Scoped effects

In the context of algebraic effects and handlers, it is easy to see that handlers have two purposes:

They determine the scope whose effects will be handled.
They define how these effects will be handled, by giving code for each effect.

The theory and practice of scoped effects is about decomposing these two aspects.

Semantics

There seem to be two approaches to the semantics of scoped effects, viz.

through indexed algebras Pirog et al. 2018
through functorial algebras Yang et al. 2022

References

The scoped effects approach was introduced in

Wu, Nicolas, Tom Schrijvers, and Ralf Hinze. “Effect Handlers in Scope.” In Proceedings of the 2014 ACM SIGPLAN Symposium on Haskell. Association for Computing Machinery, 2014. https://doi.org/10.1145/2633357.2633358.

@inproceedings{wu_effect_2014,
	title = {Effect handlers in scope},
	doi = {10.1145/2633357.2633358},
	booktitle = {Proceedings of the 2014 {ACM} {SIGPLAN} Symposium on Haskell},
	publisher = {Association for Computing Machinery},
	author = {Wu, Nicolas and Schrijvers, Tom and Hinze, Ralf},
	date = {2014}
}

Category-theoretic models of scoped effects are given in

Piróg, Maciej, Tom Schrijvers, Nicolas Wu, and Mauro Jaskelioff. “Syntax and Semantics for Operations with Scopes.” In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, 809–18. Association for Computing Machinery, 2018. https://doi.org/10.1145/3209108.3209166.

@inproceedings{pirog_2018,
	title = {Syntax and Semantics for Operations with Scopes},
	doi = {10.1145/3209108.3209166},
	pages = {809--818},
	booktitle = {Proceedings of the 33rd Annual {ACM}/{IEEE} Symposium on Logic in Computer Science},
	publisher = {Association for Computing Machinery},
	author = {Piróg, Maciej and Schrijvers, Tom and Wu, Nicolas and Jaskelioff, Mauro},
	date = {2018},
}

Yang, Zhixuan, Marco Paviotti, Nicolas Wu, Birthe Van Den Berg, and Tom Schrijvers. “Structured Handling of Scoped Effects.” In Programming Languages and Systems, edited by Ilya Sergey, 13240:462–91. Lecture Notes in Computer Science. Cham: Springer International Publishing, 2022. https://doi.org/10.1007/978-3-030-99336-8_17.

@inproceedings{yang_2022,
	location = {Cham},
	title = {Structured Handling of Scoped Effects},
	volume = {13240},
	doi = {10.1007/978-3-030-99336-8_17},
	series = {Lecture Notes in Computer Science},
	pages = {462--491},
	booktitle = {Programming Languages and Systems},
	publisher = {Springer International Publishing},
	author = {Yang, Zhixuan and Paviotti, Marco and Wu, Nicolas and Van Den Berg, Birthe and Schrijvers, Tom},
	editor = {Sergey, Ilya},
	date = {2022},
}