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I'm preparing to deliver some lectures on homological algebra and category theory, and have found lots of nice long lists of examples of functors and categories arising in every-day mathematical practice. I am interested in a similar list, but for non-examples.

I know, for instance, that the center $Z(G)=\{g\in G\,|\, hg=gh \text{ for all } h\in G\}$ of a group/ring/etc. fails to be a functor, and that the association of a Cayley graph to a group fails to be a functor from Groups to Graphs.

There was an earlier thread about this, but with the restriction that non-examples must be functions on objects and on morphisms but fail to respect morphism composition. I felt like the examples in this thread were somewhat artificial as well. I'm interested in examples where a student may expect there to be a category or functor involved, but there is not.

Mike Debellevue
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    Generally, I think there are relatively few natural examples of this. Non-examples of natural transformations are more common. – Kevin Carlson Jun 15 '20 at 01:01

1 Answers1

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Given a topological space we can consider its underlying set. This easily extends to a functor $U: \mathbf{Top} \to \mathbf{Set}$, from the category of topological spaces and continuous functions to the category of sets and functions. It simply sends a continuous function to its underlying function of sets. This makes $\mathbf{Top}$ into a concrete category (i.e. a category with a faithful functor into $\mathbf{Set}$).

Now consider $\mathbf{hTop}$, the category of topological spaces and homotopy equivalence classes of continuous functions. Once more we can simply forget the extra structure of a topological space and consider its underlying set. However, now it is impossible to turn this operation into a functor. In general there is no faithful functor $\mathbf{hTop} \to \mathbf{Set}$.

Relevant Wikipedia and nLab links.

Mark Kamsma
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    If you try the obvious thing and use choice to pick a representative $f$ of each homotopy class $[f]$, then there's no reason to believe that the representative picked for the composition of two homotopy classes $[fg]$ is the same set function as $fg$. Good example! – Mike Debellevue Jun 15 '20 at 13:03