Indexed Induction and Coinduction, Fibrationally

This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a dual to the sound induction rule for inductive types that we developed previously. That is, we present a sound coinduction rule for any data type arising as the carrier of the final coalgebra of a functor, thus relaxing Hermida and Jacobs' restriction to polynomial functors. To achieve this we introduce the notion of a quotient category with equality (QCE) that i) abstracts the standard notion of a fibration of relations constructed from a given fibration; and ii) plays a role in the theory of coinduction dual to that played by a comprehension category with unit (CCU) in the theory of induction. Secondly, we show that inductive and coinductive indexed types also admit sound induction and coinduction rules. Indexed data types often arise as carriers of initial algebras and final coalgebras of functors on slice categories, so we give sufficient conditions under which we can construct, from a CCU (QCE) U:E \rightarrow B, a fibration with base B/I that models indexing by I and is also a CCU (resp., QCE). We finish the paper by considering the more general case of sound induction and coinduction rules for indexed data types when the indexing is itself given by a fibration

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S

Indexed induction and coinduction, fibrationally.

This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobs’ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)

Characterizing Van Kampen Squares via Descent Data

Categories in which cocones satisfy certain exactness conditions w.r.t. pullbacks are subject to current research activities in theoretical computer science. Usually, exactness is expressed in terms of properties of the pullback functor associated with the cocone. Even in the case of non-exactness, researchers in model semantics and rewriting theory inquire an elementary characterization of the image of this functor. In this paper we will investigate this question in the special case where the cocone is a cospan, i.e. part of a Van Kampen square. The use of Descent Data as the dominant categorical tool yields two main results: A simple condition which characterizes the reachable part of the above mentioned functor in terms of liftings of involved equivalence relations and (as a consequence) a necessary and sufficient condition for a pushout to be a Van Kampen square formulated in a purely algebraic manner.Comment: In Proceedings ACCAT 2012, arXiv:1208.430

Triangulated categories of mixed motives

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.Comment: This is the final version. To appear in the series Springer Monographs in Mathematic

Generic Fibrational Induction

This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs' elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, a sound induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of a particular syntactic form. We establish the soundness of our generic induction rule by reducing induction to iteration. We then show how our generic induction rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The first of these lies outside the scope of Hermida and Jacobs' work because it is not polynomial, and as far as we are aware, no induction rules have been known to exist for the second and third in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set.Comment: For Special Issue from CSL 201