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I would like to know in anyone has developed method for constructing colimits in the category of algebra for a monad in the $(\infty,1)$-categorical framework, using transfinite constructions. I have …

(note: this question is essentially a reference request for the tensor product described at the end. the rest is context) It is well known that the category of chain complexes (in positive degree, wi …

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 …

In $1$-category theory, a well pointed endofunctor of a category $C$, is an endofunctor $F:C \rightarrow C$ endowed with a natural transformation $\sigma : Id \rightarrow F$ such that the two natural …

It is in general not too hard to show that maps between finite $CW$-complexes/finite simplicial sets are homotopy equivalences. Question : Can we do something similar for: quasi-categorical equival …

Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ? More precisely, is there an $\infty$-topos $BG$ s …

I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of usin …

Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e. $$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$ And we want to …

I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties: a) It attach to each small …

Hello ! If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an in …

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales). For the record, this is proved by, starting fo …

My question is : Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ? What I mean by $\infty$-connected …

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity …

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 …