2.5. Slice and Comma Categories.
In this section we introduce comma categories, which serve as a very useful categorical construction. The reason why it is so useful is because the notion of a comma category has the potential to simplify an otherwise complicated discussion. As they can be constructed in any category, and because they contain a large amount of useful data, they are frequently used as an intermediate step in more complex categorical constructions. Thus, while the concept is "simple," they nevertheless appear in all kinds of complicated discussions in category theory.
Let
- Objects. All pairs
for all and morphims . In other words, the objects are all morphisms in which originate at . - Morphisms. For two objects
and , we define
as a morphism between the objects, where
At this point you may be a bit overloaded with notation if this is the first time you've seen this before. You need to figure out how this is a category (what's the identity? composition?) and ultimately why you should care about this category. To aid your understanding, a picture might help.
We can represent the objects and morphisms of the
category
Now, how does composition work?
Composition of two composable morphisms
We can visually justify composition as well. If we have two commutative diagrams as on the left, we can just squish them together to get the final commutative diagram on the right.
Hence, we see that
One use of comma categories is to capture and generalize
the notion of a pointed category.
Such pointed categories include
the category of pointed sets
We've seen, in particular on the discussion of functors, the necessity for
pointed categories. For example, we cannot discuss "the" fundamental
group
where
Similarly, it makes no sense to talk about "the" tangent plane of
a smooth manifold. Such an association requires the selection of a point
where
Consider the category
- Objects. The objects are pairs
with a topological space and . - Morphisms. A morphism
is any continuous function such that .
Recall that the one point set
Why? Well, an object of
for some
Now, a morphism in this comma category will be of the form
In other words, if
The above example generalizes to many pointed categories, some of which are
We now briefly comment for any slice category
where on objects
is used in technical constructions involving slice categories as
it has nice properties; we will make use of it later when we discuss
limits.
Next,we introduce how we can also describe the category of an objects under another category.
Let
- Objects. All
pairs
where is a morphism in . That is, the objects are morphisms ending at . - Morphisms. For two objects
and , we define
to be a morphism between the objects to correspond to a morphism
Composition of functions
The following is a nice example that isn't traditionally seen as an example of a functor.
Let
Now for for every group
homomorphism, we may calculate
the kernal of
To see this, we have to understand what happens on the morphisms.
So, suppose we have two objects
Then we can define
What this means is that the commutativity of the above triangle forces
a natural relationship between the kernels of
In geometry and topology, one often meets the need to define a
\end{center}
For example, on the above left we can map the Möbius strip onto
On the right, we can recall that
In general, for a topological space
Hence we see that a bundle over a topological space
One particular case of interest concerns vector bundles.
Let
- 1.
is a finite-dimensional vector space over some field - 2. For each
, there is an open neighborhood and a homeomorphism
with
As we might expect, a morphism of vector bundles between
To realize this in real mathematics, we can take the classic example of the
tangent bundle on a
smooth manifold
where we recall that
This actually provides a differentiable structure on
is a continuous mapping. Hence we've satisfied both (1.) and (2.) in the the definition of a vector bundle. The other properties can be easily verified so that this provides a nice example of a vector bundle.
We can also formulate categories of objects under and over functors.
Let
- Objects. All
pairs
where such that
where
Representing this visually, we have that
Composition of the morphisms in
One can easily construct the category
Let
Then we define the comma category
- Objects. All pairs
where are objects of , respectively, such that
where
As usual, we can represent this visually via diagrams:
where in the above picture we have that
{\large Exercises \vspace{0.5cm}}
- *1.* Let
be a category with initial and terminal objects and .- i. Show that
. - i. Also show that
.
- i. Show that
- *2.* Consider again a group homomorphism
, but this time consider the image . Show that this defines a functor
where on morphisms, a morphism
is mapped to the restriction
In some sense, this is the "opposite" construction of the
kernel functor we introduced. Instead of taking the kernel of a group homomorphism,
we can take its image.
* *3. Here we prove that the processes of imposing the induced topology
and the coinduced topology are functorial. Moreover, the correct language
to describe this is via slice categories.
* i*. Let
$$
\tau_X = \{U \subset X \mid f(U) \text{ is open in }Y\}.
$$
This is called the **induced topology on** $X$.
So, we see that (by abuse of notation) the function $f: X \to U(Y)$
is now a continuous function $f: (X, \tau_X) \to (Y, \tau_Y)$.
Prove that this process forms a functor $\text{Ind}: (**Top**\downarrow U(Y)) \to (**Top**\downarrow Y)$.
* ***ii*.** This time, let $(X, \tau)$ be a topological space, $Y$ a set,
and consider a function $f: U(X) \to Y$. We can similarly impose a
topology $\tau_Y$ on $Y$:
$$
\tau_Y= \{ V \subset Y \mid f^{-1}(V) \text{ is open in }X \}.
$$
This is called the **coinduced topology on** $Y$.
Show that this is also a functorial process.
-
*4.*
- i. Let
, be topological spaces with a continuous function. Show that this induces a functor where on objects . - ii. Let
be a category. Show that we generalize (i) to define a functor
where
. * ii. Let be the pointed category of categories which we describe as * Objects. All pairs with a category and * Morphisms. Functors which preserve the objects. * *5. In this exercise we'll see that slice categories describe intervals for thin categories. * i.** Regard as a thin category, specifically as one with a partial order. For a given , describe the thin category . * ii.* Suppose is a partial order (so that and implies ). Describe in general the categories and . - i. Let