范畴论

The definition of a pseudo-dualizing complex is obtained from that of a dualizing complex by dropping the injective dimension condition, while retaining the finite generatedness and homothety isomorphism conditions. In the specific setting…

Tangent categories offer a categorical context for differential geometry, by categorifying geometric notions like the tangent bundle functor, vector fields, Euclidean spaces, vector bundles, connections, etc. In the last decade, the theory…

We show that for an extensive $1$-category $\mathcal{E}$ with pullbacks and pullback stable coequalisers in which the forgetful functor $\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E})$ has left adjoint, the…

In this article we provide a model-independent definition of the concept of lax $2$-functors from $(\infty,2)$-category theory and show that it agrees with the existing and widely used combinatorial model for those in terms of…

Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the…

We construct a category equivalent to the category $\mathbf{Mon}$ of monoids and monoid homomorphisms, based on categories with strict factorization systems. This equivalence is then extended to the category $\mathbf{Mon_s}$ of unital…

We introduce Para and coPara double categories for double categories. They rely on a horizontal action $\crta\ot$ of a horizontally monoidal double category $\Mm$ on a double category $\Dd$. We prove a series of properties, most…

We exhibit a symmetric promonoidal kernel on the simplex category $\Delta$ with Cartesian unit, yielding on representable functors a Hadamard natural transformation $\Delta^p\times\Delta^q\to\Delta^{pq}$ based on pointwise multiplication of…

A comma category, exemplified in algebraic geometry by coherent systems, combines two categories over a third through morphisms between their objects. We establish sufficient conditions for it to be abelian, compute its Grothendieck group,…

We give an internal description of constructible objects in an $\infty$-topos. More precisely, $P$-consctructible objects are locally constant objects internal to Fun($P$,An), for any noetherian poset $P$.

(Pseudo) double categories have two sorts of morphisms: tight ones which compose strictly, and loose ones which compose up to coherent isomorphism. In this paper, we consider bimodules between double categories in the loose direction. We…

We construct Grothendieck topologies on the path category of a finite graph, examining both coarse and discrete cases that offer different perspectives on quiver representations. The coarse topology declares each vertex covered by all…

We extend the theory of distributive series of monads of \cite{EC1} by extending the definition to include an $\bN$-indexed collection of monads. Under certain conditions, distributive series of monads will have a colimit in the category of…

We analyze the structure of left maps in algebraic weak factorization systems constructed using Garner's algebraic small object argument. We find that any left map can be constructed from generators in Bourke and Garner's double category of…

We examine the categorical structure of the Grothendieck construction $\Sigma_{\mathsf{C}}\mathsf{L}$ of an indexed category $\mathsf{L} \colon \mathsf{C}^{op} \to \mathsf{CAT}$. Our analysis begins with characterisations of fibred limits,…

We define a commutative monoid structure on the poset of left-exact localizations of a higher topos, that we call the acyclic product. Our approach is anchored in a structural analogy between the poset of left-exact localizations of a topos…

This paper gives a classification of classes of discrete dynamical systems (a set equipped with an endofunction) closed under finite limits and small colimits. The conclusion is simple: they bijectively correspond to the ideals of the…

Working within enriched category theory, we further develop the use of soundness, introduced by Ad\'amek, Borceux, Lack, and Rosick\'y for ordinary categories. In particular we investigate: (1) the theory of locally $\Phi$-presentable…

Let $\mathbf{C}$ be a Cauchy-complete category. The subtoposes of $[\mathbf{C}^{\mathrm{op}},\mathbf{Set}]$ are sometimes all of the form $[\mathbf{D}^{\mathrm{op}},\mathbf{Set}]$ where $\mathbf{D}$ is a full subcategory of $\mathbf{C}$.…

We introduce and study operadic categories with cardinalities in finite sets and establish conditions under which their associated theories of operads and algebras are equivalent to the standard framework introduced in 2015 by Batanin and…