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Definitions
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A relative monad $T \colon A \to E$ is much like a monad except that its underlying functor is not required to be an endofunctor: rather, it can be an arbitrary functor between categories. To even formulate such a notion, (for instance the definition of the unit), the two categories have to be related somehow, typically via a specified comparison functor $J \colon A \to E$, in which case we say that $T$ is a monad relative to $J$.
Ordinary monads are the special case of $J$-relative monads where $J$ is the identity functor.
In generalisation of the relation between adjunctions and monads, relative monads are related to relative adjunctions. Dually, relative comonads are related to relative coadjunctions.
Let $J \colon A \to E$ be a functor between categories, the root.
The following definition is a variation of the formulation of a monad in extension form:
A $J$-relative monad $T$ [ACU15, Def. 2.1] comprises:
a function $T \colon |A| \to |E|$, the underlying functor;
for each object $X \in |A|$, a morphism $\eta_X \colon J X \to T X$ in $E$, the unit;
for each morphism $f \colon J X \to T Y$ in $A$, a morphism $f^\dagger \colon T X \to T Y$ in $E$, the extension operator,
such that, for each $X, Y, Z \in |A|$, the following equations hold:
$f = f^\dagger \circ \eta_X$ for each $f \colon J X \to T Y$ (left unitality);
$(\eta_X)^\dagger = 1_{T X}$ (right unitality);
$(g^\dagger \circ f)^\dagger = \g^\dagger \circ f^\dagger$ for each $f \colon J X \to T Y$ and $g \colon J Y \to T Z$.
It follows that $T$ is canonically equipped with the structure of a functor. For each $f \colon X \to Y$ in $A$:
and that the unit $\eta$ and the extension operator $({-})^\dagger$ are then natural transformations.
In particular, in the special case that $J$ is the identity functor, Def. reduces to the definition of monad in extension form.
Monads are, by definition, monoids in monoidal categories of endofunctors. It is similarly possible to present relative monads as monoids in categories of functors. However, generally speaking, arbitrary functor categories are not monoidal. However, given a fixed functor $J \colon A \to E$, the functor category $[A, E]$ may frequently be equipped with skew-monoidal structure. The notion of a skew-monoidal category is like that of a monoidal category except that the unitors and associators are not necessarily invertible. Monoids may be defined in a skew-monoidal category analogously as to in a monoidal category, and a monoid in $[A, E]$ (equipped with the skew-monoidal structure induced by $J$) is precisely a $J$-relative monad.
(ACU15, Thm. 3.4)
Let $J \colon A \to E$ be a functor for which $\mathrm{Lan}_J \colon [A, E] \to [E, E]$ exists (e.g. if $A$ is small and $E$ cocomplete). Then $[A, E]$ admtis a skew-monoidal structure, with unit $J$ and tensor $F \circ^J G = (\mathrm{Lan}_J F) \circ G$, and a relative monad is precisely a monoid in $([A, E], J, \circ^J)$.
When $J \colon A \to E$ is a free completion of $A$ under a class $\mathcal{F}$ of small colimits, then this skew-monoidal structure on $[A, E]$ is properly monoidal, since it is equivalent to the $\mathcal{F}$-colimit preserving functors $E \to E$, and the monoidal structure is just functor composition.
More generally, if $\mathrm{Lan}_J$ does not exist, we may still define a skew-multicategory? structure on $[A, E]$. Thus, relative monads are always monoids.
(AM, Thm. 4.16)
Let $J \colon A \to E$ be a functor. Then $[A, E]$ admits a unital skew-multicategory structure, and a relative monad is precisely a monoid therein.
This skew-multicategory structure is representable just when $\mathrm{Lan}_J$ exists, recovering the result of ACU15. When $J$ is a dense functor, the above theorem simplifies.
(AM, Cor. 4.17)
Let $J \colon A \to E$ be a dense functor. Then $[A, E]$ admits a unital multicategory structure, and a relative monad is precisely a monoid therein.
An alternative useful perspective on relative monads is the following.
(AM, Thm. 4.22)
Let $J \colon A \to E$ be a dense functor. A $J$-relative monad is precisely a monad in the bicategory of distributors whose underlying 1-cell is of the form $E(J, T)$ for some functor $T \colon A \to E$.
