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Let $\mathcal{C}$ be a category. We say that $\mathcal{C}$ is locally small if $\mathrm{Hom}_{\mathcal{C}}(A,B)$ is a set for all $A$, $B$ in $\mathcal{C}$.

I can't think of any natural examples of non-locally small categories which are 'obviously' not locally small. We can take $\mathcal{C}$ to have one object, and a morphism for every $x \in V$ (say in ZFC), with composition of morphisms given by the union of two sets, but I can't think when this would ever come up 'naturally'

Are there any natural examples of non-locally small categories which are obviously not locally small?

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In general, the functor category $\mathbf C^{\mathbf D}$ is not locally small if $\mathbf C$ and $\mathbf D$ are large. –  Lord_Farin Oct 23 '12 at 18:07
    
You done stole my idea :( I suppose the example comes up naturally by turning the (set theoretic) universe into a monoid under $\cup$. –  Clive Newstead Oct 23 '12 at 18:25
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There are categories that are locally small but not obviously so – the category obtained by formally inverting all weak homotopy equivalences between topological spaces, for example. –  Zhen Lin Oct 23 '12 at 18:39
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Simple examples are given by "large monoids," for example the large monoid of sets under Cartesian product, or the large monoid of vector spaces under tensor product. If you're a fan of ordinals, the large monoid of ordinals under ordinal sum is another example. More generally you can take isomorphism classes of objects in any monoidal category under the monoidal product, e.g. a category with finite products or coproducts. (Recall that monoids are small categories with one object. This identification is perfectly natural if one is willing to think of categories both as settings for studying other mathematical objects and as mathematical objects in their own right.)

As Lord_Farin says in the comments, functor categories are also not locally small in general. These arise quite naturally.

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