12 research outputs found
Shapely monads and analytic functors
In this paper, we give precise mathematical form to the idea of a structure
whose data and axioms are faithfully represented by a graphical calculus; some
prominent examples are operads, polycategories, properads, and PROPs. Building
on the established presentation of such structures as algebras for monads on
presheaf categories, we describe a characteristic property of the associated
monads---the shapeliness of the title---which says that "any two operations of
the same shape agree". An important part of this work is the study of analytic
functors between presheaf categories, which are a common generalisation of
Joyal's analytic endofunctors on sets and of the parametric right adjoint
functors on presheaf categories introduced by Diers and studied by
Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among
the analytic endofunctors, and may be characterised as the submonads of a
universal analytic monad with "exactly one operation of each shape". In fact,
shapeliness also gives a way to define the data and axioms of a structure
directly from its graphical calculus, by generating a free shapely monad on the
basic operations of the calculus. In this paper we do this for some of the
examples listed above; in future work, we intend to do so for graphical calculi
such as Milner's bigraphs, Lafont's interaction nets, or Girard's
multiplicative proof nets, thereby obtaining canonical notions of denotational
model
Polynomial functors and polynomial monads
We study polynomial functors over locally cartesian closed categories. After
setting up the basic theory, we show how polynomial functors assemble into a
double category, in fact a framed bicategory. We show that the free monad on a
polynomial endofunctor is polynomial. The relationship with operads and other
related notions is explored.Comment: 41 pages, latex, 2 ps figures generated at runtime by the texdraw
package (does not compile with pdflatex). v2: removed assumptions on sums,
added short discussion of generalisation, and more details on tensorial
strength
Polynomial functors and polynomial monads
We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored
Stabilized profunctors and stable species of structures
We introduce a bicategorical model of linear logic which is a novel variation
of the bicategory of groupoids, profunctors, and natural transformations. Our
model is obtained by endowing groupoids with additional structure, called a
kit, to stabilize the profunctors by controlling the freeness of the groupoid
action on profunctor elements.
The theory of generalized species of structures, based on profunctors, is
refined to a new theory of \emph{stable species} of structures between
groupoids with Boolean kits. Generalized species are in correspondence with
analytic functors between presheaf categories; in our refined model, stable
species are shown to be in correspondence with restrictions of analytic
functors, which we characterize as being stable, to full subcategories of
stabilized presheaves. Our motivating example is the class of finitary
polynomial functors between categories of indexed sets, also known as normal
functors, that arises from kits enforcing free actions.
We show that the bicategory of groupoids with Boolean kits, stable species,
and natural transformations is cartesian closed. This makes essential use of
the logical structure of Boolean kits and explains the well-known failure of
cartesian closure for the bicategory of finitary polynomial functors between
categories of set-indexed families and cartesian natural transformations. The
paper additionally develops the model of classical linear logic underlying the
cartesian closed structure and clarifies the connection to stable domain
theory.Comment: FSCD 2022 special issue of Logical Methods in Computer Science, minor
changes (incorporated reviewers comments
Combinatorial Species and Labelled Structures
The theory of combinatorial species was developed in
the 1980s as part of the mathematical subfield of enumerative
combinatorics, unifying and putting on a firmer theoretical basis a
collection of techniques centered around generating
functions. The theory of algebraic data
types was developed, around the same time, in functional
programming languages such as Hope and Miranda, and is still used
today in languages such as Haskell, the ML family, and Scala. Despite
their disparate origins, the two theories have striking
similarities. In particular, both constitute algebraic frameworks in
which to construct structures of interest. Though the similarity has
not gone unnoticed, a link between combinatorial species and algebraic
data types has never been systematically explored. This dissertation
lays the theoretical groundwork for a precise—and, hopefully,
useful—bridge bewteen the two theories. One of the key
contributions is to port the theory of species from a classical,
untyped set theory to a constructive type theory. This porting process
is nontrivial, and involves fundamental issues related to equality and
finiteness; the recently developed homotopy type
theory is put to good use formalizing these issues in a
satisfactory way. In conjunction with this port, species as general
functor categories are considered, systematically analyzing the
categorical properties necessary to define each standard species
operation. Another key contribution is to clarify the role of species
as labelled shapes, not containing any data, and to
use the theory of analytic functors to model labelled
data structures, which have both labelled shapes and data associated
to the labels. Finally, some novel species variants are considered,
which may prove to be of use in explicitly modelling the memory layout
used to store labelled data structures
Stabilized profunctors and stable species of structures
We introduce a bicategorical model of linear logic which is a novel variation
of the bicategory of groupoids, profunctors, and natural transformations. Our
model is obtained by endowing groupoids with additional structure, called a
kit, to stabilize the profunctors by controlling the freeness of the groupoid
action on profunctor elements. The theory of generalized species of structures,
based on profunctors, is refined to a new theory of \emph{stable species} of
structures between groupoids with Boolean kits. Generalized species are in
correspondence with analytic functors between presheaf categories; in our
refined model, stable species are shown to be in correspondence with
restrictions of analytic functors, which we characterize as being stable, to
full subcategories of stabilized presheaves. Our motivating example is the
class of finitary polynomial functors between categories of indexed sets, also
known as normal functors, that arises from kits enforcing free actions. We show
that the bicategory of groupoids with Boolean kits, stable species, and natural
transformations is cartesian closed. This makes essential use of the logical
structure of Boolean kits and explains the well-known failure of cartesian
closure for the bicategory of finitary polynomial functors between categories
of set-indexed families and cartesian natural transformations. The paper
additionally develops the model of classical linear logic underlying the
cartesian closed structure and clarifies the connection to stable domain
theory
On the complexities of polymorphic stream equation systems, isomorphism of finitary inductive types, and higher homotopies in univalent universes
This thesis is composed of three separate parts.
The first part deals with definability and productivity issues of equational systems defining polymorphic stream functions. The main result consists of showing such systems composed of only unary stream functions complete with respect to specifying computable unary polymorphic stream functions.
The second part deals with syntactic and semantic notions of isomorphism of finitary inductive types and associated decidability issues. We show isomorphism of so-called guarded types decidable in the set and syntactic model, verifying that the answers coincide.
The third part deals with homotopy levels of hierarchical univalent universes in homotopy type theory, showing that the n-th universe of n-types has truncation level strictly n+1
A categorical framework for congruence of applicative bisimilarity in higher-order languages
Applicative bisimilarity is a coinductive characterisation of observational
equivalence in call-by-name lambda-calculus, introduced by Abramsky (1990).
Howe (1996) gave a direct proof that it is a congruence, and generalised the
result to all languages complying with a suitable format. We propose a
categorical framework for specifying operational semantics, in which we prove
that (an abstract analogue of) applicative bisimilarity is automatically a
congruence. Example instances include standard applicative bisimilarity in
call-by-name, call-by-value, and call-by-name non-deterministic
-calculus, and more generally all languages complying with a variant
of Howe's format
On the complexities of polymorphic stream equation systems, isomorphism of finitary inductive types, and higher homotopies in univalent universes
This thesis is composed of three separate parts.
The first part deals with definability and productivity issues of equational systems defining polymorphic stream functions. The main result consists of showing such systems composed of only unary stream functions complete with respect to specifying computable unary polymorphic stream functions.
The second part deals with syntactic and semantic notions of isomorphism of finitary inductive types and associated decidability issues. We show isomorphism of so-called guarded types decidable in the set and syntactic model, verifying that the answers coincide.
The third part deals with homotopy levels of hierarchical univalent universes in homotopy type theory, showing that the n-th universe of n-types has truncation level strictly n+1