Initial category having all limits and colimits?

Question

Is there a 'smallest' category having all finite limits and colimits? (In the sense of initial object in the category of categories with this property).

(For a few moments, I thought it would simply be the category of finite sets and ordinary maps, but that can't be, because in that category, sum distributes over product but not vice versa.)

Edit: the functors in the category are to preserve all these finite limits and colimits, and the empty category is excluded.

Answer 1

We need to be precise here. Consider the following category $\mathcal{M}$:

• The objects are small categories with chosen initial object, binary coproducts, coequalisers, terminal object, binary products, and equalisers.
• The morphisms are functors that strictly preserve the chosen colimits and limits.

With a bit of work one can verify that $\mathcal{M}$ is a locally presentable category – specifically, it is clear that $\mathcal{M}$ has colimits of small filtered diagrams, limits of small diagrams, and has a small generating set. (In fact, $\mathcal{M}$ is locally finitely presentable – for this, one can check that it is the category of models for a finitary essentially algebraic theory.)

In particular, $\mathcal{M}$ has an initial object, which may be constructed using Freyd's initial object lemma.

Of course, you might object that $\mathcal{M}$ is not really the category you had in mind. Indeed, it seems more likely that you are interested in the following 2-category $\mathfrak{K}$:

• The objects are small categories with limits and colimits of finite diagrams.
• The morphisms are functors that preserve limits and colimits of finite diagrams (up to isomorphism).
• The 2-cells are natural transformations.

There is an evident forgetful functor $\mathcal{M} \to \mathfrak{K}$, and it is straightforward to check that it preserves initial objects in the following sense: if $M$ is an initial object in $\mathcal{M}$ and $K$ is an object in $\mathfrak{K}$, then the hom-category $\mathfrak{K} (U M, K)$ is equivalent to $\mathbf{1}$.

In particular, $\mathfrak{K}$ has an initial object in the sense of bicategories. However, $\mathfrak{K}$ does not have an initial object in the sense of ordinary categories.

Answer 2

There can't be such a category, because if there were then it would have a unique limit/colimit preserving functor $F$ to any category with all finite limits and colimits. But such a category can have an automorphism $G$ that fixes no object, and so $GF$ is a different limit/colimit preserving functor to that category.