相关论文: Relevant Categories and Partial Functions
Categories of partial functions have become increasingly important principally because of their applications in theoretical computer science. In this note we prove that the category of partial bijections between sets as an…
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over…
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 symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This…
We show that the category of categories with pullbacks and pullback preserving functors is cartesian closed.
We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent…
For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional…
We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic.…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…
Differential categories provide an axiomatization of the basics of differentiation and categorical models of differential linear logic. As differentiation is an important tool throughout quantum mechanics and quantum information, it makes…
The present paper gives a generalization of cartesian closed categories, called cartesian closed categories with dependence, whose strict version induces categories with families that support 1-, Sigma- and Pi-types in the strict sense.…
A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to…
We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give…
Regular logic is the fragment of first order logic generated by $=$, $\top$, $\wedge$, and $\exists$. A key feature of this logic is that it is the minimal fragment required to express composition of binary relations; another is that it is…
In this paper, we define a generalization of indexed categories and contextual categories which we call contextually indexed (contextual) categories. While contextual categories are models of ordinary type theories, contextually indexed…
Various concerns suggest looking for internal co-categories in categories with strong logical structure. It turns out that in any coherent category, all co-categories are co-equivalence relations.
A new categorical setting is defined in order to characterize the subrecursive classes belonging to complexity hierarchies. This is achieved by means of coercion functors over a symmetric monoidal category endowed with certain recursion…
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…
A double category of relations is essentially a cartesian equipment with strong, discrete and functorial tabulators and for which certain local products satisfy a Frobenius Law. A double category of relations is equivalent to a double…
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here we define a related structure called a monoidal reverse…