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  1. Show that every slice category $\mathbb{Sets}/X$ is cartesian closed. Calculate the exponential of two objects $A \to X$ and $B \to X$ by first determining the Yoneda embedding $y : X \to \mathbb{Sets}/X$ , and then applying the formula for exponentials of presheaves. Finally, observe that $\mathbb{Sets}/X$ is a topos, and determine its subobject classifier.

I feel like using the equivalence relation between $\mathbb{Sets}/X$ and $\mathbb{Sets}^{X}$, but I'm curious about the hint given, regarding $y: X \to \mathbb{Sets}/X$, since the set $X$ contains no non-identity arrows when regarded as a category. How is this related to the problem?

Thanks!

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  • $\begingroup$ The hint is fine. The fact that $X$ has only identity arrows makes things very easy to calculate. $\endgroup$
    – Zhen Lin
    Commented Jul 26, 2021 at 12:47
  • $\begingroup$ @ZhenLin Could you help me a bit more? I have no idea how should I use the embedding with the arrows $A \to X$. $\endgroup$
    – AprilGrimoire
    Commented Jul 26, 2021 at 13:51
  • $\begingroup$ You are supposed to use the equivalence between $\textbf{Set}_{/ X}$ and $\textbf{Set}^X$ to apply the hint. You have seen how to calculate exponentials in categories of presheaves, haven't you? $\endgroup$
    – Zhen Lin
    Commented Jul 26, 2021 at 14:15
  • $\begingroup$ @ZhenLin Thanks, I got it! $\endgroup$
    – AprilGrimoire
    Commented Jul 26, 2021 at 15:32

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