Topos theory; show a category of presheaves is an elementary topos

[Mathmos6]

Homework Statement

Let $\mathcal{C}$ be a category such that, for each object $c \in Ob(\mathcal{C})$, the slice category $\mathcal{C}\,/c$ is equivalent to a small category, even though $\mathcal{C}$ may not be small. Show that the functor category $[ \mathcal{C}^{\text{ op}}, \bf{Set}]$ is an elementary topos.

The Attempt at a Solution

Well I know how to show this for a small category $\mathcal{C}$: we need finite limits, exponentials and a subobject classifier. For example in constructing exponentials we can apply Yoneda's Lemma to figure out how to define the exponential $G^F$ for 2 presheaves $F,\,G \in [\mathcal{C},\,\bf{Set}]$ as $G^F (c) = \mathrm{Hom}(\mathbf{y}c \times F, G)$. However, speaking in general we can't do this because we don't necessarily know everything is sufficiently small so as to correspond to sets: $\mathrm{Hom}(\mathbf{y}c \times F, G)$ may not be a set for example.

Presumably we need to use the condition on each slice category being equivalent to a small category in order to show these things are sets and then apply the standard proof (most of which can be found in MacLane and Moerdijk's Sheaves in Geometry and Logic p46-47 , but I can't seem to figure out how to do so. If anyone could suggest anything I'd be very grateful. Many thanks - M

 

[mighty2000]

Thank you for your interesting question. I agree with your approach to use Yoneda's Lemma to construct exponentials in the functor category [ \mathcal{C}^{\text{ op}}, \bf{Set}]. However, as you mentioned, we need to show that the Hom-sets in this category are sets, which may not be the case if \mathcal{C} is not small.

To solve this problem, we can use the fact that each slice category \mathcal{C}\,/c is equivalent to a small category. This means that the objects and morphisms in \mathcal{C}\,/c can be put in bijection with sets. Therefore, we can define the Hom-sets in [ \mathcal{C}^{\text{ op}}, \bf{Set}] as sets of morphisms in \mathcal{C}\,/c. In other words, for any two presheaves F, G \in [\mathcal{C},\,\bf{Set}], we can define Hom-sets as Hom-sets in the category \mathcal{C}\,/c, i.e. \mathrm{Hom}(F,G)(c) = \mathrm{Hom}_{\mathcal{C}\,/c}(F(c), G(c)).

Using this definition, we can show that the functor category [ \mathcal{C}^{\text{ op}}, \bf{Set}] has all the necessary properties to be an elementary topos. For example, we can use Yoneda's Lemma to show that this category has exponentials, as you suggested. We can also use the equivalence between \mathcal{C}\,/c and a small category to show that this category has finite limits and a subobject classifier.

I hope this helps to clarify your doubts. Please let me know if you have any further questions.