symmetric monoidal (∞,1)-category of spectra
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.
Moreover, in many cases the “internal” version of the structure can be identified with lax morphisms of categorified strurctures with terminal domain. For instance, a monoid in a monoidal category is equivalently a lax monoidal functor with terminal domain. More generally, the categorified structure can often be generalized to a “virtual” (generalized multicategory-like) structure.
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, general versions of it can be defined and proven formally. These do not generally include all examples, but often include many of them.
Baez and Dolan gave the following precise definition: for any operad $\mathcal{O}$, an $n$-categorical $\mathcal{O}$-algebra internal to an $(n+1))$-categorical $\mathcal{O}$-algebra $A$ is a lax morphism from the terminal $(n+1))$-categorical $\mathcal{O}$-algebra to $A$.
Batanin gave the following definition: for a cartesian monad $T$, an internal $T$-algebra in an algebra $A$ for the extension of $T$ to internal categories is a lax $T$-morphism from the terminal such algebra to $A$. More generally, $A$ can be an algebra for some other monad, for instance which builds in some commutativity.
Hasuo, Jacobs, and Sokolova gave the following very similar definition: for a Lawvere theory $L$, an internal $L$-algebra in a categorical $L$-algebra (an $L$-algebra in Cat) is a lax $L$-morphism from the terminal such algebra to $A$. They used this to prove a general result about the liftings of $L$-algebra structures to coalgebras.
Another such formalization was given (independently) in (Lurie) in the context of higher algebra in homotopy theory/(∞,1)-category theory:
given an (∞,1)-operad $\mathcal{O}$ we have:
an (∞,1)-category $\mathcal{C}$ equipped with algebraic structure as encoded by $\mathcal{O}$ – an $\mathcal{O}$-monoidal (∞,1)-category – is equivalently encoded by a coCartesian fibration of (∞,1)-operads $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$;
an (∞,1)-algebra over an (∞,1)-operad over $\mathcal{O}$ can be defined internal to such a $\mathcal{O}$-monoidal $(\infty,1)$-category $\mathcal{C}$ 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
Other formalizations can be found in