# Tagged Questions

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### Is there a schema category for hyperstructures?

I am completely fascinated by Niels Baas' notion of hyperstructures, chiefly because I can see how such gadgets could be used in modeling both biological and social systems, or other evolutionary ...
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### understanding the definition of $\infty$-operad of module objects

I'm just trying to understand the following definition: Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following: Let $O^\otimes$ be ...
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### Differentiation of Lie $\infty$-groupoids

I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming. A Lie $\infty$-groupoid is a ...
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### Is the Thom diagonal co-$E_\infty$?

Given a map of spaces $f:X\to BGL_1(R)$ for $R$ an $E_\infty$-ring spectrum (of course this can be done more generally) one can produce a Thom spectrum $Mf$ by a number of methods. Let's denote such a ...
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### Nonunital $E_\infty$-rings

An elementary fact of algebra is that the category of nonunital commutative rings is equivalent to that of $\mathbb{Z}$-augmented unital commutative rings, the equivalence being given by forming ...
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### Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...
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### Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
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### Higher algebra and terminology about 2-objects

It is well known that one way to build higher category theory is to use some induction process, where an $n$-category has as $0$-cells some $n-1$ categories, such that for two $0$-cells $\mathcal{A}$ ...
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### Lurie's Endomorphism Space vs. Endomorphisms

In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of ...
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### Can we construct a Baas-Sullivan presentation of TMF?

Quick Review: The Baas-Sullivan construction cones off generators $\alpha_1, ..., \alpha_n \in \pi_*(MU)$ from $MU$ to get a new spectrum $MU/(\alpha_1, ..., \alpha_n)$, which is isomorphic to some ...
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### Construction of Highly Structured Quotient Groups in Quasicategories

Suppose we have a map of $E_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there ...
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### Lower Algebra: Modules over the monoidal category of abelian groups

Proposition 6.3.2.18 of Higher Algebra identifies $Mod_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable ...
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### Infinite dimensional 2-Hilbert spaces

Is there a definition of an infinite dimensional 2-Hilbert space? Finite dimensional 2-Hilbert spaces have been discussed by Baez in http://arxiv.org/abs/q-alg/9609018 In the more recent paper by ...
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### Lurie's approach to the bar-cobar adjunction

I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...
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### A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ? I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...
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### Obstructions for $E_n$-algebras

In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure. Have the obstructions for an object ...
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### unbounded derived category of a $\infty$-topos

In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, ...
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### Exact sequence of L-infinity-algebras

We call a sequence of $L_\infty$-algebras (weak) maps $$0\to L\xrightarrow{f} M\xrightarrow{g} N\to 0$$ is exact if it is exact on the the underlying chain complexes level. Thought I don't know ...
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### generalisations of the Seifert-van Kampen Theorem?

I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available. My attention was ...
### $A_{\infty}$ structure questions
Hello, I would like explanation or clear source for some things related to $A_{\infty}$-spaces, via Stasheff's polytopes. I tried not to think about them, because they seem too complicated for me; I ...