Related papers: Kan injectivity in order-enriched categories
Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence…
We introduce the notion of Kan injectivity in 2-categories and study its properties. For an adequate 2-category $\mathcal{K}$, we show that every set of morphisms $\mathcal{H}$ induces a KZ-pseudomonad on $\mathcal{K}$ whose 2-category of…
We use Kan injectivity to axiomatise concepts in the 2-category of topoi. We showcase the expressivity of this language through many examples, and we establish some aspects of the formal theory of Kan extension in this 2-category (pointwise…
We give an elementary construction of the exact completion of a weakly lex category for categories enriched in the cartesian closed category $\mathsf{Pos}$ of partially ordered sets. Paralleling the ordinary case, we characterize categories…
Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes…
This paper contains results from two areas -- formal theory of Kan extensions and concrete categories. The contribution to the former topic is based on the extension of the concept of Kan extension to the cones and we prove that limiting…
We present a framework for characterizing injectivity of classes of maps (on cosets of a linear subspace) by injectivity of classes of matrices. Using our formalism, we characterize injectivity of several classes of maps, including…
We study Kan extensions in three weakenings of the Eilenberg-Moore double category associated to a double monad, that was introduced by Grandis and Par\'e. To be precise, given a normal oplax double monad $T$ on a double category $\mathcal…
We construct a symmetric monoidal category $LIE^{MC}$ whose objects are shifted L-infinity algebras equipped with a complete descending filtration. Morphisms of this category are "enhanced" infinity morphisms between shifted L-infinity…
We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T-algebra structure…
Finitary monads on $\mathsf{Pos}$ are characterized as the precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analagously, finitary enriched monads on…
Categories can be identified -- up to isomorphism -- with polynomial comonads on Set. The left Kan extension of a functor along itself is always a comonad -- called the density comonad -- so it defines a category when its carrier is…
Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces…
Given a monad $T$ on $\mathscr{A}$ and a functor $G \colon \mathscr{A} \to \mathscr{B}$, one can construct a monad $G_\#T$ on $\mathscr{B}$ subject to the existence of a certain Kan extension; this is the pushforward of $T$ along $G$. We…
This is a little investigation into the classification of complexes of direct sums of line bundles on projective spaces. We consider complexes on projective k-space Pk : O_Pk(-1)^a --> O_Pk^b --> O_Pk(1)^c, with the first map injective and…
We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system),…
The category $Ban$ of Banach spaces and linear maps of norm $\leq 1$ is locally $\aleph_1$-presentable but not locally finitely presentable. We prove, however, that $Ban$ is locally finitely presentable in the enriched sense over complete…
In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective $T_0$ spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true…
We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors…
Quantitative algebras are algebras enriched in the category $\mathsf{Met}$ of metric spaces so that all operations are nonexpanding. Mardare, Plotkin and Panangaden introduced varieties (aka $1$-basic varieties) as classes of quantitative…