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Kleisli category of a monad
Definition
For a monad on , the Kleisli category has the same objects as and hom-collections
The identity at is , and composition is . The category laws are established before this definition in Kleisli composition is associative and unital.
Depends on
Used by
- A Kleisli composite for the list monad computed by substitution and concatenation Example
- The Kleisli adjunction for the maybe monad is monadic but not strictly monadic Example
- The Kleisli category of the maybe monad is the category of sets and partial functions Example
- FALSE: The Kleisli and Eilenberg–Moore categories are equivalent for every monad False statement
- The Kleisli adjunction induces the given monad Theorem
Dependency tree · two levels
2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, 2nd ed., Definition 5.2.10 (standard reference, not scraped)
- B. Richter, From Categories to Homotopy Theory, Definition 6.3.1 (standard reference, not scraped)