Chapter 12 (Functors) gives as an example of a functor the “free group functor” from the category of sets () to the category of groups (). We will call this functor , and its construction, as described, is the following: For any set , the object is the free group on . Let be a…
(Unfinished) In Chapter 12, R.G. describes three possible transformations “from real to complex vector spaces and back”. Inspired by Chapter 17 (Functors), I have examined these transformations from the functor point of view. We can distinguish not two, but three categories here, the first and third of which are equivalent: : Complex vector spaces with…
Exercise 105 (Chapter 17, “Functors”): Define the identity functor from a category to that same category. What do you suppose is meant by equivalent categories? Identity functors We have encountered two flavors of identity definitions. Identity mappings on sets are defined by the way they act on set elements, namely that they don’t change them….
On an infinite-dimensional vector space one usually defines a linear combination as a finite sum; for instance, if is a basis of the -vector space , one may write, for some finite part of : I find this notation cumbersome, because it seems to make the sum dependent on the arbitrary choice of the finite…