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Results tagged with homotopy-theory
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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.
51
votes
Accepted
What are surprising examples of Model Categories?
Here is an example that surprised me at some time in the past.
Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs.
Here a directed graph is simply a $4$-tuple $(V, …
- 37.8k
45
votes
Timeline of "foundational" advances in homotopy theory?
Such a timeline is necessarily highly subjective.
With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory.
The list below concentrates o …
18
votes
Accepted
When did the Joyal model structure on simplicial sets originate?
Here is what André Joyal wrote in an email to me:
No, I have not discovered the model structure for quasi-categories in the 1980's.
I became interested in quasi-categories (without the name) around 1 …
- 37.8k
17
votes
Accepted
Classifying space BG and contractable space EG
The easiest way to construct an explicit contracting homotopy
is to observe that EG is the geometric realization of the nerve of the groupoid G//G,
which has G as its set of objects and exactly one mo …
- 37.8k
17
votes
Why do we need model categories?
Model categories provide a powerful framework for commuting (homotopy) limits and colimits,
and, more generally, for commuting left adjoint functors and (homotopy) limits,
as well as right adjoint fun …
- 37.8k
14
votes
Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
Given that Sp is better behaved than all other existing models of spectra
No, Sp is not better behaved than other models.
The reason that it seems to be is because all operations in Sp
(e.g., Ω^∞ …
13
votes
Accepted
Representation theory of higher homotopy groups
There are many results that generalize the Riemann–Hilbert correspondence from the fundamental groupoid to the fundamental ∞-groupoid, for example:
Jonathan Block, Aaron Smith. A Riemann–Hilbert cor …
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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 …
10
votes
Accepted
How to compute Homotopy Pullback
Are there any explicit ways of calculating it, similar to the methods, working for the bog down homotopy pullbacks and pushout?
Yes, in fact the same formula continues to work in this case.
Consider …
- 37.8k
9
votes
Accepted
Homotopy excision for structured ring spectra -- reference?
Theorems 1.4–1.11 in Ching and Harper's paper “Higher homotopy excision and Blakers-Massey theorems for structured ring spectra” (arXiv:1402.4775)
give higher homotopy excision and Blakers-Massey (and …
- 37.8k
9
votes
Accepted
Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) i...
if we don't assume properness, I don't even see why the first is homotopy-invariant!
The pushout of a diagram A←B→C in which all objects are cofibrant and one of the maps is a cofibration
is always …
- 37.8k
9
votes
Accepted
Can any $E_1$ algebra over $\mathbb{F}_p$ be modeled as a dg algebra?
Yes, this is precisely the content of Theorem 7.11 in arXiv:1410.5675, which should be combined with §7.4 of arXiv:1510.04969.
In fact, the cited results prove this for any nonsymmetric operad in chai …
- 37.8k
9
votes
Accepted
Practical consequences of the geometric cobordism hypothesis
My question is: does this lead to a more-or-less explicit construction of any non-trivial quantum field theories? If so, this would be extremely interesting since only a handful of interacting quantu …
- 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
8
votes
References for homotopy colimit
Chris Douglas has a nice short discussion of homotopy limits in his text “Sheaves in homotopy theory” (Chapter 5 of Topological modular forms).
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