All Questions
Tagged with deformation-theory homotopy-theory
13 questions
2
votes
0
answers
354
views
Square-zero extensions mod $p^n$
$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
- 68
14
votes
0
answers
250
views
What is the relationship between Goodwillie calculus and derived deformation theory?
Goodwillie calculus is a way of understanding a functor $F$ in terms of its Goodwillie tower, a tower whose limit approximates $F$, whose layers can be understood in terms of stable data. Derived ...
- 63.9k
5
votes
0
answers
272
views
Is Koszul duality a deformation theory when not over a field?
Let $k$ be a field. Then Thm 15.3.3.1 of Lurie's SAG says that Koszul duality, regarded as a contravariant endofunctor $\bar D$ of augmented $E_n$-algebras over $k$, is a deformation theory in the ...
- 63.9k
4
votes
0
answers
168
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Building conilpotent coalgebras from co-square-zero-extensions
Let $\mathrm{K}$ be a field of char. 0.
Given a chain complex $\mathrm{X} $ over $\mathrm{K}$ denote $\mathrm{E}(\mathrm{X})$ the co-square-zero-extension on $\mathrm{X}, $ i.e. the
cocommutative ...
- 1,361
7
votes
1
answer
493
views
Pro-representability of deformation functor associated to a DG Lie algebra
Edit : There are several satisfying proofs in the case each $L^i$ is finite-dimensional. It is proven (for example, Hinich DG coalgebras as formal stacks) that for $A$ : local Artin ring then $\...
4
votes
1
answer
291
views
Intrinsic formality versus rigidity of a differential graded Lie algebra
Let $\mathfrak g:=(V,d,[\cdot,\cdot])$ be a differential graded Lie algebra (DGLA) where $d$ is the zero differential.
Intrinsic formality: The DGLA $\mathfrak g$ will be said intrinsically formal ...
1
vote
0
answers
96
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Transformation of operad algebras
Not a good title.
Suppose we have two dg symmetric Koszul operads, say $O_1$ and $O_2$. Then their (homotopy) algebras over a dg vector space $V$ are (equivalent to) twisting morphisms
$$f\in TW(O^*...
- 2,074
14
votes
1
answer
645
views
Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory
Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...
- 25.6k
7
votes
2
answers
3k
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How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?
First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that ...
- 9,019
3
votes
1
answer
217
views
What is the definition of "the $L_\infty$ part of a $G_\infty$ morphism"?
We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G_\infty$ morphism from $HH^\...
- 9,019
22
votes
2
answers
4k
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obstruction theories in algebraic geometry
I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction ...
- 8,004
7
votes
2
answers
1k
views
Deformation theory of co-$A_\infty$ structures.
The following question is related to my previous post on co-$A_\infty$ spaces (co-$A_\infty$ spaces), but goes in a somewhat different direction.
Some Background:
In trying to classify $A_\infty$ ...
- 18.8k
18
votes
2
answers
2k
views
Why do my quantum group books avoid homotopical language?
I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.
Many notes closer to "Kontsevich ...
- 904