A cartesian presentation of weak n–categories

By Charles Rezk. 2009

174 citations recorded by the source corpus.

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/662cd4132dfc66490bd2a15b7082f70dc3c40cc4

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Homotopy theory for algebras over polynomial monads (2013, unknown): “• Rezk’s model for (∞, n)-categories [57];”
Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, methodology): “On the other hand, Rezk’s theory has the advantage that AR is a canonical model for A up to levelwise homotopy equivalence in the category of diagrams ∆o → Top.”
Derived algebraic geometry (2014, background): “It also had the effect of bringing the model category language in the setting of higher category theory (see [122]): it is interesting to note that most, if not all, of the established theory of higher categories are based on the same idea and rely in an essential way on model category theory ([123]”
Fibrations and Yoneda's lemma in an ∞-cosmos (2017, background): “Another model for (∞, n)-categories is given by the fibrant objects in a cartesian closed model structure due to Charles Rezk in the category of sSet-valued presheaves on Joyal’s category θn [22].”
Derived Algebraic Geometry (2014, background): “It also had the effect of bringing the model category language in the setting of higher category theory (see [122]): it is interesting to note that most, if not all, of the established theory of higher categories are based on the same idea and rely in an essential way on model category theory ([123]”
Model structures on commutative monoids in general model categories (2014, background): “ith Michael Batanin, we have shown that all monoidal Bousﬁeld localizations of simplicial presheaves satisfy the hy potheses of Theorem 4.13 and as a result the model structures considered by Rezk in [40] will satisfy the commutative monoid axiom. 5.5.2. Projective Model Structure. The projective mo”
A type theory for synthetic ∞-categories (2017, methodology): “For instance, using Joyal’s theory of disks [10] would presumably yield a type theory in which to study (∞, n)-categories in the style of [23].”