nLab Batanin omega-category

I

Idea

A Batanin $\omega$-category is a weak ∞-category defined as an algebra over a suitable contractible globular operad. So this is an algebraic definition of higher category.

The definition is similar to that of Trimble n-category (which is actually a special case of a Batanin $\omega$-category) and similar to the definition of Grothendieck-Maltsiniotis infinity-category.

Morphisms

When a weak $\infty$-category is modeled as a module over an $O$-operad, morphisms of modules $F : C \to D$ will correspond to strict $\infty$ functors. To get weak $\infty$-functors one has to resolve $C$.

One way to do this is described in (Garner).

References

• Michael Batanin, Monoidal globular categories as a natural environment for the theory of weak $n$-categories , Advances in Mathematics 136 (1998), no. 1, 39–103.

• Ross Street, The role of Michael Batanin’s monoidal globular categories, in Higher Category Theory, eds. E. Getzler and M. Kapranov, Contemp. Math. 230, American

  Mathematial Society, Providence, Rhode Island, 1998, pp. 99–116. (pdf)

Work towards establishing the homotopy hypothesis for Batanin $\omega$-groupoids can be found here:

• Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)

• Denis-Charles Cisinski, Batanin higher groupoids and

  homotopy types_ (arXiv:math/0604442)

A nice introduction to this subject is:

• Eugenia Cheng, Batanin omega-groupoids and the homotopy hypothesis, (recorded lecture) from the Fields Institute Workshop on Higher Categories and their Applications, January 10, 2007.

A discussion of weak $\omega$-functors between Batanin $\omega$-categories is in

• Richard Garner, Homomorphisms of higher categories (arXiv:0810.4450)

An application of Batanin weak $\omega$-groupoids to homotopy type theory appears in

• Benno van den Berg, Richard Garner, Types are weak $\omega$-groupoids (arXiv:0812.0298)

A discussion of weak $\omega$-functors between Batanin $\omega$-categories, and all kind of weak $n$-transformations in the spirit of Batanin approach, with an emphasis to the possibility to the existence of the weak $\omega$-category of the weak $\omega$-categories in Batanin’s sense appears in

• Camell Kachour, Steps toward the weak higher category of weak higher categories in the globular setting, published in Categories and General Algebraic Structures with Applications (2015). (web)

Batanin weak $\omega$-categories were further developed here:

• Tom Leinster, Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series 298, Cambridge University Press, 2004. (arXiv:math/0305049)

An approach to Batanin’s weak $\omega$-categories that also sets up a definition of Trimble infinity-category and a fundamental infinity-groupoid construction is here:

• Eugenia Cheng and Tom Leinster, Weak ∞-categories via terminal coalgebras, (arXiv:1212.5853).