Related papers: Quantaloidal Completions of Order-enriched Categor…
In this paper we carry the construction of equilogical spaces into an arbitrary category $\mathsf{X}$ topological over $\mathsf{Set}$, introducing the category $\mathsf{X}$-$\mathsf{Equ}$ of equilogical objects. Similar to what is done for…
Neeman shows that the completion of a triangulated category with respect to a good metric yields a triangulated category. We compute completions of discrete cluster categories with respect to metrics induced by internal t-structures. In…
Our work is a fundamental study of the notion of approximation in V-categories and in (U,V)-categories, for a quantale V and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below…
We define the phrase `category enriched in an fc-multicategory' and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal…
Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between…
We use the notion of universal extension in a linear abelian category to study extensions of variations of mixed Hodge structure and convergent and overconvergent isocrystals. The results we obtain apply, for example, to prove the exactness…
We define the notion of an additive model category, and we prove that any additive, stable, combinatorial model category has a natural enrichment over symmetric spectra based on simplicial abelian groups. As a consequence, every object in…
We investigate the interplay of crystal bases and completions in the sense of Enright on certain nonintegrable representations of quantum groups. We define completions of crystal bases, show that this notion of completion is compatible with…
We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere…
Complementarity was originally introduced as a qualitative concept for the discussion of properties of quantum mechanical objects that are classically incompatible. More recently, complementarity has become a \emph{quantitative} relation…
We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a new proof for…
Let $\Lambda^{\ast}$ be the free monoid of (finite) words over a not necessarily finite alphabet $\Lambda$, which is equipped with some (partial) order. This ordering lifts to $\Lambda^{\ast}$, where it extends the divisibility ordering of…
Generalizing F-nilpotent completion for a ring spectrum F we first define the notion of completion with respect to a thick subcategory in a monogenic stable homotopy category. Specializing this to the thick subcategory generated by…
We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and…
We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories…
We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy…
This rough note describes some attempts to define a notion of enriched topology (and the associated theory of enriched stacks) on a category enriched over a symmetric monoidal model category, and poses some related questions.
In this work we introduce the notion of higher $\mathbb{E}$-extension groups for an extriangulated category $\mathcal{C}$ and study the quotients $\mathcal{X}_{n+1}^{\vee}/[\mathcal{X}]$ and $\mathcal{X}_{n+1}^{\wedge}/[\mathcal{X}]$ when…
We use a generic notion of flatness in the enriched context to define various completions of metric spaces -- enrichments over [0,\infty] -- and preorders -- enrichments over 2. We characterize the weights of colimits commuting in…
In this note, we investigate the algebraic and topological representation theory of cylindric ortholattices and cylindric Boolean algebras. The first contribution demonstrates that cylindric ortholattices are closed under canonical…