Related papers: The category {\bf Rel}({\bf Nom})
Morphisms between (formal) contexts are certain pairs of maps, one between objects and one between attributes of the contexts in question. We study several classes of such morphisms and the connections between them. Among other things, we…
For every functor $\mathcal{F} : \mathcal{K} \to \mathbf{C}$, where $\mathcal{K}$ is a small category and $\mathbf{C}$ is a model category which satisfies some mild hypotheses, we define a model category $\mathbf{C}^m$ of…
Let $F$ be the category with the set of objects $\bf N$ and morphisms being the functions between the standard finite sets of the corresponding cardinalities. Let $Jf:F\rightarrow Sets$ be the obvious functor from this category to the…
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…
A cocycle category H(X,Y) is defined for objects X and Y in a model category, and it is shown that the set of morphisms [X,Y] is isomorphic to the set of path components of H(X,Y) provided the ambient model category is right proper and…
For a nice-enough category $\mathcal{C}$, we construct both the morphism category ${\rm H}(\mathcal{C})$ of $\mathcal{C}$ and the category ${\rm mod}\mbox{-}\mathcal{C}$ of all finitely presented contravariant additive functors over…
Locales have been studied as "topologies without points", mainly by tools of category theory. While traditional topology presents a space as a set of points with specified neighborhoods, localic topology presents a space as a lattice of…
We show that a functor category whose domain is a colored category is a topos.The topos structure enables us to introduce cohomology of colored categories including quasi-schemoids. If the given colored category arises from an association…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are…
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.
In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based…
Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements &…
Garret Birkhoff observed that any binary relation between two sets determines a Galois connection between the powersets, or equivalently, closure operators on the powersets, or equivalently, complete lattices of subsets that are dually…
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
For a category $\mathbb{C}$, a small category $\mathbb{I}$, and a pre-cover relation $\sqsubset$ on $\mathbb C$ we prove, under certain completeness assumptions on $\mathbb C$, that a morphism $g: B\to C$ in the functor category $\mathbb…
Relative realizability toposes satisfy a universal property that involves regular functors to other categories. We use this universal property to define what relative realizability categories are, when based on other categories than of the…
Since categories are graphs with additional "structure", one should start from fuzzy graphs in order to define a theory of fuzzy categories. Thus is makes sense to introduce categories whose morphisms are associated with a plausibility…
An algebra is said to be \emph{$\tau$-tilting finite} provided it has only a finite number of $\tau$-rigid objects up to isomorphism. We associate a category to each such algebra. The objects are the wide subcategories of its category of…