# Examples of Categories

It’s impossible to understand what a category is, let alone see why it is interesting, without examples so let’s write down a bunch of examples. Some will be categories we have seen before, and some we haven’t.

• The trivial category with no objects and no maps (pictured above).
• A discrete category, with some objects, and only the required identity morphisms $1_X$ for each $X\in Ob(\mathcal C)$.
• The category where the objects are sets, and the morphisms are functions.
• This category, with the $5$ objects and $6$ morphisms drawn in (and the $5$ identity morphisms not drawn in):

• The category with vector spaces as the objects, and linear transformations as the morphisms. Notice that we always have the identity transformation, and that composition of transformations is associative. Also notice that we sometimes have inverses, but we aren’t required to. Categories don’t require maps to be invertible.
• The category with groups as the objects, and group homomorphisms as the morphisms (see why we used the word “morphism” and the symbol $\hom$?).
• The category where the objects are smiley faces drawn on paper and the morphisms are rotations of the smiley faces.
• The category with partially ordered sets as the objects and monotone functions as the morphisms.
• The category with differentiable manifolds as the objects and smooth maps as the morphisms.
• The category with rings as objects and ring homomorphisms as morphisms.

Yeah, I know. It’s a long list. I’ve barely scratched the surface of the ones that we might care about. Notice that some are defined in terms of branches of mathematics we already encountered, and some are defined purely abstractly (like the one defined only by it’s diagram).

### 3 Responses to Examples of Categories

1. BenX says:

I still feel confused but I guess I understand what a category. My confusion might be from not knowing what categories can be used for (or what general things can even be said about categories). I look forward to the next post.

• soffer801 says:

Another way to think of categories are “an area of math.” It’s a unifying language. What can be said is coming. I don’t want to put too much in one post, so it may be coming a bit slow. But better that than too fast.

2. dawnly shien says:

i understand now what is the category……thanks i have a assignment