# Problem with Awodey's Category Theory Exercise 8.11, showing Sets/X is cartesian closed using a specific hint.

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!

• The hint is fine. The fact that $X$ has only identity arrows makes things very easy to calculate. Commented Jul 26, 2021 at 12:47
• @ZhenLin Could you help me a bit more? I have no idea how should I use the embedding with the arrows $A \to X$. Commented Jul 26, 2021 at 13:51
• 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? Commented Jul 26, 2021 at 14:15
• @ZhenLin Thanks, I got it! Commented Jul 26, 2021 at 15:32