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In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity categori …

It's been more than a year and a half since I asked this question and I had a lot of thought about it so I decided I will post my own answer. First I entirely agree with Yonatan that the main problem …

So far the discusion mostly focused on a geometric explanation, I'd like to mention the algebraic one as well: One way to formulate it involves the delooping machinery: up to delooping, $\mathbb{S}^n …

Today we understand that what we are really interested in when we talk about "homotopy theory" are in the end "$\infty$-categories". In fact I have even heard some peoples claim that maybe in the fut …

For this to work, it is best to identify connective spectrum with spaces equipped with a group-like $E_\infty$-algebra structure (these are equivalent). From this point of view: $\mathbb{Z}$ is the f …

Is there a lot of ring spectrum which are idempotent in the sense that the multiplication map $R \wedge R \rightarrow R$ is an equivalence ? The sphere spectrum $\mathbb{S}$ and the $0$ spectrum are …

Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists w …

One has a nice "folk" model structure on strict $\infty$-categories due to Yves Lafont, Francois Metayer and Krzysztof Worytkiewicz whose notion of weak equivalences seem to be the notion of weak equi …

For a model category $C$, I'm denoting $h_\infty(C)$ the associated $\infty$-category (for example its Dwyer-Kan localization at weak equivalences, or if $C$ is simplicial the simplicial nerve of the …

As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma …

Given $f:X \to Y$ a continuous map between two spaces (unpointed CW-complexes) such that $f$ induces an isomorphism in homology with integer coefficient, and $f$ induces an isomorphism on homology of …

Another thing for which Kan Ex$^{\infty}$ functor is useful is actually in the construction of the Kan-Quillen model structure. It morally gives a purely algebraic version of the simplicial approximat …

As I said in the comment, this would involve a very large number of different references! (almost one by subsection...) But to some extent, contributors to the nLab already started doing that and it i …

This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what …

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 t …