# Questions tagged [homotopy-limits]

The homotopy-limits tag has no usage guidance.

12
questions

**1**

vote

**1**answer

61 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 $...

**8**

votes

**3**answers

490 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 ...

**3**

votes

**0**answers

99 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 ...

**4**

votes

**1**answer

293 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 ...

**8**

votes

**1**answer

248 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

94 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 ...

**7**

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**0**answers

186 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 ...

**7**

votes

**1**answer

429 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 ...

**6**

votes

**2**answers

1k 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-...

**8**

votes

**0**answers

293 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

561 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

297 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^...