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Yes, there is such a model structure, at least if we are willing to pick a category of simply connected spaces that has limits and colimits. The modern proof simply applies the existence theorem for l …

The original reference for such results is Proposition 3.3.3 on page 120 in Fabien Morel, Vladimir Voevodsky, A^1-homotopy theory of schemes, Publications mathématiques de l’I.H.É.S., tome 90 (1999), …

If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group. Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$. Since $\B Ω G≃G$, this ∞-category is eq …

Since the question uses semisimplicial sets, it makes sense to point out the following rather elegant model for the classifying space $\def\B{{\sf B}}\B G$ as a semisimplicial set: declare the set of …

These are sequences not of groups, but of ∞-groups, which can be modeled as simplicial groups or topological groups, equipped with a class of weak equivalences induced from simplicial sets or topologi …

The short note Group Extensions and $H^3$ by P. J. Morandi works out the details of extensions by nonabelian groups. Any extension of $G$ by $A$ has an induced group homomorphism $\def\Out{{\sf Out}} …

The answer to Question (ii) is positive. That is to say, there is a weak equivalences between the following functors from Segal topological categories to quasicategories: the composition of the singu …

One reference is the two papers by Nikolaus–Schreiber–Stevenson: Principal ∞-bundles – General theory Principal ∞-bundles – Presentations In particular, these papers explain the equivalence between G- …

I presume “constructive” means a computational algorithm is desired. Assuming the chain complex $F$ over $\def\Z{{\bf Z}}\Z/2$ is bounded from below, we are going to construct by induction on $n$ a ba …

The curvature form descends to a genuine 2-form on the base space (unlike the connection 1-form). In fact, locally on the base space, we can pick a trivialization of the principal bundle and compute …

A gerbe on a manifold $M$ is a morphism of simplicial presheaves $$\def\tB{{\sf B}}\def\U{{\rm U}}\def\cC{{\sf\check C}}\cC(U)→\tB^2\U(1),$$ where $\{U_i\}_{i∈I}$ is an open cover of $M$, $\cC(U)$ is …

Given that the map $\def\E{{\sf E}}\def\B{{\sf B}}\E G→\B G$ is the geometric realization of a simplicial covering map (namely, the nerve of the functor $\def\sq{/\!/} G\sq G→*\sq G$), the canonical t …

In addition to the construction of (generalized) Eilenberg–MacLane spaces mentioned in the comments, there are many other examples. For example, the Dold–Kan correspondence allows us an easy perspecti …

As pointed out to me by George Raptis, a detailed treatment of Brown representability for $(n,1)$-categories ($1≤n≤∞$), stable or not, is now available in Hoang Kim Nguyen, George Raptis, Christoph S …

My question is: Is the homology of $(C_*,\partial)$ isomorphic to the singular homology of E? Yes. Observe that the singular complex functor sends Serre fibrations to Kan fibrations. Thus, the map …