ABSTRACTS-FMCS 2017
Jerome Fortier: Restriction Lambda-Calculus
One of the most general, and therefore nicest, families of categories
where morphisms behave like partial maps is the family of restriction
categories. There is also a notion of a cartesian closed restriction
category (CCRC), analogous to the usual notion of a CCC, but in the
partial world. This work is an attempt to state and prove the
Curry-Howard property for CCRC's, by developing the corresponding
syntax. Our solution is something like simply typed lambda-calculus,
with a restriction operator. The resulting logic turns out to be
substructural, and therefore very nice!
Keith O’Neill: A Connection between Noncommutative and Commutative
Codifferential Categories
Noncommutative geometry is characterized by a welter of examples of structures
which are impervious to any semblance of visualization. Therefore, a mathematician
embarking on the analysis of a noncommutative space is faced with a difficult
dilemma: whether to aimlessly wander in a morass of pure formality or to utilize
analogies to more tractable geometric structures. Whenever possible, we opt for the
latter. However, the fallibility of such an approach is indicated by our inability to
intuit the qualities of quantum phase space - a classic example of a noncommutative
space.
A solution to this dilemma is the formalization of the analogy, a task for which
category theory is quite apt. In this talk, we investigate one such for- malization
which utilizes the apparatus of differential categories. Differential categories and
their noncommutative variants distill the essential formal features of differential
calculus to produce a compact and flexible theory. Here, we exploit this
formalization to establish a tangible connection between dif- ferential structures in
noncommutative contexts and those in commutative contexts.
Darien Dewolf: An Element-based Reformulation of Restriction Monads
Last year at FMCS, I introduced restriction monads: monads in a
bicategory equipped with a restriction 2-cell satisfying axioms reminiscent of
those satisfied in a restriction category. This talk will give a reformulation
of restriction monads in bicategories with an initial object. An immediate
benefit of this reformulation are one-to-one correspondences between (i) small
restriction categories and restriction monads in Span(Set) and (ii) small
restriction categories and restriction monads in Set-Mat. These
correspondences form the motivation for defining internal restriction
categories and restriction enriched categories, respectively.
Tarmo Uustalo: Container combinatorics: monads and lax monoidal functors
Abbott et al.'s containers are a "syntax" for a wide class of set
functors in terms shapes and positions. Containers whose "denotation"
carries a comonad structure can be characterized as directed
containers, or containers where a shape and a position in it determine
another shape, intuitively a subshape of the given shape rooted by the
given position. In this talk, I will discuss similar explicit
characterizations for container functors with a monad structure and
container functors with a lax monoidal functor structure, as well as
some variations. We argue that this type of characterizations make a
tool, e.g., for enumerating the monad structures that some set functor
admits. Such explorations are of interest, e.g., in the semantics
of effectful functional programming languages.
Rory Lucyshyn-Wright: Optimal updating of database views through functorial
semantics and fibrations
Alongside basic database concepts such as tables, schemas, and queries, many
database systems include `virtual' tables called views, whose content is
automatically derived from the database by means of a fixed query. Views are
typically read-only, since the problem of implementing updatable views entails a
challenging theoretical problem that has been studied by many authors. Johnson
and Rosebrugh have studied this problem by means of a category-theoretic
formalism for database theory through functorial semantics. Therein, schemas are
certain limit-colimit sketches and hence are presentations of theories relative to a
doctrine, in the Beck-Lawvere sense, so that database states are precisely models of
these theories. Concretely, such theories are (structured) categories, and their
models are (structure-preserving) set-valued functors. Correspondingly, every view
defines a structure-preserving functor between theories and so induces a functor
between the categories of models. It is an insight of Johnson and Rosebrugh that a
view is updatable, in an optimal way, if and only if the latter functor is both a
fibration and an op-fibration. Spivak (2012) has also employed the paradigm of
functorial semantics for databases, in a setting where the theories are mere small
categories. In this talk, we observe that the problem of optimal view updating in
this setting admits a natural solution: Given any functor i:S --> T between small
categories S and T, the induced functor [T,Set] --> [S,Set] is a Street fibration if and
only if it is a Street op-fibration, if and only if i is fully faithful. Further, we apply
similar methods to study the extent to which related results are available for finite
limit theories as well as the algebraic databases of Schultz, Spivak, Vasilakopoulou,
and Wisnesky (2017).
