323 research outputs found
Sequential products in effect categories
A new categorical framework is provided for dealing with multiple arguments
in a programming language with effects, for example in a language with
imperative features. Like related frameworks (Monads, Arrows, Freyd
categories), we distinguish two kinds of functions. In addition, we also
distinguish two kinds of equations. Then, we are able to define a kind of
product, that generalizes the usual categorical product. This yields a powerful
tool for deriving many results about languages with effects
Commutativity
We describe a general framework for notions of commutativity based on
enriched category theory. We extend Eilenberg and Kelly's tensor product for
categories enriched over a symmetric monoidal base to a tensor product for
categories enriched over a normal duoidal category; using this, we re-find
notions such as the commutativity of a finitary algebraic theory or a strong
monad, the commuting tensor product of two theories, and the Boardman-Vogt
tensor product of symmetric operads.Comment: 48 pages; final journal versio
Representing Guardedness in Call-By-Value
Like the notion of computation via (strong) monads serves to classify various flavours of impurity, including exceptions, non-determinism, probability, local and global store, the notion of guardedness classifies well-behavedness of cycles in various settings. In its most general form, the guardedness discipline applies to general symmetric monoidal categories and further specializes to Cartesian and co-Cartesian categories, where it governs guarded recursion and guarded iteration respectively. Here, even more specifically, we deal with the semantics of call-by-value guarded iteration. It was shown by Levy, Power and Thielecke that call-by-value languages can be generally interpreted in Freyd categories, but in order to represent effectful function spaces, such a category must canonically arise from a strong monad. We generalize this fact by showing that representing guarded effectful function spaces calls for certain parametrized monads (in the sense of Uustalu). This provides a description of guardedness as an intrinsic categorical property of programs, complementing the existing description of guardedness as a predicate on a category
Certification of programs with computational effects
In purely functional programming languages imperative features, more
generally computational effects are prohibited. However, non-functional lan-
guages do involve effects. The theory of decorated logic provides a rigorous
for- malism (with a refinement in operation signatures) for proving program
properties with respect to computational effects. The aim of this thesis is to
first develop Coq libraries and tools for verifying program properties in
decorated settings as- sociated with several effects: states, local state,
exceptions, non-termination, etc. Then, these tools will be combined to deal
with several effects
Promonads and String Diagrams for Effectful Categories
Premonoidal and Freyd categories are both generalized by non-cartesian Freyd
categories: effectful categories. We construct string diagrams for effectful
categories in terms of the string diagrams for a monoidal category with a
freely added object. We show that effectful categories are pseudomonoids in a
monoidal bicategory of promonads with a suitable tensor product.Comment: In Proceedings ACT 2022, arXiv:2307.1551
Mackey functors on compact closed categories
We develop and extend the theory of Mackey functors as an application of
enriched category theory. We define Mackey functors on a lextensive category
\E and investigate the properties of the category of Mackey functors on \E.
We show that it is a monoidal category and the monoids are Green functors.
Mackey functors are seen as providing a setting in which mere numerical
equations occurring in the theory of groups can be given a structural
foundation. We obtain an explicit description of the objects of the Cauchy
completion of a monoidal functor and apply this to examine Morita equivalence
of Green functors
(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong (∞,n)-functors. We construct a double (∞,n)-category built out of the target (∞,n)-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of E d -algebras in a symmetric monoidal (∞,n)-category C to an (∞,n+d)-category using the higher morphisms in C
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