homotopy groups
I'm interested in something built upon the construction laid out in nlab article on Snaith's theorem
Let $(E, \mu, \iota)$ be a ring spectrum.
For $\beta \in \pi_n(E)$ an element of the $n$th stable ...
2
votes
0
answers
114
views
Are homotopy colimits strict?
Let's say we are working with a fibrant simplicially enriched category $\mathbf{B}$ that has all limits and all homotopy limits, and let $\mathbf{A}$ be a full subcategory that is closed under weak ...
4
votes
0
answers
85
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Homotopy colimits in subcategories of combinatorial model categories
We know that, in a combinatorial simplicial model category $\mathbf{M}$, we can find a regular cardinal $\lambda$ large enough that all $\lambda$-filtered homotopy colimits can be computed as strict ...
1
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0
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75
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Homotopy limits indexed by a covering
We all know that the homotopy fiber of a continuous function $f: A \to B$ has a simple expression in terms of points and paths connecting them, that is
$$ \textrm{hofib}_y(f) = \{ (x,\gamma) \in A \...
6
votes
2
answers
511
views
Deformation of a diagram preserve the homotopy limit
I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version.
Suppose ...
5
votes
1
answer
584
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Homotopy coherent colimits in chain complexes
In remark 1.2.6.2 (HTT), Lurie states that
Another possible approach to the problem of homotopy
coherence is to restrict our attention to simplicial (or topological) categories
C in which every ...
3
votes
0
answers
64
views
Homotopy limits of section spaces
Let $\mathcal{U}$ be an open cover of a topological space $B$. As we see, for example in this question, the associated Cech diagram $B_{\mathcal{U}}$ constitutes a simplicial space with the weak ...
1
vote
1
answer
140
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Why does this construction give a weak factorization system in the category of span diagrams?
In Dwyer and Spalinski's classic paper Homotopy Theories and Model Categories, they describe homotopy pushouts by defining a model structure on the category of span diagrams in a given model category $...
10
votes
3
answers
1k
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Can filtered colimits be computed in the homotopy category?
For $\mathcal{S}$ the $(\infty,1)$-category of spaces its homotopy category $h\mathcal{S}$ does not have pushouts or pullbacks. Even if it does, they won't always agree with the (homotopy) pushouts or ...
3
votes
0
answers
168
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Techniques for computing homotopy pullbacks
I am in the context of simplicial sets. I have a square diagram that I want to show to be homotopy pullback. What I can grasp is that it is a pullback up to some equivalences in the middle of my ...
4
votes
1
answer
368
views
Homotopy limit over a diagram of nullhomotopic maps
Let $I$ be a $\mathrm{Top}_*$-enriched poset and $X: I \to \mathrm{Top}_*$, and consider the homotopy limit
$$
\underset{i \in I}{\mathrm{holim}}X(i),
$$
where the maps $X(i) \to X(j)$ are ...
9
votes
1
answer
532
views
Homotopy fibers of infinity functors
Let $F: C \to D$ be an infinity functor. Is it true that the homotopy fiber at $y$ can be described as $C \times_D D^{\simeq}_{/y}$? If not, is there a simple formula resembling this one?
Beside the ...
2
votes
0
answers
135
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Why is a homotopy limit of a cosimplicial space not the ordinary limit?
I've been trying to compute a homotopy limit of a cosimplicial object $X: \Delta \to \mathscr{M}$, where $\mathscr{M}$ is some simplicial model category, we may take it to be spaces. I'm wondering ...
7
votes
0
answers
251
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Topological localization of (infinite) inverse limits
The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...
7
votes
1
answer
697
views
Fiber vs homotopy fiber in model categories: simple question
I have a concrete problem with the homotopy fiber and I am getting lost with the
literature. I state my question and, to avoid confusions, I state downwards the
definitions I am using.
Let $C$ be a ...
8
votes
3
answers
2k
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Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the Dold-...
9
votes
0
answers
371
views
Reference for maps whose pushouts are also homotopy pushouts
Consider a category C with weak equivalences, e.g., a model category.
For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) ...
7
votes
1
answer
725
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 ...
2
votes
0
answers
351
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 C^...