Algebras and modules
Model category presentations
Geometry on formal duals of algebras
Higher category theory
higher category theory
Extra properties and structure
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.
Microcosm principle: Certain algebraic structures can be defined in any category equipped with a categorified version of the same structure.
A standard example is the notion of monoid object which make sense in a monoidal category, and the notion of commutative monoid object which makes sense in a symmetric monoidal category.
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.
Formalization for algebras over (∞,1)-operads
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:
an (∞,1)-category equipped with algebraic structure as encoded by – an -monoidal (∞,1)-category – is equivalently encoded by a coCartesian fibration of (∞,1)-operads ;
an (∞,1)-algebra over an (∞,1)-operad over can be defined internal to such a -monoidal -category and is equivalently given by a section of this map.
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