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Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced functors.
In this paper we prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathcal{X}$ denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in $\mathcal{X}$ are equivalent to chain complexes in $\mathcal{X}$; (b) the observation of Church-Ellenberg-Farb that $\mathcal{X}$-valued species are equi...
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 condi...
A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protom...
The Day Reflection Theorem gives conditions under which a reflective subcategory of a closed monoidal category can be equipped with a closed monoidal structure in such a way that the reflection adjunction becomes a monoidal adjunction. We adapt this result to skew monoidal categories. The beauty of this variant is further evidence that the direction choices involved in the skew notion are important for organizing, and adding depth to, certain mathematical phenomena. We also provide conditions...
Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational algebras and weighted Rota-Baxter algebras as special monoids and special semigroups, respectively, for the same monoidal structure on the category of graphs in a monoidal additive category. Weighted derivations are lifted to the categorical level.
Distributive laws between two monads in a 2-category $\CK$, as defined by Jon Beck in the case $\CK=\mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\CK$ of monads. Steve Lack and the author defined wreaths to be monads in a 2-category $\mathrm{EM}\CK$ of monads with different 2-cells from $\mathrm{Mnd}\CK$. Mixed distributive laws were also considered by Jon Beck, Mike Barr and, later, various others, they are comonads in $\mathrm{Mnd}\CK$. Actually, as...
We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.
We make a few remarks concerning pointwise extensions in a bicategory which include the case of bicategories of enriched categories. We show that extensions, pointwise or not, can be replaced by extensions along very special fully faithful maps. This leads us to suggest a concept of limit sketch internal to the bicategory.
International audience; In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the established presentation of such structures as algebras for monads on presheaf categories, we describe a characteristic property of the associated monads---the shapeliness of the title---which says that “any two operations of the sa...
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