nLab
higher algebra

higher geometry \leftarrow Isbell duality \to 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:

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

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 (,1)(\infty,1)-Categories

The monoidal structure on stable homotopy theory

Symmetric monoidal (,1)(\infty,1)-categories and commutative algebra

Commutative ring spectra

Symmetric monoidal model categories

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

References

A comprehensive development of the theory is in

See also

Revised on November 15, 2013 01:36:18 by Urs Schreiber (188.200.54.65)