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|higher structures are by enrichment|by enrichment]] or [[higher structures by internalization|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](#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](http://golem.ph.utexas.edu/category/2006/12/this_weeks_finds_in_mathematic_4.html#c006655) (Discussion at the n-Cafe) * Tom Leinster, higher operads, higher categories, [arXiv:math/0305049 ](http://arxiv.org/abs/math/0305049) * [[nLab:higher topos theory]] * [[nLab:André Joyal]], _The theory of quasicategories and its applications_ lectures at [Simplicial Methods in Higher Categories](http://www.crm.es/HigherCategories/), ([pdf](http://www.crm.cat/HigherCategories/hc2.pdf)) * [[nLab:André Joyal]], _Notes on quasi-categories_ ([pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf)). * Eugenia Cheng, Aaron Lauda, higher-dimensional categories: an illustrated guide book, [pdf](http://www.cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf) * Ieke Moerdijk, Bertrand Toen, simplicial methods for operads and algebraic geometry * Clemens Berger, A Cellular Nerve for Higher Categories, [pdf](http://math1.unice.fr/~cberger/nerve.pdf){#Berger}