symmetric monoidal (∞,1)-category of spectra
higher geometry $\leftarrow$ Isbell duality $\to$ higher algebra
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.
Ordinary algebra concerns itself in particular with structures such as associative algebras, which are monoids internal to monoidal categories:
a monoid internal to Set is just an ordinary monoid;
a monoid internal to Ab, the category of abelian groups, is a ring;
a monoid internal to Vect is an ordinary algebra: a vector space equipped with a linear binary associative product with unit;
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
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.
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.
examples
duality between algebra and geometry in physics:
A comprehensive development of the theory is in
See also