7
$\begingroup$

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

$\endgroup$
10
$\begingroup$

$\boldsymbol{C}/(-)$ 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 $(-)_*$.

$\endgroup$
  • 1
    $\begingroup$ 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
  • 2
    $\begingroup$ $|\boldsymbol{C}|$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.