Hans-E. Porst

On tree coalgebras and coalgebra presentations

For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra At. The universal property of this family of coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique At which takes the root of t to s. Consequently, the tree coalgebras are finitely presentable and form a strong generator. Thus, these categories of coalgebras are locally finitely presentable; in particular every system is a filtered colimit of finitely presentable systems.In contrast, for transition systems expressed as coalgebras over the finite-power-set functor we show that there are systems which fail to be filtered colimits of finitely presentable (=finite) ones.Surprisingly, if λ is an uncountable cardinal, then λ-presentation is always well-behaved: whenever an endofunctor F preserves λ-filtered colimits (i.e., is λ-accessible), then λ-presentable coalgebras are precisely those whose underlying objects are λ-presentable. Consequently, every F coalgebra is a λ-filtered colimit of λ-presentable coalgebras; thus Coalg F is a locally λ-presentable category. (This holds for all endofunctors of λ-accessible categories with colimits of ω-chains.) Corollary: A set functor is bounded at λ in: the sense of Kawahara and Mori iff it is λ+-accessible.

From Varieties of Algebras to Covarieties of Coalgebras

Abstract Varieties of F-algebras with respect to an endofunctor F on an arbitrary cocomplete category C are defined as equational classes admitting free algebras. They are shown to correspond precisely to the monadic categories over C . Under suitable assumptions satisfied in particular by any endofunctor on Set and Set op the Birkhoff Variety Theorem holds. By dualization, covarieties over complete categories C are introduced, which then correspond to the comonadic categories over C , and allow for a characterization in dual terms of the Birkhoff Variety Theorem. Moreover, the well known conditions of accessibilitly and boundedness for Set -functors F, sufficient for the existence of cofree F-coalgebras, are shown to be equivalent.

On Categories of Monoids, Comonoids, and Bimonoids

The categories of monoids, comonoids and bimonoids over a symmetric monoidal category C are investigated. It is shown that all of them are locally presentable provided Cs underlying category is. As a consequence numerous functors on and between these categories are shown to be part of an adjoint situation; in particular, the category of comonoids is monoidally closed.

Every topological category is convenient for Gelfand duality

In this paper we generalize our work on Gelfand dualities in cartesian closed topological categories [42] to categories which are only monoidally closed. Using heavily enriched category theory we show that under very mild conditions on the base category function algebra functor and spectral space functor exist, forming a pair of adjoint functors and establishing a duality between function algebras and spectral spaces. Using recent results in connection with semitopological functors, we show that every (E,M)-topological category is endowed with at least oneconvenient monoidal structure admitting a generalized Gelfand duality. So it turns out that there is no need for a cartesian closed structure on a topological category in order to study generalized Gelfand-Naimark dualities.

Fundamental Constructions for Coalgebras, Corings, and Comodules

We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphasis is given to the question which properties of these categories can be seen as instances of general categorical resp. algebraic results. However, we also obtain new results concerning the existence of limits and of factorizations of morphisms.

What is concrete equivalence

The notion of concrete equivalence is introduced, based on a modification of the traditional notion of concrete functor. The discussion of examples includes a direct (i.e. not referring to any monadicity theorem) proof of the fact that monadicity is stable under concrete equivalence.

Generalizations of the Sweedler Dual

As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.

The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract The category Hopf ℂ of Hopf monoids in a symmetric monoidal category ℂ, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ℂ preserve directed colimits one has the following results: (1) If, in ℂ, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ℂ and comonadic over the category of monoids in C. (2) If, in ℂ, extremal monomorphisms are stable under tensor squaring, then Hopf ℂ is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ℂ.

The algebraic theory of order

Partially ordered sets are described in terms of partial operations with equationally defined domains and equations, thus the categoryPOS of posets is represented as a one-sorted essentially algebraic category in the sense of Freyd [7] which, in this case even can be fully embedded into a non-trivial variety. This is achieved by using the relation of a poset rather than its underlying set as the carrier set of the algebraic structure. Essentially equational descriptions of somePOS-based algebraic structures are given, and an equational characterization of Galois connections is obtained.

On Underlying Functors in General and Topological Algebra.

In the present paper we compare the concepts of monadic functors and functors with lifting and preservation properties which are essential in general algebra and in topological algebra. In doing this we show that the Linton conditions are sufficient for tripleability under milder assumptions as is known till yet. Using the concept of regular categories and regular functors [4] we give a complete generalization of theSet-based situation to a large class of base categories which admit certain factorizations.