# Monads need not be endofunctors

arXiv: https://arxiv.org/abs/1412.7148 · HTML: https://mlsanity.com/paper/1412.7148/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2014.

## Built on by (most cited first, 2 of 2)

- [NeSyCat Torch: A Differentiable Tensor Implementation of Categorical Semantics for Neurosymbolic Learning](https://mlsanity.com/paper/2606.19279/) (2026)
- [Sample-Smooth Spaces: A Convenient Category for Differentiable Probabilistic Programming](https://mlsanity.com/paper/2609.26270/) (2026)

## Abstract (arXiv)

We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed lambda-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads.