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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.
3
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2
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340
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Is the mapping cylinder a replacement for morphism by cofibration in model categories?
Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that …
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15
votes
1
answer
786
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If homotopy groups of spaces are identical, then stable ones are also identical?
Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$?
In particular, is this …
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5
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1
answer
278
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Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely pre …
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9
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1
answer
568
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What is known about the homotopy type of the classifier of subobjects of simplicial sets?
For the presheaf topos $\mathrm{PSh}(C)$, the subobject classifier is the presheaf $\Omega$ such that
For $c \in C$, $\Omega(c)$ is the set of all subobjects of the functor $\mathrm{Hom}(-, c)$
For $ …
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5
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1
answer
446
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Can we prove that any number of algebraic data cannot be a complete invariant of a homotopy ...
The statement in the title seems to be generally accepted as true, but I have not seen proof. They are?
The strict formulation I have in mind is the following. By an algebraic category we mean the cat …
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4
votes
0
answers
103
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Dugger's theorem for enriched model categories
We know that a combinatorial model category has a small presentation.
Is an enriched version of this theorem known? The closest I could find is: Guillou, May - Enriched model categories and presheaf c …
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0
votes
1
answer
144
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Examples of cartesian-closed model categories
One of the main settings of my research are Cartesian-closed model categories. I would like to know as many interesting and/or important examples of such categories as possible. "Interesting" and "imp …
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4
votes
1
answer
359
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Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?
Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic …
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3
votes
1
answer
421
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How to get by with only functorial cylindrical objects?
In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects / path spa …
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1
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3
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676
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How now to study operads in homotopy theory?
There is a great introduction by May, "The Geometry of Iterated Loop Spaces". I really enjoy reading it, but it was written 50 years ago and contains outdated technical details related to the language …
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7
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0
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305
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Can every finitely presented group be realized as a fundamental group of a compact four-dime...
Every finitely presented group is realized as a fundamental group of a two-dimensional complex (a simple exercise on Van Kampen's theorem). I was told that a two-dimensional complex can be well embedd …
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10
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1
answer
663
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Do elements of every order occur in homotopy groups of spheres?
It is known from Serre's classical result that every p-torsion occurs in the homotopy groups of every sphere. Is it known: do elements of every order occur in homotopy groups of spheres?
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1
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1
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203
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Is the decomposition of the homotopy type of a complex into a product and into a smash produ...
Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the multi …
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16
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2
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698
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Is the decomposition of the homotopy type of a complex into a bouquet unique?
Is it true that if $A_1 \vee A_2 \vee .. \vee A_n = B_1 \vee B_2 \vee .. \vee B_m$, where $A_i, B_j$ are homotopy types of complexes not decomposable into a bouquet, then the multisets $A_i$ and $B …
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7
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2
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977
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Does there exist a complete algebraic invariant of the homotopy type of a finite CW-complex?
Let $\mathrm{Cell}$ be the homotopy category of finite cell complexes. The main motive of my question
Is it true that for any algebraic category $A$ there is no fully faithful functor $F: \mathrm{Cel …
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