5 research outputs found
A correspondence between rooted planar maps and normal planar lambda terms
A rooted planar map is a connected graph embedded in the 2-sphere, with one
edge marked and assigned an orientation. A term of the pure lambda calculus is
said to be linear if every variable is used exactly once, normal if it contains
no beta-redexes, and planar if it is linear and the use of variables moreover
follows a deterministic stack discipline. We begin by showing that the sequence
counting normal planar lambda terms by a natural notion of size coincides with
the sequence (originally computed by Tutte) counting rooted planar maps by
number of edges. Next, we explain how to apply the machinery of string diagrams
to derive a graphical language for normal planar lambda terms, extracted from
the semantics of linear lambda calculus in symmetric monoidal closed categories
equipped with a linear reflexive object or a linear reflexive pair. Finally,
our main result is a size-preserving bijection between rooted planar maps and
normal planar lambda terms, which we establish by explaining how Tutte
decomposition of rooted planar maps (into vertex maps, maps with an isthmic
root, and maps with a non-isthmic root) may be naturally replayed in linear
lambda calculus, as certain surgeries on the string diagrams of normal planar
lambda terms.Comment: Corrected title field in metadat
Bifibrations of polycategories and classical multiplicative linear logic
In this thesis, we develop the theory of bifibrations of polycategories.
We start by studying how to express certain categorical structures as
universal properties by generalising the shape of morphism. We call this
phenomenon representability and look at different variations, namely the
correspondence between representable multicategories and monoidal categories,
birepresentable polycategories and -autonomous categories, and
representable virtual double categories and double categories.
We then move to introduce (bi)fibrations for these structures. We show that
it generalises representability in the sense that these structures are
(bi)representable when they are (bi)fibred over the terminal one. We show how
to use this theory to lift models of logic to more refined ones. In particular,
we illustrate it by lifting the compact closed structure of the category of
finite dimensional vector spaces and linear maps to the (non-compact)
-autonomous structure of the category of finite dimensional Banach spaces
and contractive maps by passing to their respective polycategories. We also
give an operational reading of this example, where polylinear maps correspond
to operations between systems that can act on their inputs and whose outputs
can be measured/probed and where norms correspond to properties of the systems
that are preserved by the operations.
Finally, we recall the B\'enabou-Grothendieck correspondence linking
fibrations to indexed categories. We show how the B-G construction can be
defined as a pullback of virtual double categories and we make use of
fibrational properties of vdcs to get properties of this pullback. Then we
provide a polycategorical version of the B-G correspondence.Comment: 250 pages, 15 figures, PhD thesis in the Theory Group at the Computer
Science School of the University of Birmingham under the supervision of Noam
Zeilberger and Paul Lev
Bifibrations of polycategories and classical multiplicative linear logic
In this thesis, we develop the theory of bifibrations of polycategories.
We start by studying how to express certain categorical structures as universal properties by generalising the shape of morphism. We call this phenomenon representability and look at different variations, namely the correspondence between representable multicategories and monoidal categories, birepresentable polycategories and *-autonomous categories, and representable virtual double categories and double categories.
We then move to introduce (bi)fibrations for these structures. We show that it generalises representability in the sense that these structures are (bi)representable when they are (bi)fibred over the terminal one. We show how to use this theory to lift models of logic to more refined ones. In particular, we illustrate it by lifting the compact closed structure of the category of finite dimensional vector spaces and linear maps to the (non-compact) *-autonomous structure of the category of finite dimensional Banach spaces and contractive maps by passing to their respective polycategories. We also give an operational reading of this example, where polylinear maps correspond to operations between systems that can act on their inputs and whose outputs can be measured/probed and where norms correspond to properties of the systems that are preserved by the operations.
Finally, we recall the BĂ©nabou-Grothendieck correspondence linking fibrations to indexed categories. We show how the B-G construction can be defined as a pullback of virtual double categories and we make use of fibrational properties of vdcs to get properties of this pullback. Then we provide a polycategorical version of the B-G correspondence