# What's a good introduction to category theory for someone doing analysis?

I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory.

Any recommendations?

• Is there something in particular that you need to know? In terms of general education, it may help to know just the very basics: definition of category, functor, and natural transformation, examples of limits and colimits in your chosen area, examples of universal properties. Almost any introductory text (e.g., Mac Lane) would give you most of that. – Todd Trimble Aug 16 '15 at 20:19
• Probably something that has examples from analysis would be more motivating. – Andrej Bauer Aug 16 '15 at 20:26
• I was thinking that one universal property that ought to be familiar to every analyst is that of completion: the Cauchy completion $\bar{X}$ of a metric space $X$ is characterized by the fact that given any uniformly continuous map $f: X \to Y$ into a complete metric space $Y$, there exists a unique uniformly continuous extension $\bar{X} \to Y$ of $f$, and this characterizes $\bar{X}$ up to isomorphism. Of course this applies more generally in the context of uniform spaces, such as TVS. – Todd Trimble Aug 16 '15 at 20:32
• I think Helemskii's book "Lectures and Exercises on Functional Analysis" contains a very nice intro into category theory. It may be a bit light on the algebraic side, but it certainly does contain a lot of analytic examples and motivations, and once you get the ball rolling and have some favourite examples in mind you can study any classical text on category theory at your leisure. – Anton Fetisov Aug 16 '15 at 22:13
• @AntonFetisov Please consider making your comment an answer. To the OP: perhaps you have seen LF-spaces described as colimits of Frechet spaces, and also you may have seen the category of Banach spaces (and weak linear retractions) described as a symmetric monoidal closed category. Also you may be aware that some of the theory of nuclear spaces, developed by Grothendieck, is usefully formulated in the language of category theory. This may provide some additional motivation. You can find some of this described at the nLab. – Todd Trimble Aug 16 '15 at 23:00