Life update, thoughts on monads

Because I just started this blog, I have a lot of writing material that I've been hanging onto. I will probably end up dumping a lot of old stuff here for the time being. The following is a letter I wrote to a friend with some thoughts on monads.

It's nice to look back at old writing, as it reminds you that you are making progress : I finally did end up reading that section on monads and homology in Categories for the Working Mathematician, and found it quite rewarding. I have even learned enough French to start working through the (excellent) book by Godement. The notes on monads and the Stone-Cech compactification that I mention in the email can be found here:

https://www.math.wisc.edu/~nicodemus/Monads_and_Stone_Cech_Compactification.tex
https://www.math.wisc.edu/~nicodemus/Monads_and_Stone_Cech_Compactification.pdf

~~~

Let me just give an update on my life and interests recently. Category theory was born from within homological algebra in an effort to keep track of the families of groups and coherence and naturality conditions within that field. Some ideas from category theory, it seems, are still best understood in this context. Recently, I have been slowly starting to learn the broad world of geometry in an attempt to piece together some of the historical origins and intuitions between constructions in homological algebra and algebraic topology. For example, differential geometry involves a lot of multilinear algebra (tensor algebras, exterior algebras, and bundles of these) which I suspect will help me get intuition for some constructions in commutative/homological algebra like the Koszul complex. It is where we naturally deal with vector bundles, from which interesting notions in algebraic topology arise (K theory of vector bundles, classifying spaces of principal G-bundles).  It is the origin of de Rham cohomology, which I understand to be the most geometrically intuitive notion of cohomology, which I hope will give me some more intuition for other cohomology theories in other areas; and I hope to eventually understand sheaf cohomology on a continuous spectrum from the most geometric forms (de Rham cohomology) to the least geometric and most categorical (topos cohomology, which subsumes, say, group cohomology, sheaf cohomology under the same theoretical framework, and paves the way for new notions like etale cohomology) So recently I have been diving into a bit of differential geometry out of the book by Warner, "Foundations of Differentiable Manifolds and Lie Groups." It is my hope that I will by the end of the semester develop enough of an understanding of sheaf cohomology that I am able to present some talks on topos theory from both the geometric perspective (which I understand very little) and the logical perspective (which I understand, I believe, very well)

Anyway. Monads are presented in Categories for the Working Mathematician; first in a chapter summarizing their basic properties (which is readable) and then, the crucial bit on applications to homology, summarized in a brief but highly technical section on p. 176 in the chapter on Monoids. I myself have read the first chapter, but never been able to digest or comprehend that latter one; I will probably return to it at the right time, after I have a better understanding of simplicial sets as an organizing principle for chain complexes via Dold Kan.

In Weibel's book on homological algebra, Chapter 8 - Simplicial Methods in Homological Algebra - is highly recommended reading. Section 8.6 introduces monads and comonads and describes their usage to organize the data of simplicial sets, as before. In particular, for applications to sheaf cohomology, he summarizes it in a brief example and refers the reader to the book by Godement; "Topologie Algebrique et theorie de faisceaux" ; I am sure you can find by googling around an English source for the Godement resolution of a sheaf.

Personally my perspective on monads (just one perspective; I believe there are probably other very good ones that I just don't understand yet) is restricted to monads on the category Sets, where they behave very well.

Consider the free group functor \(\mathbf{Sets}\to \mathbf{Groups}\) The construction is essentially syntactic; one has a certain formal language consisting of a family of functions and constant symbols (m for multiply, i for inverse, e for the identity) and one simply generates (for some given underlying set X) inductively the set of all grammatically valid terms in this language; that is, if x, y, z are some elements in X, then the free group has some element m(x,(m(e,i(z)). One then quotients out the set by the equivalence relation generated by the axioms of group theory, which are (conveniently) all equational, so that the axioms are easy to describe: for any term t, we have \(m(t,i(t)) = e; m(t,m(t',t'')) = m(m(t,t'),t'')\) and so on.

Now the free group functor is left-adjoint of course to the forgetful functor. If G is a group, and \(\varepsilon_G : F(U(G))\to G\) is the counit of the adjunction (F free, U forgetful (underlying)), then it is interesting to work out the content of the counit map. Let us follow it into Sets where the homomorphism structure will not distract us: U(\varepsilon): UFUG -> UG. What does it do? Essentially it sends the syntactic expressions freely generated over the elements of G, to their interpretation or evaluation in G, using G's group structure.

