The n-Category Café

Of course, I should read again Tom’s introductory paper, and the longer two it introduces.

There’s an unfinished project that bugs me ever since I left university a decade ago. Perhaps somebody has done it. I call it the Total Tensor Calculus (The mathematicians’ and physicists’ calculus get extremely clumsy e.g. when working with Stokes, doing Hilbert space operators, etc. Mathematicians don’t see the forest for all the Lie brackets, Physicists don’t for all the indices. wrrr dada …)

So, my question:
Is anybody aware of the use of shuffle comultiplication on the tensor algebra for higher covariant derivatives of tensor products? (Or, adjointly, the coshuffle multiplication).

The higher tensor product rule is perhaps coalgebraic urknall.

Simplest nontrivial application would be the tensoriality of curvature.

Thanks, David.

The talk (or my half, anyway) was designed to be comprehensible even if you don’t know what ‘flat’ means. More exactly:

• If you don’t know what flat means, it’s fine, as long as you’re willing to assume that the category called ‘$A$’ has finite limits. Flatness then turns into something more familiar. This gives less general but still interesting statements.
• If you do know what flat means, you don’t have to make any assumptions on $A$. You then get more general statements.

If you’re in the first group of people, there’s good news: you don’t need to know what flat means in order to follow the argument leading to the main result. That’s actually the less general result. Apostolos talked about the more general version.

(The main result is that — deep breath — every finitary endofunctor of a locally finitely presentable category has a terminal coalgebra. As the slides explain, this is an old theorem with a new proof, due not to me but to Panagis Karazeris, Apostolos Matzaris and Jiri Velebil. Its main virtue, I think, is that it constructs this terminal coalgebra.)

So, David, while of course I would do nothing to dissuade you from getting on top of the categorical notion of flatness, and while indeed I am envious of the delights that await you, it’s not necessary in order to comprehend the result stated above.

(And your guess about the terminal coalgebra for $X \mapsto 1 + X + X^2$ is correct. It’s the set of all finite or infinite trees with nullary, unary or binary branching.)

Here’s a short explanation of the main idea of the talk.

First we need the right point of view on modules. Let $\mathbf{A}$ and $\mathbf{B}$ be categories. A module $M$ from $\mathbf{A}$ to $\mathbf{B}$, written $$M: \mathbf{A} \Rightarrow \mathbf{B},$$ is a functor $M: \mathbf{A}^{op} \times \mathbf{B} \to \mathbf{Set}$. Really we use a crossed arrow $-+->$ rather than a double arrow $\Rightarrow$, but I can’t figure out how to get that symbol here. (And others call this a $(\mathbf{B}, \mathbf{A})$-bimodule, or a profunctor or distributor from $\mathbf{A}$ to $\mathbf{B}$, or indeed from $\mathbf{B}$ to $\mathbf{A}$). For $a \in \mathbf{A}$ and $b \in \mathbf{B}$, we write $$m: a \Rightarrow b$$ to mean $m \in M(a, b)$. Then the ‘scalar multiplication’ (functorial action) of the module looks like this: $$a' \stackrel{f}{\to} a \stackrel{m}{\Rightarrow} b    gives    a' \stackrel{m f}{\Rightarrow} b$$ (where $f: a' \to a$ is an arrow in $\mathbf{A}$), and similarly $$a \stackrel{m}{\Rightarrow} b \stackrel{g}{\to} b'    gives    a \stackrel{g m}{\Rightarrow} b'.$$ For $M$ to be a functor means that every string of arrows $$a \to \cdot \to   \cdots   \to \cdot \Rightarrow \cdot \to   \cdots   \to \cdot \to b$$ has a well-defined composite $a \Rightarrow b$.

Now we can begin. Take a category $\mathbf{A}$ and a module $M: \mathbf{A} \Rightarrow \mathbf{A}$.

For every functor $X: \mathbf{A} \to \Set$ we obtain by tensor product a new functor $M \otimes X: \mathbf{A} \to \Set$. (Explicitly, given $a \in \mathbf{A}$, an element of the set $(M \otimes X)(a)$ is an equivalence class of pairs $(m: b \Rightarrow a, x \in X(b))$, written $m \otimes x$.) So, the module $M$ induces an endofunctor $M \otimes -$ of $\mathbf{Set}^\mathbf{A}$.

The key point of the talk, as I see it, is this. Suppose that we have a coalgebra $X$ for $M \otimes -$. Then for each element $x \in X(a)$ of the coalgebra, we obtain an infinite ‘complex’ $$\cdots   \stackrel{m_3}{\Rightarrow} a_2 \stackrel{m_2}{\Rightarrow} a_1 \stackrel{m_1}{\Rightarrow} a_0 = a$$ of elements of the module. I should have given you enough information here to figure out how we obtain such a complex, but it’s also in the slides.

This complex is non-canonical — you have to make lots of choices — but under certain flatness hypotheses, it’s canonical up to a certain sensible equivalence relation.

So, each element $x$ of an $(M \otimes -)$-coalgebra gives rise to a complex of elements of the module $M$. You can think of this complex as the ‘observable behaviour’ of $x$. Now, in general we’re supposed to think of terminal coalgebras as consisting of ‘all possible behaviours’. And indeed, if you work in a certain setting where everything is suitably flat, the terminal $(M \otimes -)$-coalgebra really does consist of equivalence classes of complexes as above.

I found one set of hypotheses that makes this true, and used it in my work on self-similarity. Panagis Karazeris, Apostolos Matzaris and Jiri Velebil found another, and realized that this construction could be used to reprove the old theorem on terminal coalgebras that I mentioned above. They’ve also been taking it into new territory.

Now Panagis Karazeris, Apostolos Matzaris, and Jiri Velebil have brought out a paper – Final coalgebras in accessible categories – on this material.