In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic.
What are some good examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?
Since limits and colimits can be characterized as Kan extensions or adjunctions, we have the obvious standard constructions: (co)products, (co)equalizers, etc., but these are common to a lot of categories we work with. Are there any examples more specific to analysis and differential geometry?
Best Answer
The following is really an adjunction between $2$-categories but I am going to ignore that subtlety. This blog post discusses everything in more detail and with a few more examples.
Consider on the one hand all concrete categories, by which I mean pairs $(C, U)$ of a category $C$ and a functor $U : C \to \text{Set}$ (the "underlying set") functor, and on the other hand the inclusion of those concrete categories which arise as the categories of models of a Lawvere theory $T$. Here by a Lawvere theory I mean for simplicity a category with finite products and objects $1, x, x^2, \dots$ for a distinguished object $x$, and by the category of models of a Lawvere theory I mean the category of product-preserving functors $T \to \text{Set}$, with the underlying set functor given by evaluation at $x$. Many familiar concrete categories of algebraic objects arise in this way, e.g. groups, rings, modules.
This inclusion has a left adjoint sending a concrete category $(C, U)$ to the "closest approximation" of that concrete category by the category of models of a Lawvere theory, which is the following. The full subcategory of the category of functors $C \to \text{Set}$ on the products $1, U, U^2, \dots$ of $U$ can be thought of as the category of operations on the objects of $C$ (e.g. natural transformations $U^n \to U$ correspond to the $n$-ary operations), and these naturally form a Lawvere theory which one can take the category of models of. Moreover, there is a natural functor of concrete categories from $(C, U)$ to the category of models of the Lawvere theory $T_U$ determined by $U$; this is the unit of the adjunction.
Alright, now for some examples in analysis, Lie theory, and differential geometry.