8 research outputs found
Inversion, Iteration, and the Art of Dual Wielding
The humble ("dagger") is used to denote two different operations in
category theory: Taking the adjoint of a morphism (in dagger categories) and
finding the least fixed point of a functional (in categories enriched in
domains). While these two operations are usually considered separately from one
another, the emergence of reversible notions of computation shows the need to
consider how the two ought to interact. In the present paper, we wield both of
these daggers at once and consider dagger categories enriched in domains. We
develop a notion of a monotone dagger structure as a dagger structure that is
well behaved with respect to the enrichment, and show that such a structure
leads to pleasant inversion properties of the fixed points that arise as a
result. Notably, such a structure guarantees the existence of fixed point
adjoints, which we show are intimately related to the conjugates arising from a
canonical involutive monoidal structure in the enrichment. Finally, we relate
the results to applications in the design and semantics of reversible
programming languages.Comment: Accepted for RC 201
Axioms for the category of sets and relations
We provide axioms for the dagger category of sets and relations that recall
recent axioms for the dagger category of Hilbert spaces and bounded operators.Comment: 14 pages; corrected proof of Corollary 1.
Traced Monads and Hopf Monads
A traced monad is a monad on a traced symmetric monoidal category that lifts
the traced symmetric monoidal structure to its Eilenberg-Moore category. A
long-standing question has been trying to provide a characterization of traced
monads without explicitly mentioning the Eilenberg-Moore category. On the other
hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are
invertible. For compact closed categories, symmetric Hopf monads are precisely
the kind of monads that lift the compact closed structure to their
Eilenberg-Moore categories. Since compact closed categories and traced
symmetric monoidal categories are closely related, it is a natural question to
ask what is the relationship between Hopf monads and traced monads. In this
paper, we introduce trace-coherent Hopf monads on traced monoidal categories,
which can be characterised without mentioning the Eilenberg-Moore category. The
main theorem of this paper is that a symmetric Hopf monad is a traced monad if
and only if it is a trace-coherent Hopf monad. We provide many examples of
trace-coherent Hopf monads, such as those induced by cocommutative Hopf
algebras or any symmetric Hopf monad on a compact closed category. We also
explain how for traced Cartesian monoidal categories, trace-coherent Hopf
monads can be expressed using the Conway operator, while for traced coCartesian
monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We
also provide separating examples of traced monads that are not Hopf monads, as
well as symmetric Hopf monads that are not trace-coherent.Comment: Restructured the paper based on anonymous reviewer's suggestion
New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic
Intuitionistic logic, in which the double negation law not-not-P = P fails,
is dominant in categorical logic, notably in topos theory. This paper follows a
different direction in which double negation does hold. The algebraic notions
of effect algebra/module that emerged in theoretical physics form the
cornerstone. It is shown that under mild conditions on a category, its maps of
the form X -> 1+1 carry such effect module structure, and can be used as
predicates. Predicates are identified in many different situations, and capture
for instance ordinary subsets, fuzzy predicates in a probabilistic setting,
idempotents in a ring, and effects (positive elements below the unit) in a
C*-algebra or Hilbert space. In quantum foundations the duality between states
and effects plays an important role. It appears here in the form of an
adjunction, where we use maps 1 -> X as states. For such a state s and a
predicate p, the validity probability s |= p is defined, as an abstract Born
rule. It captures many forms of (Boolean or probabilistic) validity known from
the literature. Measurement from quantum mechanics is formalised categorically
in terms of `instruments', using L\"uders rule in the quantum case. These
instruments are special maps associated with predicates (more generally, with
tests), which perform the act of measurement and may have a side-effect that
disturbs the system under observation. This abstract description of
side-effects is one of the main achievements of the current approach. It is
shown that in the special case of C*-algebras, side-effect appear exclusively
in the non-commutative case. Also, these instruments are used for test
operators in a dynamic logic that can be used for reasoning about quantum
programs/protocols. The paper describes four successive assumptions, towards a
categorical axiomatisation of quantitative logic for probabilistic and quantum
systems
A Linear Exponential Comonad in s-finite Transition Kernels and Probabilistic Coherent Spaces
This paper concerns a stochastic construction of probabilistic coherent
spaces by employing novel ingredients (i) linear exponential comonads arising
properly in the measure-theory (ii) continuous orthogonality between measures
and measurable functions. A linear exponential comonad is constructed over a
symmetric monoidal category of transition kernels, relaxing Markov kernels of
Panangaden's stochastic relations into s-finite kernels. The model supports an
orthogonality in terms of an integral between measures and measurable
functions, which can be seen as a continuous extension of
Girard-Danos-Ehrhard's linear duality for probabilistic coherent spaces. The
orthogonality is formulated by a Hyland-Schalk double glueing construction,
into which our measure theoretic monoidal comonad structure is accommodated. As
an application to countable measurable spaces, a dagger compact closed category
is obtained, whose double glueing gives rise to the familiar category of
probabilistic coherent spaces.Comment: 31 page
On the Resolution Semiring
In this thesis, we study a semiring structure with a product based on theresolution rule of logic programming. This mathematical object was introducedinitially in the setting of the geometry of interaction program in order to modelthe cut-elimination procedure of linear logic. It provides us with an algebraicand abstract setting, while being presented in a syntactic and concrete way, inwhich a theoretical study of computation can be carried on.We will review first the interactive interpretation of proof theory withinthis semiring via the categorical axiomatization of the geometry of interactionapproach. This interpretation establishes a way to translate functional programsinto a very simple form of logic programs.Secondly, complexity theory problematics will be considered: while thenilpotency problem in the semiring we study is undecidable in general, it willappear that certain restrictions allow for characterizations of (deterministicand non-deterministic) logarithmic space and (deterministic) polynomial timecomputation