### The Status of Coalgebra

#### Posted by David Corfield

After my post on coalgebra, I’m still unsure which position to take regarding its status with regard to algebra. Here are some options:

- (1) It’s not a distinction worth making – a coalgebra for $(C, F)$ is an algebra for $(C^{op}, F^{op})$.
- (2) It
*is*a distinction worth making, but there’s plenty of coalgebraic thinking going on – it’s just not flagged as such. - (3) Coalgebra is a small industry providing a few tools for specific situations, largely in computer science, but with occasional uses in topology, etc.

As for (1), this may be true, but we tend not to think of algebras for an endofunctor on Set as coalgebras for the opposite functor on the category of complete atomic Boolean algebras. And why do we seem to privilege Set when it forms the initial algebra for the Powerset functor on classes, while we seem not to do much with Non-well-founded-set, the terminal coalgebra? Why don’t we care more about non-well-founded monoids or comonoids?

As for (2), if it’s correct, then we might expect simple coalgebraic structures to be as common as simple algebraic structures. Now, we might say that the prevalence of the natural numbers in mathematics is due to their being the initial algebra for the endofunctor on Set $F(X) = 1 + X$. If coalgebra is equally as important, we’d expect the terminal coalgebra to show up often too.

So the terminal coalgebra is the set of *extended* natural numbers $\mathcal{N} = {0, 1, 2,...} \union \{\infty\}$. And just as $\mathbb{N}$ comes with an isomorphism

$\langle 0, successor \rangle: 1 + \mathbb{N} \to \mathbb{N}$,

$\mathcal{N}$ comes with a predecessor function:

$pred: \mathcal{N} \to 1 + \mathcal{N},$

where $pred(0) = *, pred(n) = n - 1, pred(\infty) = \infty$.

Do we see this structure anywhere outside of computer science, even if only implicitly? Well, I guess the looping operation from p.13 of Categorification could be seen in this light. This, remember, takes out the lowest level of structure in an $n$-category and reindexes downwards.

So, could we say it sends an $n$-category to a $pred(n)$-category, for $n \in \mathcal{N}$? (It might need a little tweaking at the bottom end for the lowest values of $n$.)

## Re: The Status of Coalgebra

It may be that we see it in topology. We certainly see the

space$\mathbb{N} \cup \{\infty\}$ in topology: it’s the ‘walking’ or ‘generic’ convergent sequence.More precisely, topologize $\mathbb{N} \cup \{\infty\}$ as the one-point compactification of the discrete space $\mathbb{N}$; equivalently, topologize it as the subspace $\{1 - 2^{-n} : n \in \mathbb{N}\} \cup \{1\}$ of the real line. There is a functor $S: \mathbf{Top} \to \mathbf{Set}$ sending a space $X$ to the set $S(X)$ of ‘convergent sequences in $X$’; by a ‘convergent sequence in $X$’, I mean a sequence $(x_n)_{n \in \mathbb{N}}$ in $X$

together witha point $x$ to which the sequence converges. Then $S(X) \cong \mathbf{Top}(\mathbb{N} \cup \{\infty\}, X)$ naturally in $X$. In other words, the functor $S$ is representable, and the representing object is $\mathbb{N} \cup \{\infty\}$.A very basic fact is that a sequence in a space $X$ consists of a point in $X$ (the first element of the sequence) together with a sequence (the original sequence, shifted). So there’s an isomorphism $S(X) \cong U(X) \times S(X)$ natural in $X$, where $U: \mathbf{Top} \to \mathbf{Set}$ is the functor sending a space to its set of points. In other words, there’s a natural isomorphism $S \cong U \times S$. Now $U$ is also representable, with representing object $1$, so we have isomorphisms $\mathbf{Top}(\mathbb{N} \cup \{\infty\}, -) \cong S \cong U \times S \cong \mathbf{Top}(1 + \mathbb{N} \cup \{\infty\}, -)$ or equivalently, by the Yoneda Lemma, an isomorphism $\mathbb{N} \cup \{\infty\} \cong 1 + \mathbb{N} \cup \{\infty\}.$ This is David’s $pred$.