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I understand that the product in a slice category $\mathscr{C}/X$ is the pullback in the category $\mathscr{C}$ - my end goal is to prove that fact. The one thing that is (currently) making me bash my head is what the product in that slice category even looks like - I've always had trouble wrapping my mind around categories whose objects are arrows.

Would anyone be able to help me visualize the product in some meaningful, useful way?

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  • $\begingroup$ Express pointed sets as a slice category and have a look. $\endgroup$ Oct 14, 2018 at 6:10
  • $\begingroup$ Just write out the universal property of the product for an arbitrary category, then write out what that means when the arrows are arrows of a slice category. This won't help you "visualize" it (though you can write out commutative diagrams that will give you some "visual" picture), but it is a very direct approach on the problem. There are some slicker, less direct approaches, but it's probably worth just doing the direct calculation. $\endgroup$ Oct 14, 2018 at 6:34
  • $\begingroup$ If you want intuition on slice categories, I encourage you to think of objects of $\mathcal C/X$ as objects over $X$ with topological mental pictures, like the reals in spiral over the circle $S^1$ (formally $x\mapsto exp(2i\pi x)$). With this view the objects are spaces in which points live above points of another fixed space and maps between the objects are the maps that don't change above which point everything lives. With that intuition in mind, the product appear kind of naturally as 'those pairs of points living above the same point'. $\endgroup$
    – Pece
    Oct 15, 2018 at 8:09

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This is just a normal pullback square in some category, arranged a bit differently:

$$ \begin{array}{ccccc} a & \leftarrow & a \times_c b & \rightarrow & b \\ & \searrow & \downarrow & \swarrow \\ & & c \end{array}$$

Now in the slice over $c$, each arrow to $c$ gets turned into an object. Let's call them $f$, $g$ and $h$ from left to right in the above diagram. The arrows $a \times_c b \rightarrow a$ und $a \times_c b \rightarrow b$ each make a triangle with two of the arrows $f$, $g$ and $h$ commute, so in the slice we have arrows

$$(a,f) \leftarrow (a \times_c b, g) \rightarrow (b,h)$$

which are projections making $(a \times_c b,g)$ into a candidate for the product $(a,f) \times (b,h)$ in the slice. Now the universality of $(a \times_c b,g)$ can be seen by adding in another candidate for the product in the slice and transfering that into the diagram in the original category, where it becomes a candidate for the pullback $a \times_c b$.

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