nLab relative monad

Idea

A relative monad is what is to a relative adjunction as a monad is to an adjunction.

In ACU14, the authors proved that a relative monad on a functor $J:\mathbf J \to \mathbf C$ are ‘skew-monoids in the skew-monoidal category $[\mathbf J, \mathbf C]$’ (see below).

Definition

Let $\mathbf J, \mathbf C$ be categories and $J: \mathbf J \to \mathbf C$ be a functor.

Definition

A relative monad $T$ on $J$ is a functor $T:\mathbf J \to \mathbf C$ equipped with

• a unit $\eta_X : J X \to T X$ natural in $X : \mathbf J$,
• a Kleisli extension $(-)^* : \mathbf C(J X, T Y) \to \mathbf C(T X, T Y)$ in both $X,Y:\mathbf J$

such that for every $X,Y,Z:\mathbf J$ and $k:J X \to T Y$,

• (left unital law) $k = k^* \circ \eta_X$,
• (right unital law) $\eta_X^* = 1_{T X}$,
• (associativity law) $(\ell^* \circ k)^* = \ell^* \circ k^*$ for every $\ell : J Y \to T Z$.

Notice that for any $f:X \to Y$ in $\mathbf J$, one has

$$Tf = (\eta_{J Y} \circ J f)^*.$$

In computer science, monads are usually defined by a unit and an extension operator, as above, so that relative monads are very close to plain monads in that regard. Since $[\mathbf J, \mathbf C]$ lacks a monoidal structure (unlike $[\mathbf C, \mathbf C]$), to define relative monads like mathematicians define monads (as monoids in a category of endofunctors) we need to be more clever.

As skew-monoids

A skew-monoidal category is like a monoidal category except its unitors and associators are not necessarily invertible. A monoid in a skew-monoidal category is called a ‘skew-monoid’.

Composition of functors $F,G:\mathbf J \to \mathbf C$ can be defined by first extending $F$ along $J$ and then composing with $G$. The resulting composition product on $[\mathbf J, \mathbf C]$ is coherent but only laxly so, hence the need to appeal to skew-monoidal categories.

Theorem

[Theorem 3.4] Suppose $J:\mathbf J \to \mathbf C$ is such that $\mathrm{Lan}_J : [\mathbf J, \mathbf C] \to [\mathbf C, \mathbf C]$ exists (e.g. if $\mathbf J$ is small and $\mathbf C$ cocomplete). Then $[\mathbf J, \mathbf C]$ is skew-monoidal, with unit $J$ and product $F \circ^J G = (\mathrm{Lan}_J F) \circ G$, and a relative monad is a monoid in $([\mathbf J, \mathbf C], J, \circ^J)$.

When $J:\mathbf J \to \mathbf C$ is a free completion of $\mathbf{J}$ under colimits from some set $\mathcal{F}$ of indexing types, then this skew-monoidal structure on $[\mathbf J, \mathbf C]$ is properly monoidal, since it is equivalent to the $\mathcal{F}$-colimit preserving functors $\mathbf C\to\mathbf C$, and the monoidal structure is just functor composition.

Examples

• A relative monad on the embedding $J:\mathbf{FinSet} \to \mathbf {Set}$ is the same thing as an abstract clone. These are equivalent to finitary monads and single-sorted algebraic theories.

• Any monad $T$ on $\mathbf{C}$ induces a relative monad $TJ$ on $J$, for any $J:\mathbf{J}\to \mathbf{C}$.

• Fixing a category $\mathbb{V}$ with finite products, to give a Freyd category is to give a strong relative monad on the Yoneda embedding $\mathbb{V}\to [\mathbb{V}^{\mathrm{op}},\mathbf{Set}]$.

• The presheaf construction $(\mathbb{C}\mapsto [{\mathbb{C}}^{\mathrm{op}},\mathbf{Set}])$ can be regarded as a relative pseudomonad on the inclusion $\mathbf{Cat}\to \mathbf{CAT}$. (See also Yoneda structures.)

• A monad on $\mathbf C$ with arities in $\mathbf{J}\subseteq \mathbf C$ is the same thing as a relative monad for the embedding $\mathbf{J}\to \mathbf C$. (Here $\mathbf{J}\subseteq \mathbf{C}$ is required to be a dense subcategory, so that to give a functor $\mathbf{J}\subseteq \mathbf{C}$ is to give a functor $\mathbf{C}\to\mathbf{C}$ preserving $J$-absolute colimits.)

Related pages

• relative comonad

References

Discussion with an eye towards monads in computer science is in

• Thorsten Altenkirch, James Chapman, Tarmo Uustalu, Monads need not be endofunctors, Logical methods in computer science (arxiv)