A lot of disciplines in higher-level mathematics can be summarized by describing what objects they study and in what setting they are studied in. For example,
- Topology is the study of topological spaces and is set in the category $\rm\bf Top$, since we consider its foundational objects to be topological spaces.
- Differential geometry is the study of smooth manifolds and is set in the intersection of the categories $\rm\bf Diff$ and $\rm\bf Vect$, since we consider two of its most foundational objects to be smooth manifolds and tangent spaces.
Analysis is strange; it seems there is no easy way to describe what is the most foundational setting that analysis can be done in. Perhaps restricting my question to real analysis that is done in a typical undergraduate course would be easier. Would it be correct to say that this type of analysis is done on $\mathbb R$ equipped with a metrizable topology and an inner product space structure?
My question is probably summed up in the following: how can we categorize the most typical types of analysis and in which categories they are set in?