The above definition makes sense even more generally when $J$ is a distributor $E ⇸ A$, i.e. a functor $\mathbf A^{op} \times E \to Set$. Explicitly, we ask for:
a functor $T \colon A \to E$;
a unit $\eta_X \in J(X, T X)$ for each $X \in |A|$, natural in $X \in |A|$ (equivalently, an element of the end, $\eta \in \int_{X \in |A|}J(X, T X)$);
an extension operator $(-)^\dagger \colon J(X, T Y) \to \mathbf C(T X, T Y)$ natural in $X,Y \in |A|$
with essentially the same equations. We recover the previous definition by taking the corepresentable distributor $E(J-,=)$. See Remark 4.24 of AM24.
(relative monads induced from actual monads)
Given
a functor
$J \,\colon\, A \to E$,
a $\;$ monad
$T \,\colon\, E \to E$
with
unit $\;$ $\eta^T_{(-)}$
extension operator $\;$ $bind^T$
the composite
defines a $A$-relative monad (Def. ) with
unit
$\eta^{T J}_X \,\coloneqq\, \eta^T_{J(X)}$
extension operator
$bind^{T J}\big( J(X) \overset{k}{\to} T \circ J(Y) \big) \;\coloneqq\; bind^T\big( J(X) \overset{k}{\to} T \circ J(Y) \big) \,.$
This example is stated in ACU15, Prop. 2.3 (1). More generally, we can precompose any relative monad with a functor to obtain a new relative monad: see Proposition 5.36 of AM24.
The required conditions on the relative monad structure $T \circ J$ immediately reduce to those of the monad structure of $T$:
Given
we have
left unitality:
right unitality:
associativity:
A concrete instance of Exp. is spelled out in Exp. below.
A relative monad on the embedding $J \colon \mathbf{FinSet} \to \mathbf {Set}$ is equivalent to an abstract clone. These are equivalent to finitary monads and single-sorted algebraic theories.
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 category-construction $(\mathbb{C}\mapsto [{\mathbb{C}}^{\mathrm{op}},\mathbf{Set}])$ may 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 $A\subseteq \mathbf C$ is the same thing as a relative monad for the embedding $A\to \mathbf C$. (Here $A\subseteq E$ is required to be a dense subcategory, so that to give a functor $A\subseteq E$ is to give a functor $E\toE$ preserving $J$-absolute colimits.)
(linear span)
We spell out the simple but maybe instructive example of the construction which sends a set $B$ to the vector space which it spans, i.e. to the $B$-indexed direct sum of some ground field $\mathbb{K}$, regarded in $\mathbb{K}$-vector spaces.
In detail, for $\mathbb{K}$ any ground field, consider:
$A \coloneqq$ Set;
$E \coloneqq \int_{B \colon Set} Vect_B$ (or “VectBund”, for short, see there for more) the category of indexed sets of vector spaces – hence of vector bundles over sets (i.e. over discrete topological spaces)
with possibly base-changing vector bundle maps between them:
the relativization functor given by sending a set to the trivial tensor unit-bundle over it:
Notice that for each map $f \;\colon\; B \to B'$ of base sets, there is a base change adjoint triple of functors
In particular, for $S' = \ast$ the terminal singleton set, the left base change along the unique $p_X \colon S \to \ast$ is the operation which forms the direct sum of the (fiber-)vector spaces in the bundle, and regards the resulting vector space as a bundle over the point:
In view of this, we claim that the functor
which may be understood as sending a set to its $\mathbb{K}$-linear span,
carries the structure of a monad relative to the functor $J$ from (1) with
unit given by
Kleisli extension given by
This specific example may be understood as a special case of the general situation of relative monads induced from an actual monad (Exmp. ): Here the actual monad in question is:
This monad is in fact the reflective localization of the reflective subcategory-embedding of plain VectorSpaces into bundled/parameterized vector spaces:
The concept was introduced, in the context of monads in computer science, in:
A comprehension development in the context of formal category theory may be found in:
Nathanael Arkor, Dylan McDermott, The formal theory of relative monads, Journal of Pure and Applied Algebra 107676. (2024) [arXiv:2302.14014, doi:10.1016/j.jpaa.2024.107676]
Nathanael Arkor, Dylan McDermott, Relative monadicity, 2023. [arXiv:2305.10405]
Nathanael Arkor, Dylan McDermott, The pullback theorem for relative monads (2024) [arXiv:2404.01281]
Exposition:
On distributive laws for relative monads:
Last revised on October 22, 2024 at 12:33:15. See the history of this page for a list of all contributions to it.