A differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.

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### Questions wrt. definition of $L_\infty$- and dg-Lie-algebras

I am trying to understand this definition from nLab (Def. 3.2) of $L_\infty$-algebras, and the following example that is supposed to boil down to dg-Lie algebras. What is the difference between ...
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### How to show that for an element $x$ in DGA cohomology it is true that $[x]=[-x]$?

Perhaps it is a really dumb question, but I want to use this property and it seems that it must hold. However, if one takes a specific homogeneous element $x$ of a differential graded algebra, then ...
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In https://kerodon.net/tag/00PK, Lurie introduces the dg-nerve functor from differential graded categories to simplicial sets as a tool to translate statements/ constructions from the dg-context to ...
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### Suspension Operator on Graded Algebras

Given a morphism of GAs $f:A\to B$ of degree $-k$, that is, $f(A_n)\subset B_{n-k}$, I want to understand the sign conventions of commutation with the suspension operator. That is, if $s:A\to A$ is ...
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### Why $f_1$ commutes with the product up to boundaries

Let $(A,m_1)$ be a differential graded algebra, $m_2:A\otimes A\to A$ its product and $H^*A$ its homology, on which there is another product induced by $m_2$ which I'll call equally. I'm trying to ...
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Let $C^{\bullet}$ be a (cohomological) chain complex of modules with coboundary map $\delta$ and filtration $$\cdots = F_{-1}C^{\bullet} = F_{0}C^{\bullet} = C^{\bullet} \supset F_{1}C^{\bullet} \... 0answers 31 views ### DG algebra and its zeroth cohomology are derived equivalent This is slightly related to this question: Can an algebra be morita equivalent to its dg-extension? . Suppose we have a DG-algebra A, such that H^0(A) is Noetherian (both left and right) and H^\... 1answer 46 views ### Can an algebra be morita equivalent to its dg-extension? Say we have a DG algebra A=\bigoplus_{n\geq 0}A_n, let B=A_0, the 0th degree of A. Assume we have that the category of DG-modules over A is equivalent to the category of module over B. Does ... 1answer 101 views ### Definition of \mathfrak{g}-differential graded algebra I am reading Group actions on manifolds by Eckhard Meinrenken (Lecture Notes, University of Toronto, Spring 2003). In page 45, definition 5.2, author introduce the notion of \mathfrak{g}-... 0answers 64 views ### Explicit formula for the equalizer of coalgebras The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is ... 0answers 58 views ### When does a fibration f:X\rightarrow Y in a model category admit a section?. If we have a fibration f:X\rightarrow Y in a model category C, where Y is cofibrant and both X, Y are fibrant. Does f admit a section (right inverse)?. If it does not work in general, ... 0answers 43 views ### Categorical product of non-unital associative differential graded coalgebras Given two non-unital associative dg coalgebras D and C, I want to give an explicit construction of the product C\prod D, this may follow from the dual construction (coproduct of non-unital ... 1answer 61 views ### if C is a filtered coalgebra, does Gr(B\Omega C)\backsimeq B\Omega (Gr C) hold? I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects X\rightarrow Gr(X) commutes with direct sums (this seems to be ... 1answer 52 views ### The twisted tensor product BA\otimes_{\tau} A as the non-unital Hochschild complex The twisting universal morphism \tau: BA\rightarrow A induces a differential \partial_{\tau} on BA\otimes_{\tau}A, we have:$$\partial_{\tau}(x\otimes y)=\partial x\otimes y+(-1)^{\lvert x\rvert}...
Suppose we have a rationalization $\bar{X}$ of a simply connected topological space $X$. Then we can construct a CDG-algebra - a Sullivan model $S$ corresponding to $\bar{X}$. How can we interpret ...