731 research outputs found
Data types with symmetries and polynomial functors over groupoids
Polynomial functors are useful in the theory of data types, where they are
often called containers. They are also useful in algebra, combinatorics,
topology, and higher category theory, and in this broader perspective the
polynomial aspect is often prominent and justifies the terminology. For
example, Tambara's theorem states that the category of finite polynomial
functors is the Lawvere theory for commutative semirings. In this talk I will
explain how an upgrade of the theory from sets to groupoids is useful to deal
with data types with symmetries, and provides a common generalisation of and a
clean unifying framework for quotient containers (cf. Abbott et al.), species
and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan.
The multi-variate setting also includes relations and spans, multispans, and
stuff operators. An attractive feature of this theory is that with the correct
homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits,
etc. - the groupoid case looks exactly like the set case. After some standard
examples, I will illustrate the notion of data-types-with-symmetries with
examples from quantum field theory, where the symmetries of complicated tree
structures of graphs play a crucial role, and can be handled elegantly using
polynomial functors over groupoids. (These examples, although beyond species,
are purely combinatorial and can be appreciated without background in quantum
field theory.) Locally cartesian closed 2-categories provide semantics for
2-truncated intensional type theory. For a fullfledged type theory, locally
cartesian closed \infty-categories seem to be needed. The theory of these is
being developed by D.Gepner and the author as a setting for homotopical
species, and several of the results exposed in this talk are just truncations
of \infty-results obtained in joint work with Gepner. Details will appear
elsewhere.Comment: This is the final version of my conference paper presented at the
28th Conference on the Mathematical Foundations of Programming Semantics
(Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer
Science. 16p
Extending Set Functors to Generalised Metric Spaces
For a commutative quantale V, the category V-cat can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor T (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor TV on V-cat. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones.
Conceptually, this allows us to to solve the same recursive domain equation X ≅ TX in different categories (such as sets and metric spaces) and we study how their solutions (that is, the final coalgebras) are related via change of base.
Mathematically, the heart of the matter is to show that, for any commutative quantale V, the “discrete functor Set → V-cat from sets to categories enriched over V is V-cat-dense and has a density presentation that allows us to compute left-Kan extensions along D
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
Relation lifting, with an application to the many-valued cover modality
We introduce basic notions and results about relation liftings on categories
enriched in a commutative quantale. We derive two necessary and sufficient
conditions for a 2-functor T to admit a functorial relation lifting: one is the
existence of a distributive law of T over the "powerset monad" on categories,
one is the preservation by T of "exactness" of certain squares. Both
characterisations are generalisations of the "classical" results known for set
functors: the first characterisation generalises the existence of a
distributive law over the genuine powerset monad, the second generalises
preservation of weak pullbacks. The results presented in this paper enable us
to compute predicate liftings of endofunctors of, for example, generalised
(ultra)metric spaces. We illustrate this by studying the coalgebraic cover
modality in this setting.Comment: 48 pages, accepted for publication in LMC
Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version)
This paper studies the quantitative refinements of Abramsky's applicative
similarity and bisimilarity in the context of a generalisation of Fuzz, a
call-by-value -calculus with a linear type system that can express
programs sensitivity, enriched with algebraic operations \emph{\`a la} Plotkin
and Power. To do so a general, abstract framework for studying behavioural
relations taking values over quantales is defined according to Lawvere's
analysis of generalised metric spaces. Barr's notion of relator (or lax
extension) is then extended to quantale-valued relations adapting and extending
results from the field of monoidal topology. Abstract notions of
quantale-valued effectful applicative similarity and bisimilarity are then
defined and proved to be a compatible generalised metric (in the sense of
Lawvere) and pseudometric, respectively, under mild conditions
Computable Functions on Final Coalgebras
AbstractThis paper tackles computability issues on final coalgebras and tries to shed light on the following two questions: First, which functions on final coalgebras are computable? Second, which formal system allows us to define all computable functions on final coalgebras?In particular, we give a definition of computability on final coalgebras, deriving from the theory of effective domains. We then establish the admissibility of coinductive definitions and of a generalised μ-operator. This gives rise to a formal system, in which every term denotes a computable function
Change Actions: Models of Generalised Differentiation
Cai et al. have recently proposed change structures as a semantic framework
for incremental computation. We generalise change structures to arbitrary
cartesian categories and propose the notion of change action model as a
categorical model for (higher-order) generalised differentiation. Change action
models naturally arise from many geometric and computational settings, such as
(generalised) cartesian differential categories, group models of discrete
calculus, and Kleene algebra of regular expressions. We show how to build
canonical change action models on arbitrary cartesian categories, reminiscent
of the F\`aa di Bruno construction
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