From Saunders Mac Lane: Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor’ from Carnap …, and “natural transformation” from the current informal parlance. Related: […]
Haskell uses a lot of ideas from category theory, but the correspondence between Haskell and category theory can be a little hard to see at times. One difficulty is that although Haskell articles use terms like functor and monad from category theory, they seldom actually talk about categories per se. If we’ve got functors, where […]
Joseph Goguen gave seven dogmas in his paper A Categorical Manifesto. To each species of mathematical structure, there corresponds a category whose objects have that structure, and whose morphisms preserve it. To any natural construction on structures of one species, yielding structures of another species, there corresponds a functor from the category of the first […]
When I took my first abstract algebra course, we had a homework question about subgroups. Someone in the class whined that the professor hadn’t told us yet what a subgroup was. My immediate thought was “I bet you could guess. Sub things are all the same. A sub-X is an X contained inside another X. […]
Category theory has a high ratio of definitions to theorems, and it can seem like every definition depends on an infinite regress of definitions. But if you graph the definition dependencies, the graph is wide rather than deep. I picked out 65 concepts and created a visualization using GraphViz. No definition is more than six […]
The definition of a subgroup is obvious, but the definition of a normal subgroup is subtle. Widgets and subwidgets The general pattern of widgets and subwidgets is that a widget is a set with some kind of structure, and a subwidget is a subset that has the same structure. This applies to vector spaces and […]
Probability density functions are independent of physical units. The normal distribution, for example, works just as well when describing weights or times. But sticking in units anyway is useful. Normal distribution example Suppose you’re trying to remember the probability density function for the normal distribution. Is the correct form or or or maybe some other […]
Test functions are how you can make sense of functions that aren’t really functions. The canonical example is the Dirac delta “function” that is infinite at the origin, zero everywhere else, and integrates to 1. That description is contradictory: a function that is 0 almost everywhere integrates to 0, even if you work in extended […]
This article will probably only be of interest to a small number of readers. Those unfamiliar with category theory may find it bewildering, and those well versed in category theory may find it trivial. My hope is that someone in between, someone just starting to get a handle on category theory, will find it helpful. […]
The goal of this post is to unpack a remark in [1]: … we can say this in fancier terms. Fix a field k …. We say that an elliptic curve E defined over k is that functor which … Well that is fancy. But what does it mean? Looking for objects A functor is […]
