35 research outputs found
Rigidity is undecidable
We show that the problem `whether a finite set of regular-linear axioms
defines a rigid theory' is undecidable.Comment: 8 page
The Euler characteristic of a category
The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Rota's Möbius inversion from posets to categories
Strictification of weakly stable type-theoretic structures using generic contexts
We present a new strictification method for type-theoretic structures that
are only weakly stable under substitution. Given weakly stable structures over
some model of type theory, we construct equivalent strictly stable structures
by evaluating the weakly stable structures at generic contexts. These generic
contexts are specified using the categorical notion of familial
representability. This generalizes the local universes method of Lumsdaine and
Warren.
We show that generic contexts can also be constructed in any category with
families which is freely generated by collections of types and terms, without
any definitional equality. This relies on the fact that they support
first-order unification. These free models can only be equipped with weak
type-theoretic structures, whose computation rules are given by typal
equalities. Our main result is that any model of type theory with weakly stable
weak type-theoretic structures admits an equivalent model with strictly stable
weak type-theoretic structures
Innocent strategies as presheaves and interactive equivalences for CCS
Seeking a general framework for reasoning about and comparing programming
languages, we derive a new view of Milner's CCS. We construct a category E of
plays, and a subcategory V of views. We argue that presheaves on V adequately
represent innocent strategies, in the sense of game semantics. We then equip
innocent strategies with a simple notion of interaction. This results in an
interpretation of CCS.
Based on this, we propose a notion of interactive equivalence for innocent
strategies, which is close in spirit to Beffara's interpretation of testing
equivalences in concurrency theory. In this framework we prove that the
analogues of fair and must testing equivalences coincide, while they differ in
the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014
Theories of analytic monads
We characterize the equational theories and Lawvere theories that correspond
to the categories of analytic and polynomial monads on Set, and hence also the
categories of the symmetric and rigid operads in Set. We show that the category
of analytic monads is equivalent to the category of regular-linear theories.
The category of polynomial monads is equivalent to the category of rigid
theories, i.e. regular-linear theories satisfying an additional global
condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere
theories corresponding to these monads are identified via some factorization
systems.Comment: 29 pages. v2: minor correction