Connection between pointed topological spaces and pointed categories Is there any motivation behind the name "pointed category" other than the fact that the category of pointed topological spaces is an obvious example of a pointed category?
Thank you.
 A: The Wikipedia article
initial and terminal objects states

If an object is both initial and terminal, it is called a
zero object or null object. A pointed category
is one with a zero object.

It later gives examples such as

In the category of pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from $(A, a)$ to $(B, b)$ being a function
$f : A \to B$ with $f(a) = b)$, every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero object.

Notice that Set, the category of sets is not a pointed
category since, as the article states, there are no zero
objects.
The Wikipedia article
Pointed space
is about pointed topological spaces. It states

The pointed set concept is less important; it is anyway the case of a pointed discrete space.

and thus, pointed sets can be regarded as a special case
of pointed spaces. The article also states

Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map $f$ between a
pointed space $X$ with basepoint $x_0$ and a pointed
space $Y$ with basepoint $y_0$ is a based map if it is
continuous with respect to the topologies of $X$ and
$Y$ and if $f(x_0)=y_0.$ This is usually denoted
$$f:(X,x_0) \to (Y,y_0). $$

You asked

Is there any motivation behind the name "pointed category" other than the fact that the category of pointed topological spaces is an obvious example of a pointed category?

It seems that the simplest example of a pointed category
is the category of pointed sets where every singleton is
a zero object.
Next simplest example is the category of pointed spaces
where any one point space is a zero object.
Given the two examples of "points", it seems natural to
use "pointed" as an adjective for any of the many other
well-known categories which have zero objects.
