Spahn combinatorial shape

One way to define higher structures is via functors on categories of combinatorial shapes (also called categories of geometric shapes).

Other approaches to define higher structures are by enrichment? or by internalization?.

General theory

Reedy categories

Interaction with model structures, generalized- and elegant Reedy categories

Characterizations of cells

Functorial characterization

Recursive characterization

Examples

$\Delta$ the simplex category

$\Omega$ the globe category

$T$ the tree category

$\square$ the cube category

$\Theta_n$ Joyal’s (disc) category

$\Theta_A$

where $A$ is an $\omega$-operad and $\Theta_A$ is a dense subcategory of the category of $\underline A$-algebras. This is diecussed in Berger.

$\Gamma$ Segal’s category

Opetope

Poset

Functorial images of combinatorial shape categories

Constraints on the functor

Nerves, Realization and Segal conditions

Model structures and presentation of higher structures

References

• This Week’s Finds in Mathematical Physics (Week 242), web (Discussion at the n-Cafe)

• Tom Leinster, higher operads, higher categories, arXiv:math/0305049

• higher topos theory

• André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf)

• André Joyal, Notes on quasi-categories (pdf).

• Eugenia Cheng, Aaron Lauda, higher-dimensional categories: an illustrated guide book, pdf

• Ieke Moerdijk, Bertrand Toen, simplicial methods for operads and algebraic geometry

• Clemens Berger, A Cellular Nerve for Higher Categories, pdf