The n-Category Café

I first figured out an answer using intuition in a pretty lowbrow way, but let me give the slick approach that Jim then prodded me into.

The idea is to use an enriched version of Gabriel–Ulmer duality:

• Peter Gabriel and Friedrich Ulmer, Lokal Praesentierbare Kategorien, Springer Lecture Notes in Mathematics 221, Berlin, 1971.

Gabriel–Ulmer duality is about finitely complete categories. I don’t understand it terribly well, but I think the idea is this:

We can think of a finitely complete category $T$ as a kind of ‘theory’ — sometimes called a ‘finite limits theory’ or essentially algebraic theory. A ‘model’ of this theory in the category of sets is a functor $$f:T\to \mathrm{Set}$$ that preserves finite limits. We can talk about a model being ‘finitely presented’. Gabriel–Ulmer duality says that the category of finitely presentable models of $T$ in $\mathrm{Set}$ is the same as $T^{\mathrm{op}}$.

I don’t know if this works for enriched categories, so I’ll pretend it does and hope someone smart can fill in the details.

In other words, I’ll hope something like this: if $T$ is a finitely complete linear category, the category of finitely presentable models of $T$ in $\mathrm{Vect}$ is equivalent to $T^{\mathrm{op}}$. Again, there may be fine print being overlooked here, but life is short, so I’m willing to risk it.

So: let $T^{\mathrm{op}}$ be ‘the free finitely cocomplete linear category on an epi’ — the thing the puzzle is asking us to compute.Then $T$ is ‘the free finitely complete linear category on a mono’ — the sort of thing where we can try to use Gabriel–Ulmer duality.

By definition, for any finitely complete linear category $X$, a ‘model’ of $T$ in $X$ is a finite-limit-preserving linear functor $$f:T\to X$$ By saying $T$ is ‘free on a mono’, we mean such models are the same as monos in $X$. So a model of $T$ in $\mathrm{Vect}$ is just a mono $$V\hookrightarrow W$$ where $V$ and $W$ are vector spaces. In a finitely presentable model, $V$ and $W$ will be finite-dimensional vector spaces.

So: $T^{\mathrm{op}}$ is equivalent to the category of monos in the category of finite-dimensional vector spaces. Let’s call that category $\mathrm{FinVect}$.

In short, we seem to be getting this:

Answer: The free finitely cocomplete linear category on an epi is the category of monos in $\mathrm{FinVect}$.

Now this raises a big question: if the category of monos in $\mathrm{FinVect}$ is free on an epi, what is that epi?

And this raises a smaller but still mildly mind-boggling question: what is an epi in the category of monos in $\mathrm{FinVect}$?

And this raises an even tinier question: what is a morphism in the category of monos in $\mathrm{FinVect}$?

That, at least, should be obvious. It’s a commuting square $$\begin{array}{ccc}V& \hookrightarrow & W\\ \downarrow & & \downarrow \\ V\prime & \hookrightarrow & W\prime \end{array}$$ where the horizontal arrows are monos.

Then, after I gave a probably wrong answer, James told me what seems to be the right answer to the big question, “if the category of monos in $\mathrm{FinVect}$ is free on an epi, what is that epi?” Here it is: $$\begin{array}{ccc}0& \to & k\\ \downarrow & & \downarrow 1\\ k& \stackrel{1}{\to }& k\end{array}$$

Interestingly, the left-hand vertical arrow is not an epi! You might naively guess that to get an epi in the category of monos, both vertical arrows would need to be epis. But that’s not true — as you can see by pondering this example.

More interestingly, this epi is not a cokernel. In the definition of abelian category, there’s a clause saying every epi must be a cokernel. So, the category of monos in $\mathrm{FinVect}$ is not abelian.

Even more interestingly, every mono in $\mathrm{FinVect}$ determines a short exact sequence, which determines an epi — and conversely. Given the mono $$0\to U\stackrel{i}{\to }V$$ we get the short exact sequence $$0\to U\stackrel{i}{\to }V\stackrel{p}{\to }W\to 0$$ where $W$ is the cokernel of $i$, and this gives the epi $$V\stackrel{p}{\to }W\to 0$$ Conversely, we can turn this process around: since $\mathrm{FinVect}$ is abelian, every mono is a kernel and every epi is cokernel.

So, the category of monos in $\mathrm{FinVect}$ is equivalent to the category of short exact sequences in $\mathrm{FinVect}$, which is equivalent to the category of epis in $\mathrm{FinVect}$. Any one of these categories should be an equally good answer to our puzzle!

And, we see that none of these categories is abelian. This is especially interesting because, if we haven’t screwed up, the category of monos in $\mathrm{FinVect}$ should have finite colimits. Why? Because it’s the answer to the puzzle “What’s the free finitely cocomplete linear category on an epi?” (Of course I’m sure we can also easily see this directly, at least if it’s true, but I’m too tired to check that now!)

It follows that dually, the category of epis in $\mathrm{FinVect}$ should have finite limits. But since these categories are equivalent, either one is a linear category that’s finitely complete and finitely cocomplete, but still not abelian!

Saying it yet another way:

The category of short exact sequences of finite-dimensional vector spaces is a linear category with finite limits and finite colimits that’s not abelian.

And that’s nice, because most of the finitely complete and cocomplete linear categories that leap to mind are abelian, but this is a pretty natural counterexample.

Further mucking about shows that this epi in the category of monos: $$\begin{array}{ccc}0& \to & k\\ \downarrow & & \downarrow 1\\ k& \stackrel{1}{\to }& k\end{array}$$ is also a mono in the category of monos. So, we’re getting a nice example of a morphism that’s an epimorphism and a monomorphism but not an isomorphism. It looks a bit cuter if we draw it as a morphism in the category of short exact sequences: $$\begin{array}{ccccccccc}0& \to & 0& \to & k& \stackrel{1}{\to }& k& \to & 0\\ & & \downarrow & & \downarrow 1& & \downarrow \\ 0& \to & k& \stackrel{1}{\to }& k& \to & 0& \to & 0\end{array}$$

So, it was a pretty fun puzzle. Do you see any mistakes in our work? Can you fill in some of the details of enriched Gabriel–Ulmer duality? Any other comments or questions?