66 research outputs found
Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion
Motivated by the recent interest in models of guarded (co-)recursion we study
its equational properties. We formulate axioms for guarded fixpoint operators
generalizing the axioms of iteration theories of Bloom and Esik. Models of
these axioms include both standard (e.g., cpo-based) models of iteration
theories and models of guarded recursion such as complete metric spaces or the
topos of trees studied by Birkedal et al. We show that the standard result on
the satisfaction of all Conway axioms by a unique dagger operation generalizes
to the guarded setting. We also introduce the notion of guarded trace operator
on a category, and we prove that guarded trace and guarded fixpoint operators
are in one-to-one correspondence. Our results are intended as first steps
leading to the description of classifying theories for guarded recursion and
hence completeness results involving our axioms of guarded fixpoint operators
in future work.Comment: In Proceedings FICS 2013, arXiv:1308.589
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
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
Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
We describe categorical models of a circuit-based (quantum) functional
programming language. We show that enriched categories play a crucial role.
Following earlier work on QWire by Paykin et al., we consider both a simple
first-order linear language for circuits, and a more powerful host language,
such that the circuit language is embedded inside the host language. Our
categorical semantics for the host language is standard, and involves cartesian
closed categories and monads. We interpret the circuit language not in an
ordinary category, but in a category that is enriched in the host category. We
show that this structure is also related to linear/non-linear models. As an
extended example, we recall an earlier result that the category of W*-algebras
is dcpo-enriched, and we use this model to extend the circuit language with
some recursive types
A Note on "Extensional PERs"
In the paper "Extensional PERs" by P. Freyd, P. Mulry, G. Rosolini and D.
Scott, a category of "pointed complete extensional PERs" and
computable maps is introduced to provide an instance of an \emph{algebraically
compact category} relative to a restricted class of functors. Algebraic
compactness is a synthetic condition on a category which ensures solutions of
recursive equations involving endofunctors of the category. We extend that
result to include all internal functors on when is
viewed as a full internal category of the effective topos. This is done using
two general results: one about internal functors in general, and one about
internal functors in the effective topos.Comment: 6 page
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