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        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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        235 views

        $\infty$-categorical understanding of Bridgeland stability?

        On triangulated categories we have a notion of Bridgeland stability conditions. Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
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        vote
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        A homotopy problem for morphisms of dg-algebras

        Let $\mathfrak g$ be a real finite-dimensional Lie algebra, and suppose we are given two morphisms of dg-algebras $f,g: (C^\bullet(\mathfrak g),d_{CE}) \to (\Omega^\bullet(\Delta^n),d)$, such that ...
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        375 views

        What are the advantages of simplicial model categories over non-simplicial ones?

        Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...
        6
        votes
        0answers
        122 views

        Does Ex^∞ send homotopy inverse limits of ∞-categories to homotopy inverse limits of spaces

        Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the ...
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        vote
        1answer
        93 views

        The table reduction morphism of operads from Barratt-Eccles to Surjection

        The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma_n\}_{n>0}$ in groups. Berger-Fresse defined here an ...
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        3answers
        322 views

        Iterated free infinite loop spaces

        Let $Q$ denote $\Omega^\infty\circ \Sigma^\infty$ the free infinite loop space functor. Given some space $X$, we see that $QX$ carries all the stable homotopy information about $X$. Naturally I wanted ...
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        162 views

        Construction of a $K(\pi,1)$-space?

        My colleague suggested a proof of a fact which I have hard time to believe. Since I am not a topologist by training I am asking it here. Consider any CW-complex structure on the $d$-dimensional ...
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        1answer
        105 views

        Two models for the classifying space of a subgroup via the geometric bar construction

        Let $H$ be a topological group which is a subgroup of two other topological groups $G$ and $G'$. It follows (from Rmk 8.9 in May - Classifying spaces and fibrations (MSN, free)) that there exist weak ...
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        99 views

        Extension of sheaves of $\infty$-algebras

        Let $(\mathcal{C},\tau)$ be a site, whose Grothendieck topology is $\tau$ $F : \mathcal{C}\to D(k)$ a sheaf of $\infty$-$k$-algebras, with $k$ a ring and $D(k)$ the derived category of $k$-modules. ...
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        0answers
        40 views

        Tensor product of an L-infinity algebra with the cochains on the 1-simplex

        I would like to understand the $L_\infty$ structure on the tensor product of an $L_\infty$ algebra (over $\mathbb{R}$) $L$ with the normalized cochains on the one-simplex $N^*(\Delta^1)$. This latter ...
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        0answers
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        References on $HZR$ theory

        Are there references available on $HZR$ theory ? I found on ncatlab that this is "genuinely $\mathbb Z/ 2\mathbb{Z}$-equivariant cohomology version of ordinary cohomology" Found nothing on wikipedia,...
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        0answers
        166 views

        $\Gamma$-sets vs $\Gamma$-spaces

        I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set. For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...
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        1answer
        720 views

        Homology of the fiber

        Let $f:X\rightarrow Y $ be a fibration (with fiber $F$) between simply connected spaces such that $H_{\ast}(f):H_{\ast}(X,\mathbb{Z})\rightarrow H_{\ast}(Y,\mathbb{Z})$ is an isomorphism for $\ast\...
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        1answer
        138 views

        $X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a co-H-Space

        I have asked the below question on MathSE (with a 200 point bounty) but have yet to receive an answer there, and so am trying here. I am happy to remove it if it is nevertheless decided that this ...
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        0answers
        134 views

        Defining a graded ring structure on $\bigoplus_{i} \text{Ext}^i (A,A)$

        I read that, using the fact that $\text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ can by used to compute $\text{Ext}^* (A,B)$ (which I understand), we can give $\bigoplus_{i} \text{Ext}^i (A,...

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