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location Essen, Germany
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visits member for 5 years, 4 months
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Feb
15
comment Descent properties of spaces
I imagine it is possible to translate everything there to model categorical language, though I can't say for sure as I haven't read the notes. Doing so would require being careful about taking (co)fibrant replacements and so on. If you are not familiar with (infinity,1)-categorical language, you can probably just read the paper keeping in mind that you should take (co)fibrant replacements whenever necessary.
Feb
15
comment Descent properties of spaces
It looks like the author is implicitly using the language of (infinity,1)-categories.
Feb
12
revised Algebraic K-theory and Homotopy Sheaves
added 2305 characters in body
Feb
12
comment Algebraic K-theory and Homotopy Sheaves
@bananastack, in Thomason-Trobaugh this statement is Proposition 5.5.4.
Feb
12
revised Algebraic K-theory and Homotopy Sheaves
added 2305 characters in body
Feb
11
revised Lifting DG-categories to characteristic zero
added tag "dg-categories"
Feb
11
suggested approved edit on Lifting DG-categories to characteristic zero
Feb
10
comment When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?
@DavidWhite, the question of Hovey is about model structures on 2-categories, not about model structures on 2-relative categories. How is it relevant to this discussion?
Feb
10
answered When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?
Feb
7
comment motivation of filtered colimits
The term "compact object" surely deserves a place in this answer.
Feb
4
comment Fourier-Mukai transforms on stacks
arxiv.org/abs/0805.0157, arxiv.org/abs/1312.7164
Feb
4
comment When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?
Model categories enriched over SSet with the Quillen model structure are presentations of (infinity,1)-categories. Model categories enriched over SSet with the Joyal model structure are presentations of (infinity,2)-categories. See Remark 0.0.4 in [Jacob Lurie, (Infinity,2)-Categories and the Goodwillie Calculus I], arxiv.org/abs/0905.0462.
Jan
29
comment When is the cofibrant replacement of a product the product of the cofibrant replacements?
This is true for coconnective dg-algebras, according to arxiv.org/abs/1112.2360.
Jan
29
answered An example of two cofibrant dg categories whose tensor product is not cofibrant
Jan
23
comment Connection between quasifibrations and homotopy cartesian squares
Are you looking for something like this fibre-wise characterization: ncatlab.org/nlab/show/…?
Jan
21
awarded  Curious
Jan
20
asked Characterization of closed immersions at the level of perfect complexes
Jan
17
comment Motivation for cyclic (co)homology
One motivation for cyclic homology and its variants is the utility in computing algebraic K-theory. See for example the Goodwillie theorem, [T. Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. Math. 124 (1986), 347–402].
Jan
16
comment Homological algebra is linearized homotopical algebra?
Homological algebra is the homotopy theory of chain complexes. The homotopy theory of chain complexes is equivalent to the homotopy theory of modules over the Eilenberg-Mac Lane spectrum $H\mathbf{Z}$. Hence homological algebra is a stable, linear version of the homotopy theory of spaces. See [Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules, Topology 42 (2003), 103-153].
Jan
16
comment Relationship between Hochschild cohomology and Drinfeld centers
@SamuelM, according to Remark 1.5 in the paper, it seems that the Drinfeld centre of $D(A \otimes A^{op})$ will be the $\infty$-category of modules over the Hochschild chain complex of $A \otimes A^{op}$.