nLab higher algebra

Context

Higher algebra

higher algebra

universal algebra

Theorems

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.

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

Symmetric monoidal model categories

duality between algebra and geometry in physics:

References

A comprehensive development of the theory is in