Universal Arrows and Universal Elements (Pt. I)

Abstract Nonsense

Point of Post: In this post we discuss the notion of universal arrows and universal elements, this is in preparation for our discussion of particular universal problems such as limits and colimits.

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Motivation

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Category theory has the ability to bring together disparate ideas in mathematics under one blanket theory. Category theory accomplishes this in two ways. Firstly, it provides a universal vocabulary which allows us to have a coherent discussion. Secondly, it provides a way of thinking, a way of formulating questions that brings forth the truly important characteristics of an object, separating it from the confounding chaff. In this post we describe a concept that epitomizes both of these categorical methodologies–universal characterizations.  The idea behind universal characterizations are simple: in a given category what we really care about are arrows, and thus we should (when possible) define an object entirely in terms of arrows. To be…

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