38,882 research outputs found
Stabilized profunctors and stable species of structures
We introduce a bicategorical model of linear logic which is a novel variation
of the bicategory of groupoids, profunctors, and natural transformations. Our
model is obtained by endowing groupoids with additional structure, called a
kit, to stabilize the profunctors by controlling the freeness of the groupoid
action on profunctor elements.
The theory of generalized species of structures, based on profunctors, is
refined to a new theory of \emph{stable species} of structures between
groupoids with Boolean kits. Generalized species are in correspondence with
analytic functors between presheaf categories; in our refined model, stable
species are shown to be in correspondence with restrictions of analytic
functors, which we characterize as being stable, to full subcategories of
stabilized presheaves. Our motivating example is the class of finitary
polynomial functors between categories of indexed sets, also known as normal
functors, that arises from kits enforcing free actions.
We show that the bicategory of groupoids with Boolean kits, stable species,
and natural transformations is cartesian closed. This makes essential use of
the logical structure of Boolean kits and explains the well-known failure of
cartesian closure for the bicategory of finitary polynomial functors between
categories of set-indexed families and cartesian natural transformations. The
paper additionally develops the model of classical linear logic underlying the
cartesian closed structure and clarifies the connection to stable domain
theory.Comment: FSCD 2022 special issue of Logical Methods in Computer Science, minor
changes (incorporated reviewers comments
Grothendieck quasitoposes
A full reflective subcategory E of a presheaf category [C*,Set] is the
category of sheaves for a topology j on C if and only if the reflection
preserves finite limits. Such an E is called a Grothendieck topos. More
generally, one can consider two topologies, j contained in k, and the category
of sheaves for j which are separated for k. The categories E of this form, for
some C, j, and k, are the Grothendieck quasitoposes of the title, previously
studied by Borceux and Pedicchio, and include many examples of categories of
spaces. They also include the category of concrete sheaves for a concrete site.
We show that a full reflective subcategory E of [C*,Set] arises in this way for
some j and k if and only if the reflection preserves monomorphisms as well as
pullbacks over elements of E.Comment: v2: 24 pages, several revisions based on suggestions of referee,
especially the new theorem 5.2; to appear in the Journal of Algebr
Algebraic K-theory of quasi-smooth blow-ups and cdh descent
We construct a semi-orthogonal decomposition on the category of perfect
complexes on the blow-up of a derived Artin stack in a quasi-smooth centre.
This gives a generalization of Thomason's blow-up formula in algebraic K-theory
to derived stacks. We also provide a new criterion for descent in Voevodsky's
cdh topology, which we use to give a direct proof of Cisinski's theorem that
Weibel's homotopy invariant K-theory satisfies cdh descent.Comment: 24 pages; to appear in Annales Henri Lebesgu
Kan extensions and the calculus of modules for -categories
Various models of -categories, including quasi-categories,
complete Segal spaces, Segal categories, and naturally marked simplicial sets
can be considered as the objects of an -cosmos. In a generic
-cosmos, whose objects we call -categories, we introduce
modules (also called profunctors or correspondences) between
-categories, incarnated as as spans of suitably-defined fibrations with
groupoidal fibers. As the name suggests, a module from to is an
-category equipped with a left action of and a right action of ,
in a suitable sense. Applying the fibrational form of the Yoneda lemma, we
develop a general calculus of modules, proving that they naturally assemble
into a multicategory-like structure called a virtual equipment, which is known
to be a robust setting in which to develop formal category theory. Using the
calculus of modules, it is straightforward to define and study pointwise Kan
extensions, which we relate, in the case of cartesian closed -cosmoi,
to limits and colimits of diagrams valued in an -category, as
introduced in previous work.Comment: 84 pages; a sequel to arXiv:1506.05500; v2. new results added, axiom
circularity removed; v3. final journal version to appear in Alg. Geom. To
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