Terminology in category theory

A lot of the terminology of category theory has obvious antecedents in analysis: limits, completeness, adjunctions, continuous (functors), to name but a few. However, analysis and category theory seem to be at opposite poles of the spectrum.

Is there anything deep here, or is it a case of "it has wings, so let's call it a duck"?

This was partly inspired by the top-rated answer to the question What is a metric space? and by the (slightly unsatisfactory answers to the) question Can adjoint linear transformations be naturally realized as adjoint functors?. In particular, in the first case - the categorical view of metric spaces - there does seem to be an obvious route between the two worlds, do the terminologies correspond there?

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Names in category theory are often born when someone realizes that a concept in one particular topic can be generalized in a categorical way. The generally-defined concept is then named after the original narrowly-defined one.

The case of metric spaces provides a slightly notorious example. As discussed in that other question, metric spaces can be viewed as an example of enriched categories. So, given any concept in metric space theory, you can try to generalize it to the context of enriched categories. This happened with the property of completeness of metric spaces, which one might call Cauchy-completeness since it's about Cauchy sequences. This concept turns out to generalize very smoothly to enriched categories, and to be a useful and important property there.

Many people call the property "Cauchy-completeness" in the general context of enriched categories too. But a significant minority disagree with this choice, feeling that it's stretching the terminology too far. For example, when applied to ordinary (Set-enriched) categories, the property merely says that every idempotent morphism in the category splits. This doesn't "feel" like the completeness condition on metric spaces. So there are other names in currency too, such as "Karoubi complete" (especially popular in the French school).

It's true that many pieces of categorical terminology do come from analysis, but maybe all that says is that analysis is an old and venerable subject. Exact is another example. It's used to mean several slightly different things in category theory, confusingly, but the most common usage is that a functor is "left exact" if it preserves finite limits. Now that comes from homological algebra, where one talks about exact sequences; a functor between abelian categories preserves left exact sequences iff it preserves finite limits. And that in turn, I believe, comes from the terminology of differential equations.

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I was about to ask for a reference for the "Cauchy completeness" when I found one: ncatlab.org/nlab/show/Cauchy+complete+category but I wouldn't have known what to search for without this post. Thanks! – Loop Space Nov 23 '09 at 14:41
One point of connection between the exactness terminology in homological algebra and in diff equations is the recognition of the generalized Stokes theorem to really be about coclasses in de Rham cohomology - with the theorem only applying to exact closed forms. Thus cohomology is about finding the ways that exactness of closed forms can break down - which gives an origin for the homological use of exactness. – Mikael Vejdemo-Johansson Nov 23 '09 at 16:17
Sorry, Andrew, not sure what happned, but I am sure that on a previous question of your along these lines, I dropped a comment that this is all discussed on the nLab. But now I can't find that comment of mine anymore, either. In any case: we have clear evidence that some kind sould should merge material in the nLab entries on metric spaces and on Cauchy completion to something more easily findable. If you are getting interested in that now, you'd be the ideal person for that! :-) – Urs Schreiber Nov 23 '09 at 22:38

As for the limit example, it is rooted in analysis by way of topology.

A topology on some set X is a system of subsets, partially ordered by inclusion, that have arbitrary joins and finite meets (or possibly the other way around depending on whether you're looking for open or closed sets), with X and 0 members of the system.

This partial order gets us a category structure, consisting of open (closed?) sets. Limits of these turn out to be interpretable as limits in the classical sense; and colimits are the dual concept as usual.

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