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When we're teaching students about categories, we usually give lots of examples of things that are categories but very few (or no!) examples of things that are not categories.

Part of this problem is that it's hard to find classes of maps that don't satisfy composition. Continuous maps compose to continuous maps, and likewise for open, closed, proper, or quotient maps; similarly for homeomorphisms and homotopy equivalences. We can try a geometrical setting, but again the composition of $C^k$ maps for $1 \le k \le \infty$ (or even $k = \omega$, the analytic maps) is again a $C^k$ map.

One class of maps that does work is the inclusions of normal subgroups: if $H \trianglelefteq K$ and $K \trianglelefteq G$, it's not necessarily true that $H \trianglelefteq G$.

This immediately gives an example in the topological setting: the class of topological spaces and covering maps is a category, but the class of topological spaces and normal (i.e., regular or Galois) coverings is not a category since it doesn't satisfy composition.

Are there other natural examples in the topological setting? The class of constant maps is not a category since it doesn't satisfy the identity axiom, but I'm more interested in failures of the composition property. I'd also be curious about algebraic examples if folks can't come up with interesting topological ones.

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    $\begingroup$ Covering maps are not closed under composition : see this question for example. $\endgroup$ – Pece Feb 17 '17 at 21:32
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    $\begingroup$ It may not exactly be what you want, but this question could be interesting : math.stackexchange.com/questions/903277/… $\endgroup$ – Arnaud D. Feb 17 '17 at 21:50
  • $\begingroup$ @Pece Thanks very much for the correction. I suppose for finite coverings this must certainly be true, but I hadn't given any thought to the pathologies that could arise with very large covers. $\endgroup$ – dvitek Feb 18 '17 at 0:04
  • $\begingroup$ @ArnaudD That works: let $\mathrm{Hold}_{\alpha}$ be a class with objects the Euclidean spaces and morphisms the $\alpha$-Hölder continuous maps, for $0 < \alpha < 1$. Then this isn't a category! (We can presumably upgrade this to manifolds as well.) $\endgroup$ – dvitek Feb 18 '17 at 0:14

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