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Calculus is a key component in analysis that has taken on many forms and generalisations. In the following table, we begin with the common notions of the optimisation of functions in one variable, to the calculus of variations and Morse theory. This can be extended to the multivariable case with the input pluralised.

Name Objective Input Output
Calculus Optimisation of functions Real number Real number
Calculus of variations Optimisation of functionals Real function Real number
Topological calculus of variations
(cf. Morse theory)		 Optimisation of higher-order
functionals (cf. manifolds)		 Real functional Real number
... ... ... ...

Is there a continuation to this hierarchy (for example in category theory), and to what extent can these generalised theories be applicable?

TheSimpliFire
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    More as curiosities than notions that fit into your scheme, see my answers to Calculus and Category theory and Is there a such thing as an operator of operators in mathematics? – Dave L. Renfro May 23 '21 at 10:54
  • I wonder why Morse theory is not under "calculus" or "calculus of variations"? – Arctic Char May 23 '21 at 18:25
  • @ArcticChar It is a generalisation of the ideas developed there. I wanted to highlight the increasing layers that each generalisation brings. – TheSimpliFire May 23 '21 at 19:55
  • Optimization almost always involves a real-valued function on a topological space. Usually, there are assumptions on the space to make it possible to study at least one of the following: existence, uniqueness, or properties of extreme. Morse theory extends this to the study of critical points and inferring properties of the domain. The range of the function is usually the reals, since an ordering is needed to define the concept of an extremum. However, the study of critical points can be done for other spaces as the range. – Deane May 24 '21 at 00:07

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