## The Fundamental Groupoid and Group (Pt. IV)

**Point of Post: **This is a continuation of this post.

## The Fundamental Groupoid and Group (Pt. I)

**Point of Post: **In this post we describe the fundamental group and groupoid functors.

**Motivation**

We have discussed in previous posts how the notion of homotopy as being an equivalence that captures, and focuses in on, the various notions of connectivity that a space carries. Now, the first thing one does when one defines a new notion of equivalence is to try to classify all objects of interest up to this equivalence. For us, this means that we would (in a perfect world) be able to classify all spaces up to homotopy equivalence. Of course, as in most categories(subjects of study), this is an untenable goal. It is not practically possible to, say, explicitly classify all finite groups (even though we have made stupendous strides in this direction).

## Flat Modules (Pt. I)

**Point of Post: **In this post we discuss, in some detail, flat modules including equational characterizations and Lazard’s theorem.

*Motivation*

We have so far discussed two kinds of modules that make certain half-exact (i.e. right exact or left exact) functors exact. Namely, we have discussed projective modules as the modules that make the Hom functor exact, and we have discussed injective modules as the modules that make the contravariant Hom functor exact. Both of these are very important since projective and injective modules are the “approximating” modules used to define powerful notions like derived functors. That said, they aren’t the most “functional” modules that make certain functors exact. In particular, in basic module theory there are three main functors, two of which are the covariant and contravariant Hom functor. The last, of course, is the tensor functor . The tensor functor shows up everywhere in mathematics and is, dare I say it, “more important” (whatever that really means) than the two Hom functors [note: this is just my limited opinion on the matter]. Thus, it would seem to behoove us to understand what kind of modules make the functor exact (it is already right exact).

This naturally then leads us to considering flat modules, which are precisely those modules that make the relevant tensor functor exact. Flat modules show up in geometry in some pretty interesting ways (none of which I am really able to speak to at this point).

## Preordered Sets as Categories, and their Functor Categories (Pt. II)

**Point of Post: **This is a continuation of this post.

## Functorial Properties of the Tensor Product (Pt. III)

**Point of Post: **This is a continuation of this post.