nLab arrow

As a synonym for morphism

The term arrow is sometimes used as a synonym for morphism, map, and also for directed edge (in a directed graph or quiver).

Abbreviations for the class of all arrows of a category $\mathsf{C}$ used in the literature include $\mathrm{Arr}(\mathsf{C})$, $\mathrm{Ar}(\mathsf{C})$, and $\mathrm{Mor}(\mathsf{C})$. Note that this is the class of objects of the arrow category of $\mathsf{C}$, and the same notations are sometimes used for that whole category.

In computer science

In computer science it may also refer to a concept generalizing monads, see at arrow (in computer science).

Arrows may be viewed as monads relative to the Yoneda embedding, or equivalently as monads in the bicategory of profunctors.

Related concepts

For the notion in computer science, see also:

• relative monad
• Freyd category

References

For the notion in computer science:

• John Hughes, Generalising monads to arrows, Science of computer programming 37.1-3 (2000): 67-111.

• Thorsten Altenkirch, James Chapman, Tarmo Uustalu, Monads need not be endofunctors, Logical Methods in Computer Science 11 1:3 (2015) 1–40 [arXiv:1412.7148, pdf, doi:10.2168/LMCS-11(1:3)2015]

category: disambiguation