Category theory has practical applications in programming language theory, for example the usage of monads in functional programming. It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.
Now we come back to Christopher Alexander's original inspiration (I believe) in mathematics. We encounter his identity preserving morphism, which is the basis for his non-destructive (beauty preserving and enhancing) transformations.
I wonder what the intersection of linguistic triples (semantic triple) will be with Alexander's patterns and pattern languages? It certainly fells like his language has a three layered semantic structure connecting patterns to one another. Of course the pattern itself has semantics and structure. How interesting to explore.
https://en.m.wikipedia.org/wiki/Category_theory
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Category Theory Wikipedia 
https://en.m.wikipedia.org/wiki/Topological_space
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Topological Space Wikipedia