symmetric monoidal (∞,1)-category of spectra
n-category = (n,n)-category
n-groupoid = (n,0)-category
In higher algebra/higher category theory one can define (generalized) algebraic structures internal to categories which themselves are equipped with certain algebraic structure, in fact with the same kind of algebraic structure. In (Baez-Dolan 97) this has been called the microcosm principle.
The term “microcosm principle” arises from the idea that the monoid object (the ‘microcosm’) can only thrive in a category (a ‘macrocosm’) that resembles itself.
The microcosm principle is a general heuristic, but in some contexts, a general version of it can be proven formally. One such formalization was given (independently) in (Lurie) in the context of higher algebra in homotopy theory/(∞,1)-category theory:
given an (∞,1)-operad we have:
See at (∞,1)-algebra over an (∞,1)-operad for examples and further details.
The term “microcosm principle” was coined in
Discussion is in
One formalization is in