Timeline for The category of sets and "stateful" functions
Current License: CC BY-SA 3.0
8 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Mar 28, 2020 at 23:03 | comment | added | goblin GONE | @Thorsten, the idea is to regard $X^*$ as a sheaf, and hence a functor in its own right. Unpacking the definitions, we find that a natural transformation $f : X^* \rightarrow Y^*$ is basically a function $f : X^\mathbb{N} \rightarrow Y^\mathbb{N}$ with the property that for all $\alpha,\beta \in X^\mathbb{N}$, we have $$(\alpha \restriction n) = (\beta \restriction n) \rightarrow (f(\alpha) \restriction n) = (f(\beta) \restriction n).$$ | |
| Mar 25, 2019 at 21:16 | comment | added | LSpice | $\DeclareMathOperator\Hom{Hom}$@Thorsten, I think I must misunderstand your [comment](mathoverflow.net/questions/247891/…). What does "all pairs of sets $(X, Y)$ have the same hom-set" mean? Surely $\Hom_{\mathbf{Set}}(\emptyset, {*}) \ne \Hom_{\mathbf{Set}}({*}, \emptyset)$? | |
| Aug 7, 2017 at 13:27 | comment | added | Thorsten | And in programming, stateful functions are modelled using the Kleisli-Category of the state monad. | |
| Aug 7, 2017 at 13:25 | comment | added | Thorsten | It is unclear what you mean by "natural transform $X*\to Y^*$". Do you mean "natural transformations between the functors from the functor $X\mapsto X^*$ to $Y\mapsto Y^*$? If yes, note that all pairs of sets $(X,Y)$ have the same hom-set. If no, what's a natural transform? | |
| Aug 20, 2016 at 18:51 | comment | added | Musa Al-hassy | This seems to be essentially the theory of polymorphic list operations ---as in functional programming--- and there's a host of categorical results there. As for the internal logic, in computer science, lists are used as naive representations for relations and so looking in that direction may be fruitful. | |
| Aug 20, 2016 at 7:53 | comment | added | მამუკა ჯიბლაძე | As you admit yourself, the question is rather vague, so I don't know how to answer. Still, in case it helps to clarify something - your $\mathbf{Set}^*$ seems to be isomorphic to the coKleisli category of the comonad structure on $\_^*$, with $\varepsilon(x_1,...,x_n)=x_n$ and $\delta(x_1,...,x_n)=((x_1),...,(x_1,...,x_n))$. | |
| Aug 20, 2016 at 7:11 | history | asked | goblin GONE | CC BY-SA 3.0 |