nLab microcosm principle



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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.



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 A.A.

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.


The term “microcosm principle” was coined in

Discussion is in

Other formalizations can be found in

  • 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

Last revised on November 10, 2021 at 10:07:37. See the history of this page for a list of all contributions to it.