范畴论

We introduce a notion of dynamics in the setting of Segal topos, by considering the Segal category of stacks $\mathcal{X} = \text{dAff}_{\mathcal{C}}^{\, \sim, \tau}$ on a Segal category $\text{dAff}_{\mathcal{C}}=$…

In the first part of this article, we give an analysis of the free monad sequence in non-cocomplete categories, with the needed colimits explicitly parametrized. This enables us to state a more finely grained functoriality principle for…

We provide a general construction of induced $t$-structures, that generalizes standard $t$-structures for $\infty$-categories of sheaves. More precisely, given a presentable $\infty$-category $\mathcal{X}$ and a presentable stable…

Modular operads are an extension of operads. In the same way that operads, as dendroidal sets, can be considered as presheaves over the category of trees, so can modular operads be considered as presheaves over a category of graphs. This…

We demonstrate that companionships and conjunctions in double $\infty$-categories -- and more generally, in double Segal spaces -- extend to functors out of the free-living companionship and conjunction respectively. Specifically, we prove…

We show that small quasicategories embed, both simplicially and 2-categorically, into prederivators defined on arbitrary small categories, so that in some senses prederivators can serve as a model for $(\infty,1)$-categories. The result for…

Aggregation of time-series or image data over subsets of the domain is a fundamental task in data science. We show that many known aggregation operations can be interpreted as (double) functors on appropriate (double) categories. Such…

A generalization of an inverse system in a category was recently introduced, as well as that of the corresponding pro-category These so called the delay-inverse systems and delay-pro-category could potentially yield a new theory of (delay-)…

This thesis focuses on topics in 2-category theory: in particular on double categories, pseudomonads and codescent objects. In Chapter 2 we recall all the necessary notions. In Chapter 3 we show that factorization systems can be…

The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Sol\`er's theorem, a result…

We investigate how change of enriching base category via a faithful, conservative right adjoint functor interacts with enriched coverages and sheaves on a given enriched category. We prove that change of base via such a functor gives rise…

We introduce regular sequences and associated Koszul resolutions for monoids in the category of functors over an essentially small linear symmetric monoidal category. Next we define polynomials over such monoids. We compute the Hochschild…

We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our…

In this paper we introduce and study \emph{rectangular torsion theories}, i.e.\ those torsion theories $(\C,\T,\F)$ with $\C$ a pointed category, where the canonical functor $\C\to \T\times\F$ is an equivalence of categories. In particular,…

The purpose of this note is to consider in detail the construction of derived functors. The classical construction, such as in Cartan-Eilenberg or Grothendieck, is clarified, and it is shown, at the same time, that everything can be…

Through the notion of weakly sound class of weights, we recover many known dualities involving accessible categories with a chosen class of limits, as instances of a general duality theorem. These include the Gabriel-Ulmer duality for…

We present the foundational theory of condensed sets and basic condensed algebra after having introduced key concepts from category theory and homological algebra. In the later sections, we indicate the relevance of condensed mathematics to…

The article proposes a method for constructing non-standard theories based on terms from partially existing sequences of elements. The method is illustrated by the example of the theory of monoids. Predicates and terms from non-standard…

We introduce Elgot categories, a sort of distributive monoidal category with additional structure in which the partial recursive functions are representable. Moreover, we construct an initial Elgot category, the morphisms of which coincide…

We prove that the Krull-Remak-Schmidt-Azumaya unique decomposition theorem holds in idempotent complete additive categories with enough compact objects. In particular, this result applies to compactly generated triangulated categories. In…