nLab microcosm principle

Idea

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 structures 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.

Examples

• monoid objects can be defined in monoidal categories (categorified monoids), or more generally multicategories (virtual monoidal categories).
• commutative monoid objects can be defined in symmetric monoidal categories, or more generally in braided monoidal categories.
• Categorifying these examples, monoidal categories are instances of pseudomonoids, which can be defined in any monoidal bicategory. There are also versions for braided, symmetric, closed, compact closed, and so on.
• traces? can be defined in bicategories equipped with a shadow (a categorified trace).
• categories are instances of monads, which can be defined in any bicategory or more generally any double category (both a kind of categorified category) or virtual double category. This generalization includes internal categories and enriched categories.
• enriched categories are also objects of the free cocompletion of an enriched bicategory under a kind of Kleisli object.
• generalized multicategories are naturally defined internal to a virtual double category, which is itself a kind of generalized multicategory.
• double categories are instances of intermonads?, which can be defined in any intercategory (a categorified double category).
• duoids (objects with two monoidal structures) can be defined in any duoidal category (a category with two monoidal structures).
• Frobenius algebras can be defined in any star-autonomous category, and cocomplete $\ast$-autonomous posets are Frobenius algebras in the $\ast$-autonomous category Sup, while pro-$\ast$-autonomous categories are Frobenius pseudomonoids in the $\ast$-autonomous (indeed, compact closed) bicategory Prof.
• The prototypical example of a 2-fibration consists of internal fibrations in a 2-category.
• type theories can be defined inside a “2-type theory”; see adjoint type theory and logical framework.

Formalizations

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.

For $n$-categorical operad algebras

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 $A.$

For categorical algebras over cartesian monads

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.

For categorical algebras over Lawvere theories

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.

For algebras over (∞,1)-operads

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:

1. 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 $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$;

2. 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.

Reference

The term “microcosm principle” was coined in

• John Baez, James Dolan, Higher-Dimensional Algebra III (arXiv:q-alg/9702014)

Discussion is in

• The Microcosm Principle

Other formalizations can be found in

• Michael Batanin, The Eckman-Hilton argument and higher operads, arxiv:0207281

• Ichiro Hasuo, Bart Jacobs, and Ana Sokolova. (2008) The Microcosm Principle and Concurrency in Coalgebra. In: Amadio R. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2008. Lecture Notes in Computer Science, vol 4962. Springer, Berlin, Heidelberg. doi:10.1007/978-3-540-78499-9_18, paper, slides, more slides

• Ichiro Hasuo, Chris Heunen, Bart Jacobs, Ana Sokolova, Coalgebraic Components in a Many-Sorted Microcosm, International Conference on Algebra and Coalgebra in Computer Science CALCO 2009: Algebra and Coalgebra in Computer Science pp 64-80, (pdf)

• Jacob Lurie, Higher Algebra