Analysis from the categorical viewpoint
According to the conventional wisdom , the structuralist viewpoint as embodied by category theory has little or nothing to offer to analysts. I cannot agree with that assessment.
Category theory bears on analysis in two distinct ways:
- In the large, by cataloging various categories of metric spaces, normed spaces, inner product spaces, and so on, and the relationships between them. The usage of category theory here is no different than in any other branch of mathematics.
- In the small, by identifying individual metric spaces, normed spaces, and other objects as enriched categories. This appearance of category theory is more surprising.
In the large
A few textbooks attempt to present topics in analysis using categorical ideas:
- Helemskii, 2006: Lectures and exercises on functional analysis (doi)
- Helemskii, 2010: Quantum functional analysis (doi)
- Schechter, 1997: Handbook of analysis and its foundations (doi, nLab )
- Takes a structuralist viewpoint throughout, with a Part B on algebra, focusing on linearity and convexity, and even a Chapter 9 on category theory
- Geroch, 1985: Mathematical physics
- Uses category theory as a organizing principle to survey mathematics useful for physics, mainly linear algebra, topology, and functional analysis
In the small
Metrics
Lawvere metric spaces are natural enough that they crop up in applications of metric spaces with no categorical pretenses, usually under different names (extended metrics, pseudometrics, quasimetrics, etc).
- Isbell, 1964: Six theorems about injective metric spaces (doi, eudml )
- Influential early study on injective metric spaces
- Often cited as the first paper to study the category of (classical) metric spaces and contractions
- Categorical language is not explicitly used, though it is briefly in: Isbell, 1972: \(s\) admits an injective metric (doi)
- Revisited in: Willerton, 2014: Tight spans, Isbell completions and semi-tropical modules (pdf)
- Lawvere, 1973: Metric spaces, generalized logic and closed categories (pdf)
- The classic paper that started it all
- Very readable; also a good general introduction to enriched categories
- Pursued further (“taken seriously”) starting in Sec. 6 of: Lawvere, 1986: Taking categories seriously (pdf)
- n-Cat Cafe post: Enriching over a category of subsets
- Rutten, 1998: Weighted colimits and formal balls in generalized metric spaces
(doi, tech report )
- Shows that weighted colimits in Lawvere metric spaces are limits of Cauchy sequences
- Cited in n-Cat Cafe post: Enrichment and its limits
- Bosangue, Breugel, Rutten, 1998: Generalized metric spaces: Completion,
topology, and powerdomains via the Yoneda embedding (doi, tech report )
- Sec. 4: The Yoneda lemma for metric spaces
- Avery & Leinster, 2021: Isbell conjugacy and the reflexive completion (arxiv)
- About Isbell conjugacy in general but includes examples from Lawvere metric spaces (Example 3.6)
- Adamek & Rosicky, 2022: Approximate injectivity and smallness in
metric-enriched categories (doi, arxiv)
- Studies categories enriched in the category of metric spaces and nonexpansive maps
Norms and weights
Weighted categories, called normed categories by Lawvere, are categories enriched in weighted sets.