## 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).

## Homtopy and the Homotopy Category (Pt. I)

**Point of Post: **In this post we motivate, define, and discuss the notion of homotopy. We then introduce the homotopy category as a quotient category of .

**Motivation**

In topology we care about the geometry of a space–how we can describe all of the geometric properties that a space has. For example, we care if the space is compact, we care if it locally connected, we care if it . All of the properties factor into whether or not two given topological spaces are to be considered “the same”. That said, for large portions of mathematics some of the topological invariants of a space are less important than others. This can occur either because they literally matter less to us (for example, compactness is something that, while nice, isn’t an absolute necessity for a topological space to be nice). This can also be true because the subject area we are are working in contains spaces which necessarily already satisfy some of the properties (for example, metric spaces already satisfy all of the nice separation axioms).

## Natural Transformations (Pt. I)

**Point of Post: **In this post we introduce the notion of natural transformations, natural equivalences, and prove some basic theorems concerning them. We also give a few examples of natural transformations between some specific functors, as well as cover the classic natural equivalences (e.g. the natural equivalence of the double dual functor and the identity functor on ).

*Motivation*

In this post we discuss a notion in category theory which will finally make rigorous such wildly tossed around statements such as “____ is * naturally/canonically* isomorphic to ___”. That phrase, or some variant thereof, is heard so often in elementary mathematics (e.g. basic group theory, even in books as basic as Pinter or Fraleigh, as well in non-algebraic books such as Simmons) but is rarely (if ever!) actually given meaning in those settings. So, what precisely does this mean? Often times one hears explanations of ‘canonically’ as meaning “without having to choose”. For example, an isomorphism in linear algebra is ‘canonical’ if we can formulate it without having to fix a basis in the source or target space. But, of course this has no meaning when one is dealing with isomorphisms in, say, , as there are no choices ever really made. So, what does it mean for two things to be ‘naturally’ isomorphic? We begin by phrasing the problem in more categorical language. Often times one speaks of ‘natural isomorphisms’ between certain constructions. For example, in (the category of finite dimensional vector spaces over some field ) it’s often stated that “every finite dimensional vector space is ‘naturally’ isomorphic to its double dual “. Really what is being said, in a sense, is that the construction mapping to itself and the construction mapping to are “the same”. Or, putting things in phrasing we know, that there is some notion of equivalence between the identity functor and double dual functor on . What is this equivalence? While there are many explanations of the intuition behind the kind of equivalence we’re interested, there is a particularly beautiful description which, at the risk of alienating some readers, I’d like to explain. In topology given two maps we call them homotopic if there exists a continuous path from to . Said differently, if there exists a continuous map (where ) such that and . One thinks of maps as being homotopic if one can nicely slide one path onto the other path without hitting any snags/tearing. One can think of this process as be indexed by time, and at each second there is a curve, and as time progresses this goes from looking like to looking like . Intuitively, natural transformations are the homotopies of category theory. Namely, a natural transformation from a functor to a functor is a ‘continuous’ (i.e. respects the category, doesn’t “tear” anything) deformation of the functor into the functor . Of course, doesn’t “tear” anything and respects the category should be further explained. By this we shall see, as I’d like to defer to the discussion below, an associated shifting of the “image” of to the “image” of , and by not “tearing” and “respecting the category” we mean that this diagram commutes–that we can figure out information in the image of by working through the image of and this transformation between them.