# Initial category having all limits and colimits?

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.

• Perhaps you want functors in this category to preserve limits and colimits. Oct 19, 2015 at 13:18
• If I didn't make a mistake, the discrete category on one object has all finite limits and colimits. Of course it's not an initial category like that, but it's definitely the smallest. Oct 19, 2015 at 13:19
• Of course, of course (to both comments). Obviously I mean the initial object, and obviously in the category with limit- and colimit-preserving functors. Oct 19, 2015 at 13:28
• @Najib Idrissi BTW, your example is actually the terminal object, me thinks? Oct 19, 2015 at 13:29

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.

• Ok. This 'initial bicategory', is it equivalent to something that we are very familiar with from 'ordinary' mathematics? (And is there a reference for the results that you mention, so that I can spend a bit more time on it?) Oct 19, 2015 at 15:09
• I do not have an explicit description of the initial object of $\mathcal{M}$. The results I mention are covered in any textbook account of locally presentable categories – for instance, Locally presentable and accessible categories. Oct 19, 2015 at 16:12

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.

• This argument shows that there isn't a initial object in the sense of ordinary categories. But there are other ways to interpret the question... Oct 19, 2015 at 14:42