In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In par...

This paper analyses the arguments of a group of philosophers – Davidson [7, 6], Kim [9], Bennett [1] and Parsons [15] – who have worked on the semantics of actions. They argue that inference patterns involving adverbs require a semantics, formalised in first-order logic, in which actions are first-class individuals. In particular, equalities between actions must be meaningful. (We should note, ...

We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonads on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way, thus...

The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative ...

The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is...

We introduce the relational ontology log, or relational olog, a knowledge representation system based on the category of sets and relations. It is inspired by Spivak and Kent’s olog, a recent categorical framework for knowledge representation. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. In this paper, we investigat...

The main aim of the dissertation is to present a way for de ning bicategories, double categories, lax and colax functors between them, where the coherence pentagon and the lax comparison cells are implicitly encoded in the system. Besides this, we also study related structures such as Morita contexts. For bicategories we follow Tom Leinster's unbiased approach, and we use profunctors regarded a...