3,198 research outputs found
Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem
The paper uses the formalism of indexed categories to recover the proof of a
standard final coalgebra theorem, thus showing existence of final coalgebras
for a special class of functors on categories with finite limits and colimits.
As an instance of this result, we build the final coalgebra for the powerclass
functor, in the context of a Heyting pretopos with a class of small maps. This
is then proved to provide a model for various non-well-founded set theories,
depending on the chosen axiomatisation for the class of small maps
Multiplicative structure in equivariant cohomology
We introduce the notion of a strongly homotopy-comultiplicative resolution of
a module coalgebra over a chain Hopf algebra, which we apply to proving a
comultiplicative enrichment of a well-known theorem of Moore concerning the
homology of quotient spaces of group actions. The importance of our enriched
version of Moore's theorem lies in its application to the construction of
useful cochain algebra models for computing multiplicative structure in
equivariant cohomology.
In the special cases of homotopy orbits of circle actions on spaces and of
group actions on simplicial sets, we obtain small, explicit cochain algebra
models that we describe in detail.Comment: 28 pages. Final version (cosmetic changes, slight reorganization), to
appear in JPA
A Kleene theorem for polynomial coalgebras
For polynomial functors G, we show how to generalize the classical notion of regular expression to G-coalgebras. We introduce a language of expressions for describing elements of the final G-coalgebra and, analogously to Kleene’s theorem, we show the correspondence between expressions and finite G-coalgebras
Extending Structures II: The Quantum Version
Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up
to an isomorphism all Hopf algebras E that factorize through A and H: that is E
is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in
E with 1_{E} \in H and the multiplication map is bijective.
The tool we use is a new product, we call it the unified product, in the
construction of which A and H are connected by three coalgebra maps: two
actions and a generalized cocycle. Both the crossed product of an Hopf algebra
acting on an algebra and the bicrossed product of two Hopf algebras are special
cases of the unified product. A Hopf algebra E factorizes through A and H if
and only if E is isomorphic to a unified product of A and H. All such Hopf
algebras E are classified up to an isomorphism that stabilizes A and H by a
Schreier type classification theorem. A coalgebra version of lazy 1-cocycles as
defined by Bichon and Kassel plays the key role in the classification theorem.Comment: 24 pages, 3 figures. Final version, to appear in Journal of Algebr
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
Varieties of Languages in a Category
Eilenberg's variety theorem, a centerpiece of algebraic automata theory,
establishes a bijective correspondence between varieties of languages and
pseudovarieties of monoids. In the present paper this result is generalized to
an abstract pair of algebraic categories: we introduce varieties of languages
in a category C, and prove that they correspond to pseudovarieties of monoids
in a closed monoidal category D, provided that C and D are dual on the level of
finite objects. By suitable choices of these categories our result uniformly
covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer,
respectively, and yields new Eilenberg-type correspondences
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
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