Related papers: Segal Enriched Categories I

We define and study opfibrations of $V$-enriched categories when $V$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with…

We make some remarks on the foundations of the homotopy theory of enriched precategories, as exposed in Carlos Simpson's book "Homotopy theory of higher categories".

We prove that the homotopy theory of parsummable categories (as defined by Schwede) with respect to the underlying equivalences of categories is equivalent to the usual homotopy theory of symmetric monoidal categories. In particular, this…

Mackey functors provide the coefficient systems for equivariant cohomology theories. More generally, enriched presheaf categories provide a classification and organization for many stable model categories of interest. Changing enrichments…

A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…

We introduce acyclic polygraphs, a notion of complete categorical cellular model for (small) categories, containing generators, relations and higher-dimensional globular syzygies. We give a rewriting method to construct explicit acyclic…

Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal…

This paper studies the existence of model category structures on algebras and modules over operads in monoidal model categories.

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model…

In this paper we consider the problem of building rich categories of setoids, in standard intensional Martin-L\"of type theory (MLTT), and in particular how to handle the problem of equality on objects in this context. Any…

We put a Quillen model structure on the category of small categories enriched in simplicial $k$-modules and non-negatively graded chain complexes of $k$-modules, where $k$ is a commutative ring. The model structure is obtained by transfer…

Enrichment and internal categories are two different way to generalize the notion of category. As such, enriching double categories (which are categories internal to Cat) is not a clear concepts. One can look at the internal categories of…

We introduce a notion of bimodule in the setting of enriched $\infty$-categories, and use this to construct a double $\infty$-category of enriched $\infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then…

I show that the theories of enrichment in a monoidal infinity-category defined by Hinich and by Gepner-Haugseng agree, and that the identification is unique. Among other things, this makes the Yoneda lemma available in the former model.

A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a monoidal model category, one can consider…

We show that the Segal topos of derived stacks over simplicial commutative $k$-algebras, which can be used to model natural phenomena, has a subobject classifier, something we regard as being a source from which dynamics is generated. This…

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is…

We show that the basic categorical concept of an S-algebra as derived from the theory of Segal's Gamma-sets provides a unifying description of several constructions attempting to model an algebraic geometry over the absolute point. It…

The goal of this article is to emphasize the role of cubical sets in enriched categories theory and infinity-categories theory. We show in particular that categories enriched in cubical sets provide a convenient way to describe many…

Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were…