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Results tagged with homotopy-theory
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user 402
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
3
votes
Accepted
Model structure on simply-connected topological spaces in which the weak equivalences are th...
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 …
- 37.8k
9
votes
Accepted
Is the adjunction between spaces and chain complexes monadic?
This answer assumes that $\def\Ch{{\sf Ch}}\def\Z{{\bf Z}}\Ch_{≥0}(\Z)$ refers to the derived ∞-category of chain complexes, i.e., with quasi-isomorphisms inverted up to a homotopy.
Recall that the ri …
- 37.8k
4
votes
Accepted
Is the standard model structure on reduced simplicial sets cofibrantly generated?
Yes, the model structure on reduced simplicial sets is cofibrantly generated.
An explicit proof of this statement is given by Goerss and Jardine in Simplicial Homotopy Theory, the proof of Proposition …
- 37.8k
6
votes
Accepted
$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
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 …
- 37.8k
3
votes
Accepted
Bⁿ and coherence
One natural setting where the delooping operation $\def\B{{\sf B}}\B$ can be defined very quickly is Γ-spaces, described by Segal in Categories and cohomology theories.
A Γ-space is simply a functor $ …
- 37.8k
2
votes
Is there a shape-independent definition of (∞,1)-categories?
Yes. As shown in the paper The enriched Thomason model structure on 2-categories, the category of (strict) 2-categories can be equipped with a model structure that makes it Quillen equivalent to the …
- 37.8k
3
votes
Accepted
Simplicial objects in quasicategory which come from homotopy coherent nerve
It is easy to construct counterexamples already for the full subcategory $\{[0],[1]\}⊂Δ$. The category $\cal C$ can be constructed by applying the nerve functor to hom-objects of a category $D$ enric …
- 37.8k
3
votes
Accepted
A fiber-like method to show equivalence of infinity categories
An obvious necessary condition for $f$ to be a categorical equivalence is that $f$ is weakly equivalent to a (co)cartesian fibration of quasicategories, i.e., $f$ is an analogue of a Street fibration …
- 37.8k
6
votes
Accepted
Can one bypass the geometric realization in the definition of algebraic $K$-theory?
I believe there is no good notion of homotopy groups for an arbitrary simplicial set S.
It depends on what “good” means.
Kan's original definition works for arbitrary pointed simplicial sets: $$\def …
- 37.8k
3
votes
Canonical reference for dictionary between $G$-spaces and fiber bundles over $BG$?
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- …
- 37.8k
11
votes
Big list: barycentric subdivision of simplicial sets
An important theoretical application is Kan's fibrant replacement functor $\def\Ex{{\sf Ex}}\def\Exi{\Ex^{\sf\infty}}\Exi$, defined as the filtered colimit of functors $\Ex^n$ ($n≥0$), where $\Ex$ is …
7
votes
Relative category structure on (Set valued) presheaves
The usual constructions of Grothendieck homotopy theory (as presented by Maltsiniotis and Cisinski) can be easily extended to the setting of relative categories.
Recall that given a small category $A$ …
- 37.8k
2
votes
Accepted
Homotopical properties of powersets of simplicial sets
The first question has a negative answer, given by the simplicial set $\def\Exi{{\sf Ex}^{\sf\infty}}X=\Exi Y$, where $Y$ is a simplicial set generated by vertices $a,b,b',c,c',d,d'$, 1-simplices $ab, …
- 37.8k
4
votes
Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$
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 …
- 37.8k
3
votes
Reference for the Brown representability theorem in the case of locally presentable (∞,1)-ca...
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 …
- 37.8k