TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-31

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A monoid object in a small endofunctor category is exactly a monad

Statement

Let $C$ be a small category. Then the following are equivalent for an endofunctor $T : C \to C$ :

$T$ is a monad on $C$ .
In the strict monoidal category $[C, C]$ under composition, $T$ is equipped with morphisms $η : 1_{C} \Rightarrow T, μ : T \circ T \Rightarrow T$ satisfying the associativity and unit equations for a monoid object.

Facts & Assumptions

Given: A small category $C$ and an endofunctor $T : C \to C$ .

[L1]

A monad on $C$ is an endofunctor together with natural transformations $η : 1_{C} \Rightarrow T$ and $μ : T^{2} \Rightarrow T$ satisfying the monad associativity and unit equations (Monad on a category).

[L2]

The phrase "monoid object in the endofunctor category" is used only when that category exists; in particular it is valid for a small source category (The monoid description of a monad requires an endofunctor category).

[L3]

For a small category, $[C, C]$ is strict monoidal under composition (The endofunctor category of a small category is strict monoidal under composition).

Proof

technique · direct

1.1

By [L2] and [L3], the endofunctors of $C$ form a strict monoidal category whose tensor is composition and whose unit is $1_{C}$ . Thus a monoid-object structure on $T$ is exactly data $η : 1_{C} \Rightarrow T$ and $μ : T \circ T \Rightarrow T$ with the usual associative and unital diagrams, because the associator and unitors are identities.

givenL2L3

2.1

The equations described in step 1.1 are precisely $μ \circ T μ = μ \circ μ T$ and $μ \circ T η = 1_{T} = μ \circ η T$ , which are exactly the monad laws in [L1]. Hence every monoid object in $[C, C]$ is a monad.

step 1.1L1

2.2

Conversely, if $(T, η, μ)$ is a monad, then [L1] gives exactly the same two diagrams, so the same data make $T$ a monoid object in $[C, C]$ .

L1step 1.1

3.1

Therefore the two notions are equivalent for small $C$ .

step 2.1step 2.2∎

Depends on

Used by

Dependency tree · two levels

9 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

S. Mac Lane, Categories for the Working Mathematician, Chapter VI.1 (standard reference, not scraped)
E. Riehl, Category Theory in Context, Remark 5.1.2 (standard reference, not scraped)