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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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An explicit description of injective fibrations
If $M$ is a combinatorial model category and $C$ a small category, then the category $M^C$ admits an injective model structure in which the cofibrations and weak equivalences are levelwise. I would l …
5
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2
answers
218
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Reedy model structures on oplax limits
Suppose $R$ is a category and $F:R\to Cat$ is a functor (or pseudofunctor). The oplax limit of $F$ is the category whose objects consist of an object $x_r \in F(r)$ for all $r$ together with a morphi …
10
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Homotopy theory of suplattices
In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ob …
6
votes
2
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531
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Morphisms every pushout of which is a weak equivalence
Let $M$ be a category equipped with a class of weak equivalences $W$. Is there a name for a morphism $f$ such that every pushout of $f$ (including, of course, $f$ itself) is a weak equivalence?
For …
11
votes
2
answers
643
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The definition of Reedy category
The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its obje …
11
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1
answer
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Which maps of simplicial sets geometrically realize to fibrations?
If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a …
6
votes
1
answer
497
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Presheaves on a complete Segal space
Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which pre …
20
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2
answers
737
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Limitations on model-categorical presentations
In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is b …
10
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161
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Rectifying diagrams of $\infty$-categories
If $C$ is a 1-category and $D$ is a locally presentable $(\infty,1)$-category presented by a combinatorial model (1-)category $M$, then any $(\infty,1)$-functor $C\to D$ can be represented by a strict …
7
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150
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Presenting geometric morphisms by geometric morphisms
It's known that any $\infty$-topos $\mathcal{E}$ can be presented by a Quillen model category $\mathbf{E}$ that is itself a 1-topos. For instance, if $\mathcal{E}$ is a left exact localization of a p …
3
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answer
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Cofibrations of functors
Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$ i …
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A model category for descent?
Recall that an $(\infty,1)$-category $C$ is said to have descent if for any small diagram $X:I\to M$ with (homotopy) colimit $\overline{X}$, the adjunction between $C/\overline{X}$ and "equifibered" $ …