Related papers: Coherence, Homotopy and 2-Theories
Quillen defined a {\em model category} to be a category with finite limits and colimits carrying a certain extra structure. In this paper, we show that only finite products and coproducts (in addition to the certain extra structure alluded…
This text develops a homotopy theory of 2-categories analogous to Grothendieck's homotopy theory of categories developed in "Pursuing Stacks". We define the notion of "basic localizer of 2-Cat", 2-categorical generalization of…
The program of internal type theory seeks to develop the categorical model theory of dependent type theory using the language of dependent type theory itself. In the present work we study internal homotopical type theory by relaxing the…
We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of…
The two pillars of Algebraic topology - Homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and…
In this paper, we establish a theorem that proves a condition when an inclusion morphism between simplicial sets becomes a weak homotopy equivalence. Additionally, we present two applications of this result. The first application…
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and…
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the…
We study a variation of Turaev's homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space $X$ and define an $\cS_X$-structure to be a monoidal 2-functor from this to the 2-category…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or…
We study the notion of a bifibration in simplicial sets which generalizes the classical notion of two-sided discrete fibration studied in category theory. If $A$ and $B$ are simplicial sets we equip the category of simplicial sets over…
This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids…
We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra \pi_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology…
We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's…
There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, whereas the other also allows classical…
We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for…
Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This…
If all objects of a simplicial combinatorial model category \cat A are cofibrant, then there exists the homotopy model structure on the category of small functors $\sS^{\cat A}$, where the fibrant objects are homotopy functors, i.e.,…