Questions tagged [homotopy-limits]

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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_{\...
Andrea Marino's user avatar
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 "...
MikeTrooper's user avatar
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0 answers
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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 ...
Alessandro Nanto's user avatar
1 vote
1 answer
90 views

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 ...
Andrea Marino's user avatar
4 votes
1 answer
167 views

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 ...
Andrea Marino's user avatar
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 ...
Excalibur's user avatar
  • 301
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 ...
Giulio Lo Monaco's user avatar
4 votes
0 answers
85 views

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 ...
Giulio Lo Monaco's user avatar
1 vote
0 answers
75 views

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 \...
Andrea Marino's user avatar
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 ...
Andrea Marino's user avatar
5 votes
1 answer
584 views

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 ...
Andrea Marino's user avatar
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 ...
Lukas Miaskiwskyi's user avatar
1 vote
1 answer
140 views

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 $...
Doron Grossman-Naples's user avatar
10 votes
3 answers
1k views

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 ...
Jan Steinebrunner's user avatar
3 votes
0 answers
168 views

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 ...
Andrea Marino's user avatar
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 ...
Niall Taggart's user avatar
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 ...
Andrea Marino's user avatar
2 votes
0 answers
135 views

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 ...
Maanroof's user avatar
  • 213
7 votes
0 answers
251 views

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 ...
Tyrone's user avatar
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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 ...
Tintin's user avatar
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8 votes
3 answers
2k views

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-...
Dmitri Pavlov's user avatar
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) ...
Dmitri Pavlov's user avatar
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 ...
Dmitri Pavlov's user avatar
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^...
Ma Ming's user avatar
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