The idea behind monads is as follows: sufficiently nice maps UFX-> X equip X with a group structure. That is, there are certain axioms (all nicely diagrammatic, so they make sense in any category) we can lay down for a map h: UFX->X such that a map satisfies the axioms iff it makes X into a group, where h would be the evaluation map with respect to that group. The notion of group homomorphism can also be coded diagrammatically into this system: if (X, h: UFX ->X) and (Y, h' UFY -> Y) are two groups, then a group homomorphism h -> h' is equivalent to a map phi: X -> Y such that h' \circ UF(phi) = phi\circ h. So the category of groups can be realized, up to isomorphism of categories, as a category constructed out of Sets using the functor UF : Sets -> Sets (this functor UF is called a monad) together with certain notions of what it means to be an "algebra" of the functor. Intuitively, this demonstrates that the category of Groups is "built" out of the category Sets, equipped with additional algebraic structure.

Every adjunction F: C->D, G: D->G gives rise to a monad G\circ F :C->C. In this sense, the example above is not special.  But given a monad T: C-> C, there is not necessarily a unique category D and adjunction F, G : C->D, D->G such that G\circ F = T. There are, however, two canonical choices, which are called the Kleisli adjunction and Eilenberg-Moore adjunction. The Eilenberg Moore adjunction associated to a monad is very interesting because, in my opinion, it is very well behaved and has sufficiently similar properties to what you would hope for a "free-forgetful adjunction" that one could, as a working definition of the informal notion of "free-forgetful adjunction", take the formal notion of Eilenberg Moore adjunction. Basically this Eilenberg Moore adjunction sets up an adjunction between C, considered as an underlying category of objects of some class of algebraic structures, and D, a special class of objects of C equipped with certain additional algebraic structure by the monad T. This is what I was trying to get at in the previous paragraph: the adjunction Sets -| Groups is an Eilenberg-Moore adjunction.

Here's one interesting theorem which motivates the definition. If (F,G) : C-| D is an Eilenberg Moore adjunction, call G : D-> C monadic. So essentially a monadic functor is, like I said, the "official" definition of a forgetful functor with a free left adjoint. Theorem: if we have some diagram in D (say, some indexing category I, and some functor J : I-> D tracing out the diagram) and you want to construct its limit, then you can send J along the forgetful functor G (which would just be the composition G\circ J) and you can construct its limit there in C (if it exists), and then lift it up to a unique object "above" it in D. The concrete example of this is that if you have some diagram of groups and you want to take their limit, you are going to take the limit of the underlying diagram of sets and then there's a unique group structure you can impose on that which makes the set maps of the cone into homomorphisms, and makes it into the limit of the diagram of groups.

There is an approach to algebra ("universal algebra") where one pursues this more generally: if Omega is an arbitrary family of function symbols and constants, and E is some family of equations over the function symbols of Omega, there is a functor T: Sets -> Sets which takes any set X and returns the free set of terms generated by Omega over X, quotiented out by the equivalence relations generated by T. T is a monad, and the book "Algebraic Theories" by E. G. Manes tries to develop as much of universal algebra as possible through this monad perspective.

Here's a bizarre fact: the forgetful functor from compact hausdorff spaces to sets is monadic. Its left adjoint is the functor which takes each set X (regarded as a discrete space where all sets are open) to its Stone Cech compactification. This not only gives us nice theorems about CH spaces (like to construct the limit of a diagram you have to take the limit of the underlying sets and then you can uniquely impose a CH topology on it making it the limit of the diagram) but it also suggests that in some sense, compact hausdorff spaces are essentially algebraic objects - and this should strike you as absolutely mind blowing! What the hell does that mean? Indeed, you'll have to dig into it to find out, but the trick is that to make it work you have to generalize the universal algebra stuff I mentioned above to admit function symbols with infinite arities - a function symbol in the language can admit a family of inputs indexed by some cardinal number.) These infinitary function symbols send families of points to their unique "limit point" in the space, in a sense I make clear in these notes.

Also I would like to recommend, as a general guide to sheaf theory, Grothendieck topologies, and toposes, the book "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk. It proves some nice results which Hartshorne really should have mentioned early on in Chapter II. But it does not mention cohomology (the main reason for which topos theory was invented originally was to provide a setting for etale cohomology) So you will have to look else where for that.