Example 4.2.13 (001G)—The Stacks project

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 (5)

Comment #2541 by Zili Zhang on May 13, 2017 at 22:13

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 Johan on May 25, 2017 at 19:25

In this example, given objects $a : Y \to X$ and $b : Z \to X$ of $\mathcal{X}/X$ we have $\Mor_{\mathcal{C}/X}((a : Y \to X), (b : Z \to X)) = \{c \in \Mor_\mathcal{C}(Y, Z) \mid a = b \circ c\}$ That is all.

Comment #6707 by Shubhrajit Bhattacharya on November 09, 2021 at 11:28

Here is an undefined control sequence \Mor. Because of that we can't see what's written. Is it possible to make necessary changes?

Comment #6710 by Johan on November 09, 2021 at 19:48

$\mathop{Mor}_{\mathcal{C}/X}((a : Y \to X), (b : Z \to X)) = \{c \in \mathop{Mor}_\mathcal{C}(Y, Z) \mid a = b \circ c\}$

Comment #6711 by Bhattacharya on November 10, 2021 at 08:18

MorC/X((a:Y→X),(b:Z→X))={c∈MorC(Y,Z)∣a=b∘c}

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