C/c: modding out an object

Question

Given a category $\mathbf{C}$ and some object $c \in \mathbf{C}$, what's the meaning of $\mathbf{C}/c$?

I know that for a congruence relation $R$ on $\mathbf{C}$, the expression $\mathbf{C}/R$ denotes the quotient category of equivalence classes. But I don't know how to similarly "mod out" an object $c$.

Answer 1

$\mathbf{C}/c$ denotes a slice category or over category of arrows pointing to $c$. On nLab it is defined like so:

The slice category or over category $\mathbf{C}/c$ of a category $\mathbf{C}$ over an object $c \in \mathbf{C}$ has

• objects that are all arrows $f \in \mathbf{C}$ such that $cod(f) = c$, and

• morphisms $g: X \to X' \in \mathbf{C}$ from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$.

$$ C/c = \left\lbrace \array{ X &&\stackrel{g}{\to}&& X' \\ & {}_f \searrow && \swarrow_{f'} \\ && c } \right\rbrace $$

Slice categories ($\mathbf{C}/c$) don't seem to have a very obvious relation to quotient categories ($\mathbf{C}/R$); they just use the same syntax. The alternative syntax $\mathbf{C} \downarrow c$ avoids this potential confusion.

There is also the dual notion of a coslice category or under category of arrows pointing away from $c$, denoted $c/\mathbf{C}$ or $c \downarrow \mathbf{C}$.