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...ogy]] and [[homotopy theory]] into other domains, such as in [[commutative algebra]] and [[algebraic geometry]]. All [[Magma (algebra)|magmas]] with [[Unity (mathematics)|units]] are H-objects in the category ...

{{Short description|Algebra concept}} == Relation with homotopical connectivity == ...

...ote>where <math>\mathbb{Z}_{(p)}</math> is the [[Localization (commutative algebra)|localization]] of <math>\mathbb{Z}</math> by the prime ideal <math>(p)</ma *[[Steenrod algebra]] ...

...s<ref>{{Cite book|last=Quillen|first=Daniel G.|date=1967|title=Homotopical Algebra|url=https://link.springer.com/book/10.1007/BFb0097438|series=Lecture Notes ...+1}</math>, where the categories <math>C_n</math> keep track of the higher homotopical information up to level <math>n</math>. Such a structure was later called a ...

...ub.io}}</ref>^{pg 193} which are a new framework for [[homotopical algebra]]. [[Category:Homotopical algebra]] ...

In mathematics, an [[Associative algebra|algebra]] such as <math>(\R,+,\cdot)</math> has multiplication <math>\cdot</math> w ...ity]], we actually get equality after "compressing" the information in the algebra. ...

...Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose. These homotopy algebras are us There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations more than others. The most ...

...(02)00135-4 | mr=1935979 | year=2002 | journal=Journal of Pure and Applied Algebra | volume=175 | issue=1 | pages=207–222| doi-access= }} ...hor1-link=André Joyal | last2=Tierney | first2=Myles | title=Categories in algebra, geometry and mathematical physics | arxiv=math.AT/0607820 | publisher=Amer ...

...athematics)|sheaves]] on <math>X</math> using the methods of [[homotopical algebra]]. ...cdot b_n</math></blockquote>Iterating this construction gives a simplicial algebra<blockquote><math>\cdots \to A[A[A[B]]] \to A[A[B]] \to A[B] \to B</math></b ...

...tion are abelian categories, hence they are suitable for doing homological algebra. [[Category:Homotopical algebra]] ...

that is, a smash product is an analog of a [[tensor product]] in abstract algebra (see [[tensor-hom adjunction]]). Explicitly, <math>X \wedge Y</math> is the ...pical algebra |website=nLab |url=https://ncatlab.org/nlab/show/homotopical+algebra }} ...

..., which provide local charts, are replaced by either [[differential graded algebra]]s (over <math>\mathbb{Q}</math>), [[simplicial commutative ring]]s or [[Hi ...geometry is fundamentally the study of geometric objects using homological algebra and homotopy. Since objects in this field should encode the homological and ...

...] (including both previously existing mathematics and new mathematics that homotopical types make possible); and the [[Formal proof|formalization]] of each of the ...ategory of [[groupoid]]s. This was the first truly "[[Homotopical algebra|homotopical]]" model of type theory, albeit only "1-[[dimension]]al" (the traditional m ...

...constructions and properties being expressed in the setting of categorical algebra<math>^{[1]}</math>. This approach emphasizes the combinatorial role of cubi ...ted d-spaces.<ref>{{Cite journal |last=Gaucher |first=P. |year=2009 |title=Homotopical interpretation of globular complex by multipointed d-space |url=http://www. ...

...anged (<math>1\times n = n\times 1 = n</math>). As a result, the [[Square (algebra)|square]] (<math>1^2=1</math>), [[square root]] (<math>\sqrt{1} = 1</math>) ...ups (those that are [[Homotopical connectivity|path-connected]] with no '[[Homotopical connectivity#Definition using holes|holes]]').{{sfn|Gaitsgory|Lurie|2019|pp ...

...1.1.5.2718}}</ref> interpreting persistence in the language of commutative algebra.<ref name=Zomorodian04 /> ...fication theorem]] interpreting persistence in the language of commutative algebra appeared in 2005:<ref name=Zomorodian04>{{Cite journal|title = Computing Pe ...