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 ...

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### Two model structures of some category

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### Deformation sublevel sets of functions which preserve boundary

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### Are HOTT evaluated as alternatives to foundations of mathematics completely free from Gödel's incompleteness theorem?

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### What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

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### Homotopy orbits, spectra and infinite loop spaces

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### Product-preserving fibrant replacement functor for the Joyal model structure

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### How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$

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### Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$

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### Deforming a section to a section without zeros

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### Simplicial homotopy groups - reference request

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### Basic questions on spectra

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### $X^K$ a Kan complex, without model structure or anodyne extensions

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### Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?

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### When is a bisimplicial set diagonal fibrant

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### Minimal infinity categories in the Segal space picture

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### Quillen equivalence, fibrant objects

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### What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?

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### Relationship between the Betti numbers $b_i(M;\mathbb{Q})$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}\pi_i(M)\otimes\mathbb{Q}$

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### Proof of MacPherson's result about set-valued constructible sheaves and exit paths

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### Products of representables are regular on a regular skeletal Reedy category?

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### Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity?

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### What is the best reference for motives?

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### Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness

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### Classification of combinatorial model categories presentable by simplicial presheaves on a Reedy category

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### When does a map of spaces deloop a closed subgroup inclusion?

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**1**answer

### Model category of diagrams with the colimit detecting the weak equivalences

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**1**answer

### Trivial cohomology with free module coefficient

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### Homotopy pushout independent of factorization and symmetric in cofibration category

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### Nonabelian Poincare duality

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### Floer cohomology from mapping spaces of $\infty$ categories

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### Homotopy pullback of a homotopy pushout is a homotopy pushout

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### Eckmann-Hilton argument / Grothendieck

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### Asphericity of hypersurface complement in ${\mathbb C}^n$

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### Cofibrations in a category of fibrant objects

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### Localizing $\mathrm{CombModCat}$ at the Quillen equivalences

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**1**answer

### Which motivic spectra are dualizable?

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### A projective (or free) $\mathbb{Z}\pi_1$-module

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**2**answers

### Pushout of spaces

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### pushout and homotopy

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**2**answers

### Are cofibrations accessible?

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### Comparing two simplicial resolution in a model category

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### Is there an “injective version” of the Bergner model structure?

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### What is the structure required to construct this homotopy of maps between mapping cones?

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### Contracting the rational bar cocomplex for a finite group G

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### Formality of the 2nd ordered configuration space of a closed Riemann surface

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### Is algebraic $K$-theory a motivic spectrum?

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### Applications of spectral Artin representability?

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### A question regarding generalized cohomology and spectra : proof of $E^{\ast}(S)\otimes\mathbb{R} = H^{\ast}(S;\pi_{\ast}E\otimes \mathbb{R})$

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### On the paradox that $n$-coskeletal simplicial sets model all homotopy types

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