# Questions tagged [model-categories]

A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.

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### Do dg schemes have derived points?

Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced ...

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### Homotopy fibre sequence and left Bousfield localization

Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, ...

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### Are constant $\infty$-sheaves constant on connected components?

Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty\text{Sh}(C, J)$. There is a
natural geometric morphism to $\infty\text{Grpd}$ whose ...

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### Does a homotopy sheaf functor commute with group completion

Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_n^{\tau}$ be the $\tau$-homotopy sheaves defined as follow
Does $\pi_n^{\tau}$ commute with ...

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### Does every span have a homotopy pushout?

Suppose we have an arbitrary span in an arbitrary model category
$$ B \leftarrow A \to C $$
Is it always possible to complete it to a homotopy pushout square?
$$ \require{AMScd} \begin{CD}
A @>>&...

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### Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by “simplicial decomposition”

I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow:
The argument works by showing that ...

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### Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?

In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about ...

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### Examples of non-proper model structure

I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...

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### When is the localization of all hypercovers equivalent to that of ?ech covers?

In Theorem A.6 of Dugger's paper, it is shown that a few localizations are equivalent to the localization of ?ech covers.
The Nisnevich localization
at all hypercovers is equivalent to the ...

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### Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization

Let $L_{Nis}(sPre(Sm_S))$ be the Nisnevich localization of the category of simplicial presheaves,
how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times_S X\to X$ are closed under ...

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### The model category structure on $\mathbf{TMon}$

I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer.
I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...

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### Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...

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517 views

### Non-small objects in categories

An object $c$ in a category is called small, if there exists some regular cardinal $\kappa$ such that $Hom(c,-)$ preserves $\kappa$-filtered colimits.
Is there an example of a (locally small) ...

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### Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...

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### When do zigzags of weak equivalences detect isomorphisms in the localization?

The usual way to prove that two model categories are equivalent is to construct a zigzag of Quillen equivalences between them, but is it always possible? We can ask a more general question.
...