Questions tagged [homotopy-limits]

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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
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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4 votes
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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 ...
Giulio Lo Monaco'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
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
2 votes
0 answers
92 views

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
2 votes
0 answers
115 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
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
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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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1 vote
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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 \...
Andrea Marino's user avatar
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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