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).
  • Problems 2 and 3 are about adjoint pairs.
  • Problems 4 and 5 are further formal properties of (co)products.
  • 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!

🤓 Feel free to ask a question if you are stuck! 🤓

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