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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.

19 votes
2 answers
1k views

A model category of abelian categories?

Let $\mathcal{M}$ be the following category: The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels. The morphisms are functors that preserve the struct …
Zhen Lin's user avatar
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17 votes
1 answer
1k views

The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent …
Zhen Lin's user avatar
  • 15.9k
16 votes
1 answer
916 views

The state of the art in the rectification of homotopy-coherent structures

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a …
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6 votes
0 answers
283 views

What are the Čech-local equivalences of (simplicial pre)sheaves?

Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield …
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6 votes
1 answer
247 views

Left Bousfield localisation of trivial model structures

Let $\mathcal{M}$ be a category, let $\mathcal{M}^\circ$ be a reflective (full) subcategory of $\mathcal{M}$, and let $L : \mathcal{M} \to \mathcal{M}^\circ$ be the reflector. Question. Does there exi …
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5 votes
1 answer
196 views

Schwänzl and Vogt, Cofibration and fibration structures in enriched categories

In [Schwänzl and Vogt, Strong cofibrations and fibrations in enriched categories], the authors refer to an earlier preprint, [Schwänzl and Vogt, Cofibration and fibration structures in enriched catego …
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