Exercises:

1. Explicitly construct a homotopy between the loop $\gamma(t) = (\cos 2\pi t, \sin 2\pi t, 0)$ on the sphere $S^2$ and the constant loop based at $(1, 0, 0)$. Note that an explicit formula is being demanded here.
2. Show that a loop in $X$ based at a point $x_0 \in X$ may be regarded as a continuous map $f:S^1\longrightarrow X$ such that $f(1) = x_0$. Show that if $f$ is homotopic to the constant loop $\varepsilon_{x_0}$ then $f$ extends as a continuous map from the closed unit disc to $X$.
3. Show that if $\gamma$ is a path starting at $x_0$ and $\gamma^{-1}$ is the inverse path then prove by imitating the proof of the reparametrization theorem (that is by taking convex combination of two functions) that $\gamma*\gamma^{-1}$ is homotopic to the constant loop $\epsilon_{x_0}$.
4. Prove theorems (7.2) and theorem (7.6) using Tietze's extension theorem.
5. Suppose $\phi : [0, 1] \longrightarrow [0, 1]$ is a continuous function such that $\phi(0) = \phi (1) = 0$ and $\gamma$ is a closed loop in $X$ based at $x_0 \in X$. Is it true that $\gamma\circ \phi$ is homotopic to the constant loop $\varepsilon_{x_0}$?
6. Show that the group isomorphism in theorem (7.8) is natural namely, if $f : X \longrightarrow Y$ is continuous and $x_1, x_2\in X$ then
   $$h_{[f\circ \sigma]} \circ f^{\prime}_* = h_{[\sigma]} \circ f^{\prime\prime}_*$$
   where, $y_1 = f(x_1),\; y_2 = f(x_2)$ and $\sigma$ is a path joining $x_1$ and $x_2$. The maps $f^{\prime}_*$ and $f^{\prime\prime}_*$ are the maps induced by $f$ on the fundamental groups. This information is better described by saying that the following diagram commutes:
   $$\begin{CD} \pi_1(X, x_1) @> f_*^{\prime} >> \pi_1(Y, y_1) \\ @... ...gma]}}VV \\ \pi_1(X, x_2) @> f_*^{\prime\prime} >> \pi_1(Y, y_2) \\ \end{CD}$$

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Lecture VIII - Categories and Functors

Note that one often works with several types of mathematical objects such as groups, abelian groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family of all topological spaces. These entities are huge and do not qualify to be sets. We shall call them families or classes and their individual members as objects. Between two objects of a family say between two topological spaces $X$ and $Y$ one is interested in the class of all continuous functions. Instead if we take two objects $G$ and $H$ from the class of all groups we are interested in the set of all group homomorphisms from $G$ into $H$. Abstracting from these examples we say that a category consists of a family of objects and for each pair of objects $X$ and $Y$ we are given a family of maps $X\rightarrow Y$ called the set of morphisms Mor$(X, Y)$ subject to the following properties:

(i) To each pair Mor$(X, Y)$ and Mor$(Y, Z)$ there is a map

$$(X, Y)\times (Y, Z)\longrightarrow (X, Z)$$

$$(f, g) \mapsto g\circ f \phantom{XXXX}$$

such that for $f \in$   Mor$(X, Y), g\in$   Mor$(Y, Z)$ and $h \in$   Mor$(Z, W)$,

$$(h\circ g)\circ f = h\circ (g\circ f)$$

(ii) To each object $X$ there is a unique element id$_X \in$   Mor$(X, X)$ such that for any $f \in$   Mor$(X, Y)$ and $g \in$   Mor$(Z, X)$

$$f\circ$$   id$$_X = f,$$   id$$_X\circ g = g$$