范畴论

Let $(1)$ be an automorphism on an additive category $\mathcal{B}$, and let $\eta\colon (1)\to {\rm Id}_{\mathcal{B}}$ be a natural transformation satisfying $\eta_{X(1)}=\eta_X(1)$ for any object $X$ in $\mathcal{B}$. We construct a new…

We show that a trivial case of Janelidze's categorical Galois theorem can be used as a key step in the proof of Joyal and Tierney's result on the representation of Grothendieck toposes as localic groupoids. We also show that this trivial…

We establish the universal properties of the bicategory of polynomials, considering both cartesian and general morphisms between these polynomials. A direct proof of these universal properties would be impractical due to the complicated…

The scientific and practical needs of the twenty-first century lead humankind to convergence of the specialized and diverse branches of science and technology. This convergence reveals the need for new mathematical theories capable of…

Prompted by an example related to the tensor algebra, we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable. We test this notion on several functors traditionally connected to the…

Beck's distributive laws provide sufficient conditions under which two monads can be composed, and monads arising from distributive laws have many desirable theoretical properties. Unfortunately, finding and verifying distributive laws, or…

This is an account of some work of Markus Rost and his students Dominik Boos and Susanne Maurer. We adapt it to the braided monoidal setting.

The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation…

For each positive integer $n$ we introduce the notion of $n$-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. We characterize which $n$-exangulated categories are $n$-exact in the…

Given a family of model categories $\cal E \to \cal C$, we associate to it a homotopical category of derived, or Segal, sections $DSect(\cal C,\cal E)$ that models the higher-categorical sections of the localisation $L\cal E \to \cal C$.…

The Fa\`a di Bruno construction, introduced by Cockett and Seely, constructs a comonad $\mathsf{Fa{\grave{a}}}$ whose coalgebras are precisely Cartesian differential categories. In other words, for a Cartesian left additive category…

We use geometric ideas coming from certain classic algebraic constructions to associate, to every classical field theory, a symmetric monoidal double functor from the double category of cobordisms with corners to a certain symmetric…

Let $(\mathcal C,\otimes,\mathbb 1)$ be an abelian symmetric monoidal category satisfying certain exactness conditions. In this paper we define a presheaf $\mathbb P^{n}_{\mathcal C}$ on the category of commutative algebras in $\mathcal C$…

Presentations for unbraided, braided and symmetric pseudomonoids are defined. Biequivalences characterising the semistrict bicategories generated by these presentations are proven. It is shown that these biequivalences categorify results in…

Tate objects have been studied by many authors. They allow us to deal with infinite dimensional spaces by identifying some more structure. In this article, we set up the theory of Tate objects in stable $(\infty,1)$-categories, while the…

Let $R$ be an associative ring with unit. Given an $R$-module $M$, we can associate the following covariant functor from the category of $R$-algebras to the category of abelian groups: $S\mapsto M\otimes_R S$. With the corresponding notion…

Lindenhovius has studied Grothendieck topologies on posets and has given a complete classification in the case that the poset is Artinian. We extend his approach to more general posets, by translating known results in locale and domain…

Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize…

We construct an operad $\mathrm{Phyl}$ whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of $\mathrm{Com}$, the operad for commutative semigroups, and $[0,\infty)$, the operad with unary…

Passive linear networks are used in a wide variety of engineering applications, but the best studied are electrical circuits made of resistors, inductors and capacitors. We describe a category where a morphism is a circuit of this sort with…