We provide axioms for the dagger category of sets and relations that recall recent axioms for the dagger category of Hilbert spaces and bounded operators.

Bhm and S tefan developed a general method of construction of cyclic objects from (op)algebras over distributive laws of monads.The goal of this note is to show that all the cyclic sets resulting from the twisted nerve of a group G arise from the Bhm-S tefan construction.It is a pleasure to thank BANGA-Africa for funding this project.I would also like

We characterize strongly finitary monads on categories Pos, CPO and DCPO as precisely those preserving sifted colimits.Or, equivalently, enriched finitary monads preserving reflexive coinserters.We study sifted colimits in general enriched categories.For CPO and DCPO we characterize varieties of continuous algebras as precisely the monadic categories for strongly finitary monads.

We develop a string-net construction of a modular functor whose algebraic input is a pivotal bicategory; this extends the standard construction based on a spherical fusion category.An essential ingredient in our construction is a graphical calculus for pivotal bicategories, which we express in terms of a category of colored corollas.The globalization of this calculus to oriented surfaces yields the bicategorical string-net spaces as colimits.We show that every rigid separable Frobenius functor between strictly pivotal bicategories induces linear maps between the corresponding bicategorical stringnet spaces that are compatible with the mapping class group actions and with sewing.Our results are inspired by and have applications to the description of correlators in two-dimensional conformal field theories.Contents 1 Introduction 474 2 Graphical calculus for pivotal bicategories 478 3 String-net models based on pivotal bicategories 511 4 Application: Universal correlators in RCFT 532We thank Christian Blanchet, Nils Carqueville, Lukas Mller and Lukas Woike for helpful discussions.

We consider Cauchy completeness in the double categories of toposes, topological spaces, locales, and other suplattice based settings.We also present a uniform approach to the relationship between adjoints and projectivity in double categories with applications to (not-necessarily commutative) rings, rigs, and quantales.

Following ideas of Lawvere and Linton we prove that classical varieties are precisely the exact categories with a varietal generator.This means a strong generator which is abstractly finite and regularly projective.An analogous characterization of varieties of ordered algebras is also presented.We work with order-enriched categories, and introduce the concept of subcongruence (corresponding to congruence in ordinary categories): it is a relation which is order-reflexive and transitive.Varieties of ordered algebras are precisely the categories with effective subcongruences and a subvarietal generator.This means a strong generator which is abstractly finite and subregularly projective.

The goal is to show how a 1978 paper of Richard Wood on monoidal comonads and exponentiation relates to more recent publications such as Pastro et alia [30] and Bruguires et alia [7].In the process, we mildly extend the ideas to procomonads in a magmal setting and suggest it also works for algebras for any club in the sense of Max Kelly [18,19].

The diagonal lemma asserts that if a map of bisimplicial sets is a levelwise weak equivalence in the Kan-Quillen model structure, then it induces a weak equivalence of the diagonal simplicial sets.In this paper, we observe that the standard proof of this fact works in greater generality, namely that of (elegant) Reedy categories.

In this paper we further the study of arrow algebras, simple algebraic structures inducing toposes through the tripos-to-topos construction, by defining appropriate notions of morphisms between them which correspond to morphisms of the associated triposes.Specializing to geometric inclusions, we characterize subtriposes of an arrow tripos in terms of nuclei on the underlying arrow algebra, recovering a classical locale-theoretic result.As an example of application, we lift modified realizability to the setting of arrow algebras, and we establish its functoriality.

Drazin inverses are a fundamental algebraic structure which have been extensively deployed in semigroup theory, ring theory, and matrix theory.Drazin inverses can also be defined for endomorphisms in any category.However, beyond a paper by Puystjens and Robinson from 1987, there has been almost no further development of Drazin inverses in category theory.Here we provide a survey of the theory of Drazin inverses from a categorical perspective.We introduce Drazin categories, in which every endomorphism has a Drazin inverse, and provide various examples including the category of matrices over a field, the category of finite length modules over a ring, and finite set enriched categories.We also introduce the notion of expressive rank and prove that a category with expressive rank is Drazin.Moreover, we not only study Drazin inverses in mere categories, but also in additive categories and dagger categories.In an arbitrary category, we show how a Drazin inverse corresponds to an isomorphism in the idempotent splitting, as well as explain how Drazin inverses relate to Leinster's notion of eventual image duality.In additive categories, we consider core-nilpotent decompositions, imagekernel decompositions, and Fitting decompositions.We also develop the notion of Drazin inverses for pairs of opposing maps, generalizing the usual notion of Drazin inverse for endomorphisms.As an application of this new kind of Drazin inverse, for dagger categories, we provide a novel characterization of the Moore-Penrose inverse in terms of being a Drazin inverse of the pair of a map and its adjoint.