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      Questions tagged [higher-algebra]

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      The relationship between representations of groups and evaluation and coevaluation maps for $vect_{G}$ module categories

      Let $G$ be a finite group and $vec_{G}$ be the monoidal category of finite dimensional $G$-graded vector spaces. Given any $vec_{G}$ module category $\mathcal{M}$ we can define a dual module category ...
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      0answers
      47 views

      Concerning the definition of a 2-crossed module

      Question: Is there some generalization of the definition of crossed-module which appropriately fits into the holonomy-considerations I am interested in and has, as an example, the generalization of ...
      6
      votes
      3answers
      342 views

      Definition of $E_n$-modules for an $E_n$-algebra

      The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more ...
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      votes
      0answers
      97 views

      Cyclic version of Lie algebra cohomology

      Lie algebra cochains have a differential $d$ where $d^2 =0$ because of the Jacobi identity, which can be written in the cyclic form or the Leibniz form. $L_\infty$ algebra cochains have a ...
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      2answers
      500 views

      Describing fiber products in stable $\infty$-categories

      Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times_Z Y$ ...
      4
      votes
      0answers
      168 views

      Examples of Lurie tensor product computations

      I am interested in examples of computing the Lurie tensor product. For example, if $A$ and $B$ are connective DG algebras (over $\mathbb{Z}$, say), then I think there is an equivalence $A\text{-mod} \...
      5
      votes
      0answers
      83 views

      Compact Generation of Co-Module Categories

      Let $\mathcal{C}$ be a compactly generated stable $\infty$-category, linear over a field of characteristic $0$ (i.e., so that it is in particular a dg-category). Let $A$ be a co-monad acting on $\...
      11
      votes
      4answers
      2k views

      Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

      This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there). The answer ...
      12
      votes
      1answer
      876 views

      Is the $\infty$-category of spectra “convenient”?

      A 1991 paper of Lewis, title “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$: There is a symmetric monoidal smash ...
      11
      votes
      1answer
      1k views

      How should one approach reading Higher Algebra by Lurie?

      A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT ...
      4
      votes
      1answer
      334 views

      Is it possible to define linear $A_\infty$-categories as special $\infty$-categories?

      A functor $N\colon\mathrm{Cat}_{A_\infty}\longrightarrow\mathrm{Cat}_\infty$ is constructed in a paper [1] by Faonte. This gives a way to get an $\infty$-category by starting with an $A_\infty$-...
      6
      votes
      1answer
      189 views

      Quillen equivalent module categories

      Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes_{A}B, f_{\ast})$ which is an equivalence. In this situation, $(-\otimes_{A}B, f_{\ast})...
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      votes
      0answers
      101 views

      Free symmetric monoidal category of compactly generated category is compactly generated

      Let $k$ be a field and let $\mathcal{C}=\mathbf{StLin}_k$ be the $\infty$-category of stable infinity categories enriched over the $\infty$-category $\mathbf{Vect}_k$, regarded as a symmetric monoidal ...
      5
      votes
      2answers
      247 views

      Is the underlying vector space of the minimal model of an $A_{\infty}$-algebra canonical?

      On the page 4 of these notes it is stated that an $A_{\infty}$-algebra $A$ is necessarily is quasi-isomorphic to an $A_{\infty}$-algebra $HA$ with trivial differential. Moreover, $HA$ is unique up to ...
      3
      votes
      0answers
      155 views

      Is $Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))$ for an additive category $\mathcal{C}$?

      Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category. ...

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