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Yes: take the product $\def\K{{\rm K}} \def\Z{{\bf Z}} A=∏_{k≥0}\K(\Z,n)$ of Eilenberg–MacLane spaces. Then for each $n≥0$ there is a map $f_n\colon A→A$ given by identities on all factors with index …

For simplicial commutative rings this is proved in Lemma 3.1.2 of Schwede's “Spectra in model categories and applications to the algebraic cotangent complex”, and the proof there immediately extends …

What exactly does 3. describe? Are these virtual vector bundles that admit numerable trivializations? Virtual vector bundles, when defined as formal differences (i.e., elements in the homotopy gr …

The claim about weak equivalences follows as soon as one proves that the cocartesian squares generated by U←U∩V→V and U'←U'∩V'→V' are also homotopy cocartesian. To this end one can use Lurie's Seifer …

A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis. Very roughly, Ω^∞MG is a simplicial set whose n-simp …

Yes. Any rank zero element x in K(X) is nilpotent by https://ncatlab.org/nlab/show/virtual%20vector%20bundle, hence 1+x is invertible.

Here is an excerpt from a paper by Simons and Sullivan (Axiomatic Characterization of Ordinary Differential Cohomology), which seems to answer the question: Fact 2.1: Let K in M denote the compact im …

First, an elementary manipulation of homotopy colimits shows that the sequential homotopy colimit can be replaced by the homotopy coequalizer of the identity map and the shift map on the coproduct of …

Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement. Computing the fibrant replacement for simp …

There is a sense in which the relation between moduli stacks and classifying spaces can be formalized, at least when we use smooth manifolds as parametrizing objects. (Topological manifolds and PL-man …

Is this isomorphism given by the Chern-Weil homomorphism? Yes, see Theorem 7.20 in the paper of Freed and Hopkins, which computes the de Rham complex of B∇G as C[g]G equipped with the zero differ …

This fact admits a much easier proof. To show that for any simplicial fibrant sheaf F and open sets U⊆V the map F(V)→F(U) is a fibration it suffices to show that F(V)→F(U) has a right lifting property …

Any presentation of a given space as a CW-complex immediately gives rise to a presentation of the fundamental groupoid, and hence also the fundamental group. Specifically, given such a presentation a …

There are many ways to define weak equivalences of simplicial sets without referring to topological spaces. A morphism f is a weak equivalence of simplicial sets if and only if one of the following e …

When is an A_∞ structure of this type - i.e. is there always an equivalent strict version? Risking unsolicited self-advertising, I would like to point out Proposition 10.1.1 (and the more general …