Related papers: Functors induced by comma categories
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the \emph{comma} categories and were introduced…
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…
The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of…
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…
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,…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
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…
In this paper, we consider categories with colored morphisms and functors such that morphisms assigned to morphisms with a common color have a common color. In this paper, we construct a morphism-colored functor such that any…
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…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…
The concept of category from mathematics happens to be useful to computer programmers in many ways. Unfortunately, all "good" explanations of categories so far have been designed by mathematicians, or at least theoreticians with a strong…
Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks, and…
We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S,…
We present a doctrinal approach to category theory, obtained by abstracting from the indexed inclusions (via discrete fibrations and opfibrations) of the left and of the right actions of X in Cat in categories over X. Namely, a "weak…
Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category $\mathsf{Lens}_F$ for any category $\mathcal{C}$ and functor $F\colon…
The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we…
We lay out an infinity categorical interpretation of reconstruction theorems which are germane to the symmetric monoidal perspective of noncommutative algebraic geometry, present sufficient conditions which allow for the factorization of…
There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely…