For a commutative quantale $mathcal{V}$, the category $mathcal{V}-cat$ can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor $T$ (formalised as an endofunctor on sets) can be extended in
a canonical way to a type constructor $T_{mathcal{V}}$ on $mathcal{V}-cat$. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones. Conceptually, this allows us to to solve the same recursive domain equation $Xcong TX$ in different categories (such as sets and metric spaces) and we study how their solutions (that is, the final coalgebras) are related via change of base. Mathematically, the heart of the matter is to show that, for any commutative quantale $mathcal{V}$, the `discrete functor $D:mathsf{Set}to mathcal{V}-cat$ from sets to categories enriched over $mathcal{V}$ is $mathcal{V}-cat$-dense and has a density presentation that allows us to compute left-Kan extensions along $D$.
Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)dashv (R,L)$ is a linearly distributive adjunction
, if and only if $Ldashv U$ is a Hopf adjunction and $Udashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $Ldashv U$, to have as right adjoint a twisted version of the left adjoint $L$. In particular, the resulting adjunction will be (left) coHopf. One step further, we prove that if $L$ is precomonadic and $Lmathbf I$ is a Frobenius monoid (where $mathbf I$ denotes the unit object of the monoidal category), then $Ldashv Udashv L$ is an ambidextrous adjunction, and $L$ is a Frobenius monoidal functor. We transfer these results to Hopf monads: we show that under suitable exactness assumptions, a Hopf monad $T$ on a monoidal category has a right adjoint which is also a Hopf comonad, if the object $Tmathbf I$ is dualizable as a free $T$-algebra. In particular, if $Tmathbf I$ is a Frobenius monoid in the monoidal category of $T$-algebras and $T$ is of descent type, then $T$ is a Frobenius monad and a Frobenius monoidal functor.
Frobenius monoidal functors preserve duals. We show that conversely, (co)monoidal functors between autonomous categories which preserve duals are Frobenius monoidal. We apply this result to linearly distributive functors between autonomous categories.
Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and a
re built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunns result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on functors on the categories $mathsf{Set}$ of sets and $mathsf{Pos}$ of posets: Every functor $mathsf{Set} to mathsf{Pos}$ has a $mathsf{Pos}$-enriched left Kan extension $mathsf{Pos} to mathsf{Pos}$. Functors arising in this way are said to have a presentation in discrete arities. In the case that $mathsf{Set} to mathsf{Pos}$ is actually $mathsf{Set}$-valued, we call the corresponding left Kan extension $mathsf{Pos} to mathsf{Pos}$ its posetification. A $mathsf{Set}$-functor preserves weak pullbacks if and only if its posetification preserves exact squares. A $mathsf{Pos}$-functor with a presentation in discrete arities preserves surjections. The inclusion $mathsf{Set} to mathsf{Pos}$ is dense. A functor $mathsf{Pos} to mathsf{Pos}$ has a presentation in discrete arities if and only if it preserves coinserters of `truncated nerves of posets. A functor $mathsf{Pos} to mathsf{Pos}$ is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets. A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves $mathsf{Pos}$-enriched sifted colimits.
We extend Barrs well-known characterization of the final coalgebra of a $Set$-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a $Set$-monad $mathbf{M}$ for functors arising as liftings. As an appli
cation we introduce the notion of commuting pair of endofunctors with respect to the monad $mathbf{M}$ and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.
The notion of crossed product by a coquasi-bialgebra H is introduced and studied. The resulting crossed product is an algebra in the monoidal category of right H-comodules. We give an interpretation of the crossed product as an action of a monoidal c
ategory. In particular, necessary and sufficient conditions for two crossed products to be equivalent are provided. Then, two structure theorems for coquasi Hopf modules are given. First, these are relative Hopf modules over the crossed product. Second, the category of coquasi-Hopf modules is trivial, namely equivalent to the category of modules over the starting associative algebra. In connection the crossed product, we recall the notion of a cleft extension over a coquasi-Hopf algebra. A Morita context of Hom spaces is constructed in order to explain these extensions, which are shown to be equivalent with crossed product with invertible cocycle. At the end, we give a complete description of all cleft extensions by the non-trivial coquasi-Hopf algebras of dimension two and three.
The notions of Galois and cleft extensions are generalized for coquasi-Hopf algebras. It is shown that such an extension over a coquasi-Hopf algebra is cleft if and only if it is Galois and has the normal basis property. A Schneider type theorem is p
roven for coquasi-Hopf algebras with bijective antipode. As an application, we generalize Schauenburgs bialgebroid construction for coquasi-Hopf algebras.
If H is a finite dimensional quasi-Hopf algebra and A is a left H-module algebra, we prove that there is a Morita context connecting the smash product A#H and the subalgebra of invariants A^{H}. We define also Galois extensions and prove the connection with this Morita context, as in the Hopf case.
