Quotients of Commutative, Affine and Relevant Monads

We have seen that nice properties such as being commutative, affine, or relevant can be transferred to strong submonads. A natural question to ask is whether something similar applies to quotients? In fact, we can apply the tricks we’ve already seen to get straight to some answers.

Commutative Monads

As strong monad morphisms commute with double strengths, if

φ:(𝕋,𝗌𝗍)⇒(ℚ,𝗌𝗍)\varphi : (\mathbb{T}, \mathsf{st}) \Rightarrow (\mathbb{Q}, \mathsf{st})

is a strong monad morphism, and \mathbb{T} is commutative, we can immediately show

𝖽𝗌𝗍ℚ⋅φ⊗φ=𝖽𝗌𝗍ℚ′⋅φ⊗φ\mathsf{dst}^{\mathbb{Q}} \cdot \varphi \otimes \varphi = \mathsf{dst}^{\mathbb{Q}’} \cdot \varphi \otimes \varphi

If \varphi \otimes \varphi is component-wise epimorphic, then

𝖽𝗌𝗍ℚ=𝖽𝗌𝗍ℚ′\mathsf{dst}^{\mathbb{Q}} = \mathsf{dst}^{\mathbb{Q}’}

and \mathbb{Q} is commutative. Even if we restrict attention to where the monoidal structure is products, this condition is not as straightforward as that for monomorphisms. Fortunately there are many special cases where \varphi being a component-wise epimorphism implies \varphi \times \varphi is. In particular, this works nicely for \mathsf{Set}-monads. As all natural transformation are strong in that case, we have

Quotients of commutative monads are commutative.

Affine, Relevant and Cartesian Monads

If

φ:(𝕋,𝗌𝗍)⇒(ℚ,𝗌𝗍)\varphi : (\mathbb{T}, \mathsf{st}) \Rightarrow (\mathbb{Q}, \mathsf{st})

is a strong monad morphism, and \mathbb{T} is affine, then using the same calculational properties as we did for submonads, but in reverse, we can conclude

⟨ℚπ1,ℚπ2⟩⋅𝖽𝗌𝗍ℚ⋅φ×φ=φ×φ\langle \mathbb{Q}\pi_1, \mathbb{Q}\pi_2 \rangle \cdot \mathsf{dst}^{\mathbb{Q}} \cdot \varphi \times \varphi = \varphi \times \varphi

and so if \varphi \times \varphi is component-wise epimorphic

⟨ℚπ1,ℚπ2⟩⋅𝖽𝗌𝗍ℚ=𝗂𝖽\langle \mathbb{Q}\pi_1, \mathbb{Q}\pi_2 \rangle \cdot \mathsf{dst}^{\mathbb{Q}} = \mathsf{id}

If we restrict attention to \mathsf{Set}-monads, using the result of the previous section, we can conclude

Quotients of affine monads are affine.

A very similar argument allows us to conclude that for \mathsf{Set}-monads

Quotients of relevant monads are relevant.

Combining both these observations gives

Quotients of Cartesian monads are Cartesian.

Algebraic Interpretation

As usual, it pays to see if we use algebraic intuition to justify our conclusions. If we consider \mathsf{Set}-monads presented by operations and equations, being commutative, affine, relevant or Cartesian requires that certain equations between terms hold. We can think of quotient monads as imposing extra equations between terms, so it is unsurprising quotient monads continue to have these nice properties.

Summary

The conclusions above are less general than for submonads. This is because epimorphisms do not interact as nicely with products as monomorphisms do. In the case of \mathsf{Set}-monads everything was about as well-behaved as we could possibly hope.

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