442 research outputs found
The Axiom of Choice in Cartesian Bicategories
We argue that cartesian bicategories, often used as a general categorical algebra of relations, are also a natural setting for the study of the axiom of choice (AC). In this setting, AC manifests itself as an inequation asserting that every total relation contains a map. The generality of cartesian bicategories allows us to separate this formulation from other set-theoretically equivalent properties, for instance that epimorphisms split. Moreover, via a classification result, we show that cartesian bicategories satisfying choice tend to be those that arise from bicategories of spans
The low-dimensional structures formed by tricategories
We form tricategories and the homomorphisms between them into a bicategory,
whose 2-cells are certain degenerate tritransformations. We then enrich this
bicategory into an example of a three-dimensional structure called a locally
cubical bicategory, this being a bicategory enriched in the monoidal 2-category
of pseudo double categories. Finally, we show that every sufficiently
well-behaved locally cubical bicategory gives rise to a tricategory, and
thereby deduce the existence of a tricategory of tricategories.Comment: 41 pages; v2: final journal versio
Shadows and traces in bicategories
Traces in symmetric monoidal categories are well-known and have many
applications; for instance, their functoriality directly implies the Lefschetz
fixed point theorem. However, for some applications, such as generalizations of
the Lefschetz theorem, one needs "noncommutative" traces, such as the
Hattori-Stallings trace for modules over noncommutative rings. In this paper we
study a generalization of the symmetric monoidal trace which applies to
noncommutative situations; its context is a bicategory equipped with an extra
structure called a "shadow." In particular, we prove its functoriality and
2-functoriality, which are essential to its applications in fixed-point theory.
Throughout we make use of an appropriate "cylindrical" type of string diagram,
which we justify formally in an appendix.Comment: 46 pages; v2: reorganized and shortened, added proof for cylindrical
string diagrams; v3: final version, to appear in JHR
Reasoning about Unreliable Actions
We analyse the philosopher Davidson's semantics of actions, using a strongly
typed logic with contexts given by sets of partial equations between the
outcomes of actions. This provides a perspicuous and elegant treatment of
reasoning about action, analogous to Reiter's work on artificial intelligence.
We define a sequent calculus for this logic, prove cut elimination, and give a
semantics based on fibrations over partial cartesian categories: we give a
structure theory for such fibrations. The existence of lax comma objects is
necessary for the proof of cut elimination, and we give conditions on the
domain fibration of a partial cartesian category for such comma objects to
exist
A 2-Categorical Analysis of the Tripos-to-Topos Construction
We characterize the tripos-to-topos construction of Hyland, Johnstone and
Pitts as a biadjunction in a bicategory enriched category of equipment-like
structures. These abstract concepts are necessary to handle the presence of
oplax constructs --- the construction is only oplax functorial on certain
classes of cartesian functors between triposes. A by-product of our analysis is
the decomposition of the tripos-to-topos construction into two steps, the
intermediate step being a weakened version of quasitoposes
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