# Questions tagged [homotopy-theory]

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.

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### Notation A(g,n) related to Eilenberg MacLane spaces [on hold]

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### Descent properties of topological Hochschild homology

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### Definition of hypercover for simplicial presheaves and hypercovering in $\infty$-topos

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### Homotopy fibre sequence and left Bousfield localization

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### Is the inclusion $\Delta^{op} \to \Gamma^{op} = Fin_\ast$ homotopy cofinal?

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### Descent in the injective model structure and descent for simplicial presheaves

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### Derived base change in étale cohomology

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### Does a homotopy sheaf functor commute with group completion

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### Koszulness of some DG-algebras and a paper by Kohno and Oda

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### The homotopy theory presented by a Waldhausen category

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### Which spaces are most naturally presented simplicially?

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### $K_3(\mathbb{Z})$ and $\pi ^S_3$

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### A particular pushout of homologicaly rational spaces

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### Open subspaces of CW complexes

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### Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by “simplicial decomposition”

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### Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?

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### Homology of a loop-suspension space and action of $\mathcal{D}_1$-operad

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### Which homotopy types can be realized as the classifying space of a right-cancellative discrete monoid?

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### Are simplicial finite CW complexes and simplicial finite simplicial sets equivalent?

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### Group completion of $E_k$-algebras

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### Are the real and complex Adams operations compatible under the inclusions $U(n) \rightarrow SO(2n)$?

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### Weak homotopy equivalence of sites

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### Cyclic homotopies of quotients of $S^3$

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### Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization

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### In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?

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### In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?

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### In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

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### Why does every chain complex have a map into its cone?

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### Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

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### Stable splitting of products

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### Equivalent definitions of Thom spectra

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### $L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

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### Diffeomorphism type of the added sphere in simply connected surgery

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### Homotopy equivalence of $K$-theory and $G$-theory

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### What is the intuition for higher homotopy groups not vanishing?

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### Pushforward of an internal category along a functor

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### Replacing the Fibre of a Fibration

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### Is there a weak homotopy equivalence between Sp(2n,ℂ)/U(n) and SU(n)?

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### Crystals and nilpotence

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### Categorical Significance of Fibrations

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### Topological invariants of a certain “stratified” manifold, with pieces of different “dimensions”

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### Homotopy pullbacks and pushouts in stable model categories

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### Cyclic version of Lie algebra cohomology

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### Obstruction to homotopy, cohomology operations and Dold-Whitney theorem

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### Cardinalities associated to the Bousfield lattice

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### Other homotopy invariants?

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### Homotopy colimits of long sequences

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### Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?

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### Dold-Kan correspondence in the category of symmetric spectra

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