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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 ...
0answers
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### 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 ...
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 ...
0answers
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### 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 ...
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 $X \times_Z Y$ ...
0answers
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4answers
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### 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 ...
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 ...
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 ...
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  by Faonte. This gives a way to get an $\infty$-category by starting with an $A_\infty$-...
1answer
189 views

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, $(-\otimes_{A}B, f_{\ast})... 0answers 101 views ### Free symmetric monoidal category of compactly generated category is compactly generated Let$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$, regarded as a symmetric monoidal ... 2answers 247 views ### Is the underlying vector space of the minimal model of an$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$is unique up to ... 0answers 155 views ### Is$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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