Comprehension and Quotient Structures in the Language of 2-Categories

Lawvere observed in his celebrated work on hyperdoctrines that the set-theoretic schema of comprehension can be elegantly expressed in the functorial language of categorical logic, as a comprehension structure on the functor p:E? B defining the hyperdoctrine. In this paper, we formulate and study a strictly ordered hierarchy of three notions of comprehension structure on a given functor p:E? B, which we call (i) comprehension structure, (ii) comprehension structure with section, and (iii) comprehension structure with image. Our approach is 2-categorical and we thus formulate the three levels of comprehension structure on a general morphism p:??? in a 2-category K. This conceptual point of view on comprehension structures enables us to revisit the work by Fumex, Ghani and Johann on the duality between comprehension structures and quotient structures on a given functor p:E?B. In particular, we show how to lift the comprehension and quotient structures on a functor p:E? B to the categories of algebras or coalgebras associated to functors F_E:E?E and F_B:B?B of interest, in order to interpret reasoning by induction and coinduction in the traditional language of categorical logic, formulated in an appropriate 2-categorical way

Quotient completion for the foundation of constructive mathematics

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.Comment: 32 page

What can co-speech gestures in aphasia tell us about the relationship between language and gesture?: A single case study of a participant with Conduction Aphasia

Cross-linguistic evidence suggests that language typology influences how people gesture when using ‘manner-of-motion’ verbs (Kita 2000; Kita & Özyürek 2003) and that this is due to ‘online’ lexical and syntactic choices made at the time of speaking (Kita, Özyürek, Allen, Brown, Furman & Ishizuka, 2007). This paper attempts to relate these findings to the co-speech iconic gesture used by an English speaker with conduction aphasia (LT) and five controls describing a Sylvester and Tweety1 cartoon. LT produced co-speech gesture which showed distinct patterns which we relate to different aspects of her language impairment, and the lexical and syntactic choices she made during her narrative

Towards a constructive simplicial model of Univalent Foundations

We provide a partial solution to the problem of defining a constructive version of Voevodsky's simplicial model of univalent foundations. For this, we prove constructive counterparts of the necessary results of simplicial homotopy theory, building on the constructive version of the Kan-Quillen model structure established by the second-named author. In particular, we show that dependent products along fibrations with cofibrant domains preserve fibrations, establish the weak equivalence extension property for weak equivalences between fibrations with cofibrant domain and define a univalent classifying fibration for small fibrations between bifibrant objects. These results allow us to define a comprehension category supporting identity types, $\Sigma$-types, $\Pi$-types and a univalent universe, leaving only a coherence question to be addressed.Comment: v3: changed the definition of the type Weq(U) of weak equivalences to fix a problem with constructivity. Other Minor changes. 31 page

Boolean Coverings of Quantum Observable Structure: A Setting for an Abstract Differential Geometric Mechanism

We develop the idea of employing localization systems of Boolean coverings, associated with measurement situations, in order to comprehend structures of Quantum Observables. In this manner, Boolean domain observables constitute structure sheaves of coordinatization coefficients in the attempt to probe the Quantum world. Interpretational aspects of the proposed scheme are discussed with respect to a functorial formulation of information exchange, as well as, quantum logical considerations. Finally, the sheaf theoretical construction suggests an opearationally intuitive method to develop differential geometric concepts in the quantum regime.Comment: 25 pages, Late

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