Relative homotopy

 1. A visualization of relative homotopy

   Relative homotopy can be defined formally as follows. Let X, Y be topological spaces and f, g : X -> Y be continuous maps that agree on a subset A of X. We say that f and g are homotopic relative A if there exists a continuous map H : [0, 1] x X -> Y such that H(0, x) = f(x), H(1, x) = g(x) and H(t, a) is independent of t for all a in A. 

   To visualize this concept, I would take X = [0, 1], which means that f, g are paths. We "imagine" that f, g are ropes and a homotopy is a way to move one rope over and over the space so that it overlaps with the other rope. By picking a subset A of X and requiring "homotopy relative A", we "tight" or "glue"/"attach" the corresponding points (to the wall or a solid pillar made from iron, for example) so that it can't move anywhere else. Relative homotopy, from this point of view, might be considered as a way to "organize" the homotopy so that it does not seem too chaotic. 

   For example, the following picture shows the difference between a homotopy between f, g : [0, 1] -> X relative {0, 1} and a homotopy that is not relative {0, 1}. For the homotopy relative {0, 1}, if you uniformly divide [0, 1] and draw the positions of the rope at each point of division, you will get some stuff that is similar to a guitar (see the picture on the left). Meanwhile, the homotopy in the usual sense (shown on the right) might behave too freely and it is hard to call its "shape" (the blue and pink lines show where certain points on the ropes go from time to time). 

2. Relative homotopy appears when we talk about fundamental groups

   The fundamental group of a based space (X, x) is defined as the homotopy class of all loops based at x, in which two loops w, w* : [0, 1] -> X with w(0) = w(1) = w*(0) = w*(1) = x are homotopic if and only if there exists a continuous map H : [0, 1] x [0, 1] -> X such that H(0, t) = w(t), H(1, t) = w*(t) and H(t, 0) = H(t, 1) = x for all t in [0, 1]. If we don't require H(t, 0) = H(t, 1) = X for all t in [0, 1], several known and useful results will not be true. In this case, we require that the two loops w, w* must be homotopic relative {0, 1} instead of the homotopy in the usual sense. The following example shows a well-known result that will not be true if relative homotopy is not required. 

   Example: A basic fact in (algebraic) topology is that the fundamental group of a circle is the additive group of integers. Without assuming relative homotopy, the loop won't be "tighted" at the basepoint. A (non-relative) homotopy from w : [0, 1] -> S^1 satisfying w(t) = (cos(2\pi*t), sin(2\pi*t)) to w* : [0, 1] -> S^1 that is constant at (1, 0) can be constructed as follows: H(t, u) = (cos(2\pi*ut), sin(2\pi*ut)). The map H is obviously continuous. This means that going one round does not matter and the so-called "fundamental group" must be trivial, we can't do anything with the "fundamental group". 

3. Fundamental groupoid

   A groupoid is a category in which every morphism has its inverse. For example, let G be a group, the category with Object = {*} and Morphism{*, *} = G (composition is given by the corresponding operation on G) is a groupoid since: 
   - Each morphism is an element of G, which has an inverse
   - Associativity is induced from G
   - The identity element of G corresponds to the identity morphism on *

   The fundamental groupoid of a topological space X is also defined via relative homotopy. Its object set is X. Meanwhile, for any two elements x, y of X (possibly x = y), the morphism set Mor(x, y) is the set of all relative homotopy classes of paths from x to y in X. With this setting, Mor(x, x) is the fundamental group of the based space (X, x).