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(that is, I'm not sure it's anywhere else called "the successor monad"; it's the monad for pointed sets in Set, I suppose... )
Handy link: monad
I disagree that this is a monad on the (large) poset of all (material) sets under inclusion. The multiplication transformation $\mu$ is not $\subseteq$; that is, $S S X$ is not a subset of $X$.
So we can only say that $S$ is
I’ll rewrite this in a bit.
PS: I like the name ‘successor monad’. I have found one previous usage (PostScript gzip), and it agrees with yours.
Perhaps it should be on a page of its own?
Also, your query is more or less answered at simplex category, or in Categories Work – $\Delta_a$ is generated by the monoid object $[0] \to [1] \leftarrow [2]$.
Perhaps it should be on a page of its own?
Well, the list monad has a page of its own, so the successor monad could too. In the meantime, I’ve redirected it.
your query is more or less answered […]
I’ve put that in.
In the context of monadic functional programming, the successor monad is also called the maybe monad or the “error monad”. Here you’re thinking about the Kleisli category, which as pointed out in the example is equivalent to sets+partial functions – hence the idea that a Kleisli morphism is something that “maybe” gives you an output, or gives you an output unless it fails with an error.
Well, thanks, all! I see there are now entries for Kleisli and Eilenberg-Moore. :-)
Toby,
I disagree … The multiplication transformation $\mu$ is not $\subseteq$ …
OK, I must have missed that seminar, or maybe should I read “Moore Closure”?
Finn,
Also, your query is more or less answered …
While some will agree with the “more” bit, I’m feeling “less” enlightened than I perhaps should… let me muse on that a bit… monad == monoid made of a $1$-cell in $End(*)$ of a $0$-cell in a $2$-cat … maybe I’ll get it soon.
If $X$ is an object of a monoidal category $C$, then $(- \otimes X)$ is a monad if and only if $X$ is a monoid object. The restriction of $S$ to $\Delta_a$ is $(- \oplus [1])$. So I think (but do correct me if I’m wrong) that the content of your observation is essentially that $\Delta_a$ is generated by [1], as explained under Universal properties at simplex category, or CWM VII.5.
The multiplication transformation $\mu$ is not $\subseteq$
OK, I must have missed that seminar, or maybe should I read “Moore Closure”?
Well, you certainly could read Moore closure; I like to think that it’s a very nice article, since I wrote most of it!
But that shouldn’t be necessary. You wrote that the monad was a monad on the poset of (presumably material) sets under inclusion, and in that case every relevant morphism has to be a morphism in this category, and $\mu_X\colon S S X \to X$ is not. So that’s why I changed it to a monad on the category of sets and functions instead.
By the way, your links will work better if you make them like [[Moore closure]]
and [[Kleisli category]]
instead of like [[Moore Closure]]
and [[Kleisli Category]]
.
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