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

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]\mathcal{C}[/itex] be a category such that, for each object [itex]c \in Ob(\mathcal{C})[/itex], the slice category [itex]\mathcal{C}\,/c[/itex] is equivalent to a small category, even though [itex]\mathcal{C}[/itex] may not be small. Show that the functor category [itex] [ \mathcal{C}^{\text{ op}}, \bf{Set}][/itex] is an elementary topos.

    3. The attempt at a solution
    Well I know how to show this for a small category [itex]\mathcal{C}[/itex]: 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 [itex]G^F[/itex] for 2 presheaves [itex]F,\,G \in [\mathcal{C},\,\bf{Set}] [/itex] as [itex]G^F (c) = \mathrm{Hom}(\mathbf{y}c \times F, G)[/itex]. 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: [itex]\mathrm{Hom}(\mathbf{y}c \times F, G)[/itex] 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
     
  2. jcsd
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