nLab
Batanin omega-category

Idea

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

The definition is similar to that of Trimble n-category.

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, (pdf) BAD LINK

• Denis-Charles Cisinski, Batanin higher groupoids and homotopy types, (pdf)

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.

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

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