Related papers: Balanced Category Theory II

In this paper, we study compactness and finiteness of an $\infty$-category $\mathcal{C}$ equipped with a conservative functor to a finite poset $P$. We provide sufficient conditions for $\mathcal{C}$ to be compact in terms of strata and…

If $\mathcal P$ is a family of filters over some set $I$, a topological space $X$ is \emph{sequencewise $\mathcal P$-\brfrt compact} if, for every $I$-indexed sequence of elements of $X$, there is $F \in \mathcal P$ such that the sequence…

We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are…

Products in double categories, as found in cartesian double categories, are an elegant concept with numerous applications, yet also have a few puzzling aspects. In this paper, we revisit double-categorical products from an unbiased…

We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less…

We study two classes of morphisms in infinite type: tamely presented morphisms and morphisms with coherent pullback. These are generalizations of finitely presented morphisms and morphisms of finite Tor-dimension, respectively. The class of…

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…

We establish a formal correspondence between resource calculi an appropriate linear multicategories. We consider the cases of (symmetric) representable, symmetric closed and autonomous multicategories. For all these structures, we prove…

We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…

In this paper, we unify various approaches to generalized covering space theory by introducing a categorical framework in which coverings are defined purely in terms of unique lifting properties. For each category $\mathcal{C}$ of…

Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full…

We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible…

For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen…

We focus on two factorization systems for opfibrations in the 2-category Fib(B) of fibrations over a fixed base category B. The first one is the internal version of the so called comprehensive factorization, where the right orthogonal class…

For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in…

Let $(\mathscr{X}$, $\mathscr{Y})$ be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to $(\mathscr{X}$, $\mathscr{Y})$, and then give some equivalent characterizations when a relative…

We construct an iterative method for factorising small strict n-categories into a unique (up to isomorphism) collection of small 1- categories. Following this we develop the theory to include a large class of $\infty$-categories. We use…

We show that factorization systems, both strict and orthogonal, can be equivalently described as double categories satisfying certain properties. This provides conceptual reasons for why the category of sets and partial maps or the category…

We show that a compact rigid balanced braided monoidal category with enough compact projective objects gives rise to a system of mapping class group representations compatible with the gluing along marked intervals. A motivation to consider…