nLab
microcosm principle

Context

Higher algebra

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

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 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.

Examples

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 nn-categorical operad algebras

Baez and Dolan gave the following precise definition: for any operad 𝒪\mathcal{O}, an nn-categorical 𝒪\mathcal{O}-algebra internal to an (n+1))(n+1))-categorical 𝒪\mathcal{O}-algebra AA is a lax morphism from the terminal (n+1))(n+1))-categorical 𝒪\mathcal{O}-algebra to AA.

For categorical algebras over cartesian monads

Batanin gave the following definition: for a cartesian monad TT, an internal TT-algebra in an algebra AA for the extension of TT to internal categories is a lax TT-morphism from the terminal such algebra to AA. More generally, AA 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 LL, an internal LL-algebra in a categorical LL-algebra (an LL-algebra in Cat) is a lax LL-morphism from the terminal such algebra to AA. They used this to prove a general result about the liftings of LL-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 𝒪\mathcal{O} we have:

  1. an (∞,1)-category 𝒞\mathcal{C} equipped with algebraic structure as encoded by 𝒪\mathcal{O} – an 𝒪\mathcal{O}-monoidal (∞,1)-category – is equivalently encoded by a coCartesian fibration of (∞,1)-operads 𝒞 𝒪 \mathcal{C}^\otimes \to \mathcal{O}^\otimes;

  2. an (∞,1)-algebra over an (∞,1)-operad over 𝒪\mathcal{O} can be defined internal to such a 𝒪\mathcal{O}-monoidal (,1)(\infty,1)-category 𝒞\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

Discussion is in

Other formalizations can be found in

  • Ichiro Hasuo, Bart Jacobs, and Ana Sokolova. The Microcosm Principle and Concurrency in Coalgebra. paper, slides, more slides
Revised on November 18, 2017 18:06:38 by Mike Shulman (76.167.222.204)