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I have recently learned a bit about higher category theory. And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory?

Is it possible to get new insight into this theory using higher categories?

In particular, it seems interesting to me to reformulate banach space interpolation theorems in categorical language.

For example, how can I rewrite the reiteration theorem in terms of a functor acting in one category? Here, reiteration theorem is a theorem in banach space interpolation theory which states that there is no use of applying the interpolation functor twice.

I know my questions may seem vague or fuzzy. But anyway,I hope to hear to get an answer soon.

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Have you had a look at Bergh-Löfström or Brudnyi-Krugljak? – András Bátkai Apr 28 '14 at 7:07
Why are you diving straight into higher category theory and not 1-category theory? Try looking up work of Kaijser and Pelletier – Yemon Choi Apr 28 '14 at 11:06
To András Bátkai: Yes, I have. But their exposition is almost category-free except for the first chapter. It is believed that passing to 2-categories can enrich 1-category theory. My question was motivated the article "From finite sets to Feynman diagramm" In this paper John C. baez and James Dolan give a lot of arguments in favour of higher category theory. – Rauan Akylzhanov May 1 '14 at 3:36
"Arguments in favour of higher category theory" do not necessarily become "arguments in favour of doing everything with higher category theory", and they are certainly not "arguments for ignoring 1-categorical work done by people on a given subject" – Yemon Choi May 4 '14 at 17:20

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