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  "body": "\n## Summary\n\nA differential graded algebra, or dg algebra, is a [[graded algebra|8KB4SQGC0C]] with a differential $d$ of degree one.\nThe differential squares to zero and obeys a signed Leibniz rule.\nForgetting the product it is a [[chain complex|WQ82C0M3QB]], and its cohomology is again a graded algebra.\nThe de Rham complex of a smooth manifold, the singular cochains of a space with the cup product, and Koszul complexes are standard examples.\nDg algebras are the basic objects of homological algebra with products and of rational homotopy theory.\n\n## Explanations\n\n### L1 human\n\nDifferential forms on a manifold can be multiplied by the wedge product and differentiated by $d$.\nApplying $d$ twice always gives zero, and $d$ of a product follows a product rule with a sign.\nA differential graded algebra is any algebra with these two features.\nA small example is polynomials $f(x)$ in degree $0$ together with the forms $g(x)\\,dx$ in degree $1$, with $d f = f'(x)\\, dx$.\n",
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