Related papers: A Note on Categories about Rough Sets
In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in $\mathbf{Cat}$. This approach, while conceptually evident, has not been made fully explicit or systematically described in the…
We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is…
The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…
We propose a new framework for the study of homological properties for (compactly generated) triangulated categories such as regularity, finiteness of global or finitistic dimension, gorensteinness or injective generation and the relation…
This article is a sequel to hep-th/9411050, q-alg/9412017. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number $\zeta$ an abelian artinian category $\FS$. We call its objects {\em finite…
We explore functors between operator space categories, some properties of these functors, and establish relations between objects in these categories and their images under these functors, in particular regarding injectivity and injective…
This article aims to provide a novel formalization of the concept of computational irreducibility in terms of the exactness of functorial correspondence between a category of data structures and elementary computations and a corresponding…
Recurrent neural networks (RNNs) have been shown to perform better than feedforward architectures in visual object categorization tasks, especially in challenging conditions such as cluttered images. However, little is known about the exact…
The category ${\rm Rel}(\mathcal{C})$ may be formed for any category $\mathcal{C}$ with finite limits using the same objects as $\mathcal{C}$ but whose morphisms from $X$ to $Y$ are binary relations in $\mathcal{C}$, that is, subobjects of…
We introduce idris-ct, a Idris library providing verified type definitions of categorical concepts.idris-ct strives to be a bridge between academy and industry, catering both to category theorists who want to implement and try their ideas…
Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…
We introduce a model category of spaces based on the definable sets of an o-minimal expansion of a real closed field. As a model category, it resembles the category of topological spaces, but its underlying category is a coherent topos. We…
It is characterized when coarsening functors between categories of graded modules preserve injectivity of objects, and when they commute with graded covariant Hom functors.
One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of…
The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every…
Categorical semantics of type theories are often characterized as structure-preserving functors. This is because in category theory both the syntax and the domain of interpretation are uniformly treated as structured categories, so that we…
To any model category $\mathcal{M}$, we associate a modular model category, a functor of points $\mathcal{M}[-]:$ Cat $\rightarrow$ Cat, that associates to any small category $\mathcal{C}$ a functor category $\mathcal{M}[\mathcal{C}] =…
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…