Adjoint Lifting Theorems for Categories of Algebras

Johnstone, Peter

doi:10.1112/blms/7.3.294

Cited by 93 publications

(56 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It is a standard result that in presence of a distributive law like n: (-)* o M ^ M o (-)* the composite M o (-)* is again a monad, see for instance [6,19,4]. Moreover, the multiset monad M can be lifted to a monad M on the category of (-) * -algebra (monoids), such that the algebras of the composite monad M o (-)* are the same as M-algebras.…”

Section: The Semiring Monadmentioning

confidence: 99%

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Jacobs

2006

Algebra, Meaning, and Computation

View full text Add to dashboard Cite

To Joseph Goguen on the occasion of his 65th birthday1.Abstract. This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten's description of the Brzozowski automaton structure in a coalgebraic framework. We enlarge the framework to a so-called bialgebraic one, by including algebras together with suitable distributive laws connecting the algebraic and coalgebraic structure of regular expressions and languages. This culminates in a reformulated proof via finality of Kozen's completeness result. It yields a complete axioma tisation of observational equivalence (bisimilarity) on regular expressions. We suggest that this situation is paradigmatic for (theoretical) computer science as the study of "generated behaviour".

show abstract

Section: The Semiring Monadmentioning

confidence: 99%

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Jacobs

2006

Algebra, Meaning, and Computation

View full text Add to dashboard Cite

show abstract

“…Proof It is a standard result due to Johnstone [11] that a mixed distributive law AC → C A determines a functor C :…”

Section: Be a Morphism Of Monads On K Assume That A And T Preserve Cmentioning

confidence: 99%

Pre-torsors and Galois Comodules Over Mixed Distributive Laws

Böhm

Menini

2009

Appl Categor Struct

View full text Add to dashboard Cite

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street's bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions (N A , R A ) and (N B , R B ) on one hand, and the category of regular comonad arrows (R A , ξ) from some equalizer preserving comonad C to N B R B on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras. Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad D and a coregular comonad arrow from D to N A R A , such that the comodule categories of C and D are equivalent.

show abstract

“…The following "folklore" result gives an alternative description of distributive laws in terms of liftings to Kleisli and Eilenberg-Moore categories, see also [16] or [1]. Proposition 1 ("laws and liftings").…”

Section: Liftings To Kleisli and Eilenberg-moore Categoriesmentioning

confidence: 99%

Trace semantics via determinization

Jacobs

Silva

Sokolova

2015

Journal of Computer and System Sciences

123

View full text Add to dashboard Cite

Abstract. This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category. This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). More recently, it has been noticed that trace semantics can also arise by first performing a determinization construction. In this paper, we develop a systematic approach, in which the two approaches correspond to different orders of composing a functor and a monad, and accordingly, to different distributive laws. The relevant final coalgebra that gives rise to trace semantics does not live in a Kleisli category, but more generally, in a category of Eilenberg-Moore algebras. In order to exploit its finality, we identify an extension operation, that changes the state space of a coalgebra into a free algebra, which abstractly captures determinization of automata. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.

show abstract

Adjoint Lifting Theorems for Categories of Algebras

Cited by 93 publications

References 0 publications

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Pre-torsors and Galois Comodules Over Mixed Distributive Laws

Trace semantics via determinization

Contact Info

Product

Resources

About