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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?.
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.
This Week’s Finds in Mathematical Physics (Week 242), web (Discussion at the n-Cafe)
Tom Leinster, higher operads, higher categories, arXiv:math/0305049
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