Polina Vinogradova: Formalizing abstract computability: Turing Categories in Coq
The concept of a recursive function has been extensively studied using traditional
tools of computability theory. However, with the development of category-theoretic
methods it has become possible to study recursion in a more abstract sense. The
particular model this paper is structured around is known as a Turing category. The
structure within a Turing category models the notion of partiality as well as
recursive computation, and equips us with the tools of category theory to study
these concepts. The goal of this work is to build a formal language description of this
categorical model. Specifically, we use the Coq proof assistant to formulate informal
definitions, propositions and proofs pertaining to Turing categories in the
underlying formal language of Coq, the Calculus of Co-inductive Constructions (CIC).
Furthermore, we instantiate the more general Turing category formalism with a CIC
description of the category which explicitly models the language of partial recursive
functions.
Geoff Cruttwell- General connections in tangent categories
Connections are a fundamental tool of differential geometry. However, there are
various formulations of the connection notion; in particular, there are definitions of
connections on both vector bundles and on the more general fibre bundles.
In the axiomatic abstract setting of tangent categories, previous work has shown
how to define connections on differential bundles (the analog of vector bundles). In
this talk, we discuss how to define connections in tangent categories in the more
general sense (that is, on the analog of fibre bundles) and look at the example of
connections on G-principal bundles in this setting.
Jonathan Gallagher: Coherently closed tangent categories
There are two developments of type theory which describe logic for
reasoning about smooth maps. The first is synthetic differential
geometry; here one uses the type theory of a smooth topos to reason about
microlinear spaces. The second is the differential lambda calculus; here an
explict type theory was developed for reasoning in smooth models of linear
logic(Kˆthe sequence spaces, convenient vector spaces).
It turns out that these two models are intimately related, and this
talk will give a direct link. In particular, we will show that the
the differential lambda calculus is the logic in the category of
euclidean or differential bundles over an object, in a coherent,
locally cartesian closed tangent category. This then describes a
dependently typed variant of the differential lambda calculus for these
settings. For example, this means that one can use the differential lambda
calculus to reason about the Euclidean vector bundles of SDG.
Jason Parker: Isotropy Groups of Algebraic Theories
Every first-order geometric theory T has a classifying topos C, which contains a
canonical group object called its ‘isotropy group’, which we may call the isotropy
group of the theory T. In this talk I will present several new results about the
isotropy groups of equational algebraic theories. After reviewing the fact that the
isotropy group of a geometric theory is isomorphic to the automorphism group of its
universal model, I will explain how we can use this characterization to obtain an
even more concrete description of the isotropy group of an equational algebraic
theory. I will then illustrate how to compute the isotropy groups of several popular
algebraic theories, including the theories of (commutative) monoids, (abelian)
groups, and commutative rings. Finally, I’ll touch upon some current results and
future questions about how to compute the isotropy groups of algebraic theories
that have been constructed by means of operations on theories (e.g. disjoint union
and tensor product).
Dorette Pronk: What are orbifolds?
An orbifold, also called a V-manifold, was originally defined by Satake as a
paracompact Hausdorff space with an atlas of charts consisting of open subsets of
Euclidean space with an action of a finite group and a homeomorphism from the
orbit space to an open subset of the underlying space. Two atlases were considered
equivalent of they had a common refinement. The problem with this definition was
that it was far from obvious what the definition of a map between orbifolds should
be. This was partially resolved when a new representation in terms of proper etale
groupoids was introduced, with both groupoid homomorphisms and HilsumSkandalis bimodules, or equivalently, generalized maps, as maps.
Describing these maps in terms of atlases has resulted in rather technical
descriptions that are not easy to work with. For differential geometers who are
interested in the local structure of maps like embeddings and immersions, this has
proven to be very challenging. Furthermore, with the change to not necessarily
effective group actions, there is added confusion about what atlases should be.
I will introduce a new notion of atlas for orbifolds which are the objects of a pseudo
double category with horizontal maps given in terms of profunctors and vertical
maps in terms of functors. We will show that this pseudo double category is weakly
equivalent to the pseudo double category of orbigroupoids with Hilsum Skandalis
maps as horizontal maps and groupoid homomorphisms as vertical maps.
If there is time I will add some examples of notions of suborbifolds and how they
can be expressed in this formalism. This is joint work with my student, Alanod Sibih.
Priyaa Srinivasan: Categorical description of completely positive maps
In my talk, I would be presenting categorical descriptions of completely positive
maps in dagger compact closed categories and dagger symmetric monoidal
categories. Completely positive maps have applications in physics, for example, they
are used to model quantum channels. Completely positive maps between
endomorphism monoids were first categorically represented by Selinger. Later,
Coecke et. al. described completely positive maps between any normalizable
Frobenius Algebras in a dagger compact closed category. In my talk, I will present
completely positive maps between normalizable Frobenius algebras in dagger
symmetric monoidal categories. The material I present is adapted from the work of
Vicary and Heunen. In my talk, I will also present CP construction on dagger
symmetric monoidal categories.
