Exercises:

1. Recast the notion of homotopy of paths in terms of morphisms of the category ${\bf Top^2}$.
2. Define a binary operation on $\mathbb{Z}\times \mathbb{Z}$ as follows
   $$(a, b)\cdot (c, d) = (a+c, b+(-1)^ad)$$
   Show that this defines a group operation on $\mathbb{Z}\times \mathbb{Z}$ and this group is called the semi-direct product of $\mathbb{Z}$ with itself. The standard notation for this is $\mathbb{Z} \ltimes \mathbb{Z}$. Compute the inverse of $(a, b)$, compute the conjugate of $(a, b)$ by $(c, d)$ and the commutator of two elements. Determine the commutator subgroup and the the abelianization of $\mathbb{Z} \ltimes \mathbb{Z}$.
3. A morphism $\phi \in$   Mor$(X, Y)$ in a category is said to be an equivalence if there exists $\psi\in$   Mor$(Y, X)$ such that $\phi\circ\psi =$   id$_Y$ and $\psi\circ\phi =$   id$_X$. In a category whose objects are sets and morphisms are maps, show that if $g\circ f$ is an equivalence for $f \in$   Mor$(X, Y)$ and $g \in$   Mor$(Y, Z)$ then $g$ is surjective and $f$ is injective.
4. We say a category ${\cal C}$ admits finite products if for every pair of objects $U, V$ in ${\cal C}$ there exists an object $W$ and a pair of morphisms $p:W \longrightarrow U$, $q: W \longrightarrow V$ such that the following property holds. For every pair of morphisms $f: Z \longrightarrow U$, $g: Z \longrightarrow V$ there exists a unique morphism $f\times g \in$   Mor$(Z, W)$ such that
   $$p\circ(f\times g) = f,\quad q\circ(f\times g) = g.$$
   Show that the categories ${\bf Top}$, ${\bf Gr}$ and ${\bf AbGr}$ admit finite products and in fact the usual product of topological spaces/groups serve the purpose with $p$ and $q$ being the two projection maps.
5. Discuss arbitrary products in a category generalizing the preceding exercise and discuss the existence of arbitrary products in the categories Top, Gr and AbGr.
6. We say a category ${\cal C}$ admits finite coproducts if for every pair of objects $U, V$ in ${\cal C}$ there exists an object $W$ and a pair of morphisms $p: U \longrightarrow W$, $q: V \longrightarrow W$ such that the following property holds. For every pair of morphisms $f: U \longrightarrow Z$, $g: V \longrightarrow Z$ there exists a unique morphism $f\oplus g \in$   Mor$(W, Z)$ such that
   $$(f\oplus g)\circ p = f,\quad (f\oplus g)\circ q = g.$$
   Show that the category ${\bf AbGr}$ admits finite coproducts and in fact the usual product of groups serves the purpose where the maps $p$ and $q$ are the canonical injections:
   

   $$p:G\longrightarrow G\times H,\quad q:H \longrightarrow G\times H$$

   $$p(g) = (g, 1),\quad q(h) = (1, h)$$

   
   What happens when this (naive construction) is tried out in the category ${\bf Gr}$ instead of ${\bf AbGr}$? In the context of abelian groups the coproduct is referred to as the direct sum.
7. Discuss the coproduct of an arbitrary family of objects in the category AbGr. It is referred to as the direct sum of the family.
8. Suppose that $X$ and $Y$ are two topological spaces, form their disjoint union $X\sqcup Y$ which is the set theoretic union of their homeomorphic copies $X\times\{1\}$ and $Y\times\{2\}$. A subset $G$ of $X\sqcup Y$ is declared open if $G\cap (X\times\{1\})$ and $G\cap (Y\times\{2\})$ are both open. Check that this defines a topology on $X\sqcup Y$ and the maps
   

   $$p:X \longrightarrow X\sqcup Y,\quad q:Y \longrightarrow X\sqcup Y$$

   $$p(x) = (x, 1), \quad q(y) = (y, 2)$$

   
   are both continuous. Show that the category ${\bf Top}$ admits finite coproducts.

in

Lecture IX - Functorial Property of the Fundamental Group

We now turn to the most basic functor in algebraic topology namely, the $\pi_1$ functor. Recall that the fundamental group of a space involves a base point and according to theorem (7.8) the fundamental group of a path connected space is unique upto isomorphism. However, this isomorphism is not canonical as theorem 7.9 shows and isomorphism classes of groups do not form a category. To get around this difficulty and to obtain a well-defined functor, we introduce the category of pointed topological spaces.