symmetric monoidal (∞,1)-category of spectra
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.
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.
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.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
A comprehensive development of the theory is in
A introductory survey is in