Functional Analysis was one of the key origins of categorical concepts and outlook, for example that the functionals themselves should be collected into a single object (Voltera-Hadamard) leads to the Hom functor,etc. This was also one of the roads followed by students in the 1950s, for example from J. L. Kelley's "galactic" treatment of M. H. Stone's functor C. However in North America (as distinct from Europe) more recent functional analysists have accepted categorical methods only grudgingly, and hence piecemeal. On the other hand, students who are not specializing in analysis are often woefully ignorant of the basics of functional analysis that are part of what every mathematician should know. To combat that ignorance in my Advanced Graduate Algebra course I often devoted several weeks to topics from functional analysis. It is a source of examples both interesting and essential. To begin to try to answer Andrej's question, I rapidly recall some examples, and hope others will also comment: The double dual functor on Banch spaces is a protype example of a composite of adjoints becoming a monad. The EM algebras for this monoid were computed by Fred Linton, in an exercise that should be better known. It also illustrates the "descent" principle that C. Houzel cited couple of months ago (what I called semantics of structure of a given functor in my thesis) : Objects constructed by a given functor tend to have, by virtue of that, more structure than originally contemplated in its codomain , hence a lifted version of the functor comes closer to being invertible. As Peter Johnstone just recalled, if we consider commutative monoids with zero and hom them into the particular object of reals, the resulting set is "actually" a compact space whose C-algebra reveals by adjointness that the opposite of the spaces form a full subcategory of the monoids with zero. Again a good exercise, related to Kelley's "square root lemma". Students might wonder why contiuous linear operators are traditionally called "bounded" (when they are not even). For many linear spaces (roughly those where sequentiality suffices) , preserving sequential limits is equivalent to preserving boundedness of sequences (for a linear map). George Mackey started to functorize this crucial observation before categories were fully explicit. Now we can consider the category of all presheaves on the category of all countable sets, define an "underlying" functor from Banach spaces to it, and verify that it actually lands in the subtopos of sheaves for the finite-disjoint-covering topology. Indeed it not only gives abelian group objects in the latter topos, but modules over R, the Dedekind reals of the topos, and a FULL subcategory of those. The above construction has an analogue using instead Johnstone's coherent topos of sheaves on countable compact spaces. ETC Bill On Tue Mar 4 10:20 , Andrej Bauer sent:
This semester I am teaching rudimentary category theory at graduate level. It is somewhat scary that I should be doing this, but other faculty members do not seem to do much general category theory.
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