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I'm reading the text Category theory in context from Emily Riehl, and having trouble to find an example asked on exercise $1.3.viii.$: prove that functors not need to reflect isomorphisms, i.e., find a functor $F:C\rightarrow D$ and a morphism $f$ in $C$ so that $Ff$ is an isomorphism in $D$ but is not an isomorphism in $C$.

I know that a non conservative functor from $Top$ to $Set$ might work but can't find the adequate morphism.

Any suggestion is apreciated.

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  • $\begingroup$ Consider a functor from $F:\mathbb{Grp}\to \mathbb{Set}$. Suppose we have a morphism $f:\Bbb{Z}\to \Bbb{Z}$ defined by $f:x\to x+1$. Clearly this is not an isomorphism (it is not even a homomorphism) in $\mathbb{Grp}$. However, $Ff$ is an isomorphism in $\mathbb{Set}$. $\endgroup$
    – user67803
    Mar 10, 2018 at 21:16
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    $\begingroup$ @AyushKhaitan Your example does not work because as you say $f$ is not a homomorphism, so it is not a morphism in the category $\mathbb{G}rp$. $\endgroup$ Mar 10, 2018 at 21:18
  • $\begingroup$ @WardBeullens- Yes you're right. $\endgroup$
    – user67803
    Mar 10, 2018 at 21:24

4 Answers 4

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A very generic counterexample:

Take for $\mathcal{D}$ the category with one object $O$ and one morphism $f$. There is a functor $F$ from any category to $\mathcal{D}$ that sends every object to $O$ and any morphism to $f$.

Since $f$ is an isomorphism this generates a counterexample for every category which contains at least one morphism which is not an isomorphism.

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Just take the forgetful functor $F$ from $\mathit{Top}$ to $\mathit{Set}$. Then, take a bijection $f$ between two topological spaces which is not a homeomorphism. Of course, $Ff$ will be an isomorphism.

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    $\begingroup$ In particular, you could look at $f:[0,1)\to S^1$. This map is bijective and continuous, and hence a morphism in Top. However, its inverse is not continuous, because of which $f$ is not an isomorphism (in other words, a homeomorphism). Clearly, $Ff$ is an isomorphism in $\mathcal{Set}$ though. $\endgroup$
    – user67803
    Mar 10, 2018 at 21:25
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Homology and homotopy functors are very natural examples of this.

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Hint: look at any continuous bijection that fails to be a homeomorphism.

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