This sheet is based on Lectures 40, 41, and 42.

• Problem 1 asks you to show that (co)products are commutative and associative (when they exist).
• Problem 6 will be used in Lecture 43 to show that if $X$ is an $H$-cogroup and $Y$ is an $H$-group then the two group multiplications on $[X,Y]$ coincide.
• Problem 7 shows that spheres are cogroup objects in $\mathsf{hTop}_*$. Remark: The (easier) claim that $S(S^n) \cong S^{n+1}$ was Problem 4 on the Algebraic Topology I exam!