higher algebra in nLab
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Context
Higher algebra
Algebraic theories
Algebras and modules
Higher algebras
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symmetric monoidal (∞,1)-category of spectra
Model category presentations
Geometry on formal duals of algebras
Theorems
higher geometry Isbell duality higher algebra
Contents
Idea
The notion of higher algebra or homotopical algebra refers to generalizations of algebra in the context of homotopy theory and more general of higher category theory.
General
Ordinary algebra concerns itself in particular with structures such as associative algebras, which are monoids internal to monoidal categories:
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a monoid internal to Set is just an ordinary monoid;
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a monoid internal to Ab, the category of abelian groups, is a ring;
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a monoid internal to Vect is an ordinary algebra: a vector space equipped with a linear binary associative product with unit;
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a monoid in a category of chain complexes is a differential graded algebra;
etc.
Of course, there are other aspects to algebra such as those resulting from non-associative theories such as Lie algebras and there are many aspects such as questions within Galois theory, and representation theory for which the above is too limited a view, but for the moment let it stand.
Higher algebra (or homotopical algebra) is similarly, but in particular, the study of monoids internal to higher categories.
A central motivating example for - or special case of the study of higher algebra was
- monoids internal to the stable (infinity,1)-category of spectra – called commutative ring spectra
The “higher algebra” embodied by commutative ring spectra has been called brave new algebra by F. Waldhausen.
More generally, algebra is partially about algebraic theories, about monads and about operads. All these have higher analogs in higher algebra.
Other aspects of higher algebra?
The parts of algebra that we set aside at the end of the idea are not outside the possible range of higher algebra, they just have not yet been that developed and it is not always clear in what directions they most naturally ‘should’ be developed. To take an example, Lie infinity-algebroid is clearly a higher algebraic analogue of a Lie algebra, and is a ‘multi-object’ one as well. Questions in representation theory are often phrased in terms of monoidal categories, and their higher algebraic analogues have new structural facets that look very interesting and useful. Finally Galois theory naturally falls into the context of Grothendieck’s extensive work both on higher stacks but also the Grothendieck-Teichmuller theory. Here the theory is awaiting clear indications what higher Galois theory might mean.
Concepts
Monads, algebraic theories, operads
Algebras and modules
Monoidal -Categories
The monoidal structure on stable homotopy theory
Symmetric monoidal -categories and commutative algebra
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examples
Commutative ring spectra
Symmetric monoidal model categories
duality between algebra and geometry
in physics:
References
A comprehensive development of the theory is in
See also
- Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, Rings, modules and algebras in stable homotopy theory, Mathematical surveys and monographs 47, American Mathematical Society, 1997
Last revised on August 8, 2020 at 17:48:33. See the history of this page for a list of all contributions to it.