Example 4.2.13. Given a category $\mathcal{C}$ and an object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define the category of objects over $X$, denoted $\mathcal{C}/X$ as follows. The objects of $\mathcal{C}/X$ are morphisms $Y\to X$ for some $Y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Morphisms between objects $Y\to X$ and $Y'\to X$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that make the obvious diagram commute. Note that there is a functor $p_ X : \mathcal{C}/X\to \mathcal{C}$ which simply forgets the morphism. Moreover given a morphism $f : X'\to X$ in $\mathcal{C}$ there is an induced functor $F : \mathcal{C}/X' \to \mathcal{C}/X$ obtained by composition with $f$, and $p_ X\circ F = p_{X'}$.

## Comments (2)

Comment #2541 by Zili Zhang on

In 33.4.8 (which refers to 7.24 and then refers here), the big etale site is defined to be $Sch_{etale}/S$. So I wonder in this example do we require the morphism $Y\to X$ belongs to $\textrm{Mor}(C)$? Since according to the comment after definition 50.27.3, objects in the big etale site over scheme $S$ are not required to be etale over $S$.

Comment #2574 by on

In this example, given objects $a : Y \to X$ and $b : Z \to X$ of $\mathcal{X}/X$ we have That is all.

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• 2 comment(s) on Section 4.2: Definitions

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