范畴论

This is the first part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -…

We study liftings of abelian model structures to categories of chain complexes and construct a realization functor from the derived category of a Grothendieck abelian category equipped with a cofibrantly generated, hereditary abelian model…

In this paper we introduce the notion of weak non-asssociative Doi-Hopf module and give the Fundamental Theorem of Hopf modules in this setting. Also we prove that there exists a categorical equivalence that admits as particular instances…

We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to…

We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general…

For a complete and cocomplete category $\mathcal{C}$ with a well-behaved class of `projectives' $\bar{\mathcal{P}}$, we construct a model structure on the category $s\mathcal{C}$ of simplicial objects in $\mathcal{C}$ where the weak…

We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the…

The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category.…

We build a symmetric monoidal and compact closed bicategory by combining spans and cospans inside a topos. This can be used as a framework in which to study open networks and diagrammatic languages. We illustrate this framework with Coecke…

A popular graphical calculus for monoidal categories makes computations tactile and intuitive. Complicated diagram chases can be expressed in a few pictures and discovered by playing with a shoelace. Joyal and Street's proof of the…

Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to…

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…

We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous $\infty$-category. The heart of the proof is the description of the…

We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; this is, moreover, implicit in the…

We outline the theory of reflections for prederivators, derivators and stable derivators. In order to parallel the classical theory valid for categories, we outline how reflections can be equivalently described as categories of fractions,…

We define a simple kind of higher inductive type generalising dependent $W$-types, which we refer to as $W$-types with reductions. Just as dependent $W$-types can be characterised as initial algebras of certain endofunctors (referred to as…

We develop a model structure on presheaves of small simplicially enriched categories on a site $\mathscr{C}$, for which the weak equivalences are 'stalkwise' weak equivalences for the Bergner model structure. This model structure is right…

Many interesting classes of maps from homotopical algebra can be characterised as those maps with the right lifting property against certain sets of maps (such classes are sometimes referred to as cofibrantly generated). In a more…

Although the notion of crossed groups was originally introduced only in the simplicial case, the definition makes sense in the other categories. For instance, Batanin and Markl studied crossed interval groups to investigate symmetries on…

Projectivity and injectivity are fundamental notions in category theory. We consider natural weakenings termed semiprojectivity and semiinjectivity, and study these concepts in different categories. For example, in the category of metric…