Category of (small) categories has finite limits

Question

I'm trying to prove that the category of (small) categories $\mathcal{Cat}$ contains finite limits. I know that this is equivalent to saying that $\mathcal{Cat}$ contains finite products and equalizers, but I'm not sure how this could be utilized here... I'm just looking for a few hints on how to get started on this one?

Answer 1

The constructions are 'just as you would expect'. For products, this is already outlined on nLab (product category). The empty product is obviously the one-point category (just one object with only the identity arrow). So all that is left are equalizers. Given two parallel functors $F, G: \mathcal{C} \to \mathcal{D}$ we can define a subcategory $\mathcal{E}$ of $\mathcal{C}$ as:

• Objects: all objects $X$ in $\mathcal{C}$, such that $F(X) = G(X)$.
• Arrows: all arrows $f$ in $\mathcal{C}$, such that $F(f) = G(f)$.

Of course, you have to check that $\mathcal{E}$ is indeed a category (exercise!). Then the inclusion of $\mathcal{E}$ into $\mathcal{C}$ will give our equalizer. Again you should check this, but this is again standard (exercise!).

Answer 2

Since the theory of categories is essentially algebraic, general results show that it has all limits and colimits. Those results furthermore show that filtered colimits and all limits are calculated based on the corresponding (co)limits of the (pairs of) carrier sets. "Essentially algebraic" more or less lives up to its name, and this result applies to things like the category of groups or rings or modules. (There are some exceptions, e.g. the category of fields.)

While the above result is surprisingly easy to prove – the only part that's somewhat tricky is showing that the non-filtered colimits exist but you don't need the colimit stuff – you're probably working through prerequisites to it right now.

What you can take from the result is what the limits look like. What the result states is e.g. that a product of groups is made by taking the products of the underlying sets. For (small) categories, this means that the product of categories has a set of objects that is the product of the objects of the two categories and similarly for the set of arrows. It also states that the identities and composition are component-wise. You can do the same thing to work out what the terminal object and equalizers look like. Make sure to verify that the results do, in fact, satisfy the corresponding universal properties.