Cole Comfort: The Category CNOT
We exhibit a complete set of identities by CNOT, the symmetric monoidal category
generated by the controlled-not gate, the swap gate, and the computational ancillae.
We prove that CNOT is a discrete inverse category. Moreover, we prove that CNOT is
equivalent to the category of partial isomorphisms of finitely-generated non-empty
commutative torsors of characteristic 2. This is equivalently the category of affine
partial isomorphisms between $\mathbb{Z}_2$ vector spaces.
Francisco Rios- A categorical model for a quantum circuit description language
Quipper is a practical programming language for describing families of quantum
circuits. In this talk, we formalize a small, but useful fragment of Quipper called
Proto-Quipper-M. Unlike its parent Quipper, this language is type-safe and has a
formal denotational and operational semantics. Proto-Quipper-M is also more
general than Quipper, in that it can describe families of morphisms in any symmetric
monoidal category, of which quantum circuits are but one example. We design
Proto-Quipper-M from the ground up, by first giving a general categorical model of
parameters and states. After finding some interesting categorical structures in the
model, we then define the programming language to fit the model. We cement the
connection between the language and the model by proving type safety, soundness,
and adequacy properties. This is joint work with Peter Selinger.
Zamdzhiev-A DCPO-enriched linear/non-linear model
Rios and Selinger have recently proposed a categorical model for the quantum
programming lan- guage Proto-Quipper-M, which is an important fragment of the
Quipper language. In this work, we describe an extension to their categorical
model with the additional property that it is DCPO- enriched, bringing us closer to
modeling general recursion in the language. Similar to their model, our model
exhibits a symmetric monoidal adjunction between a cartesian closed category and
a sym- metric monoidal closed category that previously has been shown to be a
sound categorical model for a mixed linear/non-linear (and not necessarily
quantum) programming language by Benton. Our model is built upon a
generalisation of the well-known families construction on categories, which
retains some crucial properties from the theory of fibrations which allows us to
prove our model is complete and cocomplete.
Prashant Kumar: Introduction to MPL Programming Language
MPL(Message Passing Language) is a concurrent, functional and strictly typed
language with message passing as the concurrency primitive. Concurrent MPL
programs comprise of processes with channels connecting them. Processes
communicate by passing messages along the channels. MPL brings the convenience
of type safety to the concurrent world by typing the channels. This is achieved by
associating every channel with a protocol/coprotocol which deteremines
permissible actions on a channel. Protocol/Coprotocol can be considered as the data
types of the councurrent world. MPL also has a sequential side which resembles a
strictly typed functional programming language like Haskell along with the
additional facilities of defining and using codata types and writing disciplined
recursive programs using folds and unfolds.
In the talk, the basic constructs used to develop councurrent programs in MPL will
be described with examples that will be run on the MPL’s compiler.
Leila Leganeh: Pullback complements in partial morphism categories
In my thesis I investigate pullback complements in partial morphism categories. I
characterize the existence of pullback comple- ments in a category in terms of the
existence of exponentials in an associated slice category. The existence of pullback
complements in partial morphism categories can be characterized in terms of the
ex- istence of pullback complements in the base category. Finally, for an admissible
match I compare partial pullback complement rewriting and sesqui-pushout,
double- pushout and single pushout rewritings.
Ben Macadam: Storage tangent categories
In classical differential geometry, a vector bundle associates each point of a
manifold to a vector space. This structure was first explored in tangent categories as
differential bundles by Robin Cockett and Geoff Cruttwell. In this talk, we consider
the linear-logical structure of the category of vector bundles and how this relates to
classical constructions in differential geometry, such as differential forms and their
exterior algebra.
Bob Rosebrugh- Universal updates for symmetric lenses
A "symmetric lens" between two model domains has state synchronization data and
resynchronization operations while an "asymmetric lens" provides a strategy to lift
updates in a model domain along a morphism of model domains. When the model
domains are categories we call them delta-lenses (or d-lenses). We showed that
(certain equivalence classes of) spans of asymmetric d-lenses represent symmetric
d-lenses. The c-lenses are a special case of asymmetric d-lenses whose update lifting
satisfies a universal property that makes them "least change". If we define
(equivalence classes of) spans of c-lenses to be symmetric c-lenses, then we
naturally hope that they characterize the symmetric d-lenses satisfying a "least
change" universal property. However, we'll explain why we do not expect this.
Instead we consider cospans of c-lenses and show that they do generate symmetric
d-lenses with a universal property. We also explore how to characterize those
symmetric d-lenses that arise from cospans of c-lenses.