I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What are some simple examples of spans $Z\leftarrow X\rightarrow Y$ and cospans $Z\rightarrow X\leftarrow Y$ that cannot be completed to pushout and pullback squares, respectively?
Best Answer
I'll work in the based category, and consider $S^1$ as $\{z\in\mathbb{C}:|z|=1\}$. Consider the maps $$\text{point}\xleftarrow{}S^1\xrightarrow{f}S^1, $$ where $f(z)=z^2$. Suppose that there is a pushout $P$. We would then have a natural isomorphism $[P,X]=\text{Hom}(\mathbb{Z}/2,\pi_1(X))$. On the other hand, the fibration $$ S^1 = B\mathbb{Z} \xrightarrow{f} S^1 \to \mathbb{R}P^\infty = B(\mathbb{Z}/2) \to \mathbb{C}P^\infty = BS^1 \xrightarrow{Bf} \mathbb{C}P^\infty $$ gives an exact sequence $$ [P,S^1] \to [P,\mathbb{R}P^\infty] \to [P,\mathbb{C}P^\infty]. $$ Now $\pi_1(S^1)=\mathbb{Z}$ and $\pi_1(\mathbb{R}P^\infty)=\mathbb{Z}/2$ and $\pi_1(\mathbb{C}P^\infty)=0$, so if $[P,X]=\text{Hom}(\mathbb{Z}/2,\pi_1(X))$ then we have an exact sequence $0\to\mathbb{Z}/2\to 0$, which is impossible.