Reading Awodey [p.16-17], he states the following:

The slice category $\boldsymbol{C}/C$ of a cateogry $\boldsymbol{C}$ over an object $C\in\boldsymbol{C}$ has [definition of slice category follows] (...)
If $g: C\to D$ is any arrow, then there is a composition functor, $g_{*}: \boldsymbol{C}/C \to \boldsymbol{C}/D$ defined by $g_{*}(f) = g\circ f$, and similarly for arrows in $\boldsymbol{C}/C$.
Indeed, the whole construction is a functor, $\boldsymbol{C}/(-): \boldsymbol{C} \to \boldsymbol{\operatorname{Cat}}$ as the reader can easily verify.

So I have a few questions:

  1. What does the $(-)$ symbol mean?
  2. What does the author mean by the expression "the whole construction"?
  3. And how is that "construction" a functor? To my best understanding slice category was a "category" and not a functor.

P.S.: Please let me know if it's not clear, and I'll expand/clarify.
P.S.S.: My mathematics level: newbie


$\boldsymbol{C}/(-)$ is category theory notation jargon for the mapping which takes an object $X$ from $|\boldsymbol{C}|$ and yields the slice category $\boldsymbol{C}/X$.

Since slice category is a category, it can be thought of as an object in $\boldsymbol{Cat}$, which is the category whose objects are categories. So the mapping $\boldsymbol{C}/(-)$ takes objects in category $\boldsymbol{C}$ and yields objects in category $\boldsymbol{Cat}$.

Together with the corresponding mapping of arrows that takes an arrow $g$ (from category $\boldsymbol{C}$) and yields the arrow $g_*$ (from category $\boldsymbol{Cat}$), the mapping $\boldsymbol{C}/(-)$ forms a functor from $\boldsymbol{C}$ to $\boldsymbol{Cat}$. It has the necessary ingredients: a mapping between the objects of two categories (namely $\boldsymbol{C}$ and $\boldsymbol{Cat}$) and a mapping between the arrows of those categories.

(You should now verify that these mappings do in fact form a functor: They must respect sources and targets of arrows, and identities and compositions.)

By "the whole construction is a functor", Awodey means "the slice construction (for objects), together with the $g_*$ construction I just described (for arrows) is a functor."

The "$(-)$" notation is common in category theory; for some reason they don't use the $\mapsto$ notation that one would expect. The mapping that turns $g$ into $g_*$ might be written as $(-)_*$.

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    $\begingroup$ Sorry, but what is $|\boldsymbol{C}|$? In particular the vertical bars notation I'm not familiar with. $\endgroup$ – Andriy Drozdyuk Oct 26 '12 at 4:53
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    $\begingroup$ $|\boldsymbol{C}|$ is the set of objects of category $\boldsymbol{C}$. $\endgroup$ – MJD Oct 26 '12 at 11:29
  • $\begingroup$ So would you write the mapping $\boldsymbol{C}/(-)$ as being applied to some $X$ in $\boldsymbol{C}$ as follows: $\boldsymbol{C}/(-)(X)$ to yield an object in $Cat$? Thanks, your answer is very good at explaining this. $\endgroup$ – Andriy Drozdyuk Oct 26 '12 at 16:44
  • $\begingroup$ So that I could, for example, write: $g^{*}: \boldsymbol{C}/(-)\to \boldsymbol{C}/(-)$ to avoid saying "for any arrow $g:C\to D$ ..." $\endgroup$ – Andriy Drozdyuk Oct 26 '12 at 16:48
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    $\begingroup$ Kind of silly, since people don't write $f(-)$ for a function and then apply it as $f(x)$... Also, how would you write the dom and codom of $g^{*}$? Thanks! $\endgroup$ – Andriy Drozdyuk Oct 26 '12 at 20:00

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