internalization and categorical algebra
algebra object (associative, Lie, …)
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A relative monad is much like a monad except that it is not an endofunctor on one category, but more generally a functor between two different 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 , in which case we say that is a monad relative to . Ordinary monads are then the special case where is the identity functor.
Let be categories and be a functor.
A relative monad on is a functor equipped with
a Kleisli extension in both
such that for every and the following equations hold:
(left unitality) ,
(right unitality) ,
(associativity) for every and .
Notice that for any in , one has
(ordinary monads as Kleisli triples/extension systems)
In the special case that and the identity functor, Def. reduces to the definition of ordinary monads, but in the guise known as “Kleisli triples” or “extension systems” which is nominally different from (though equivalent to) the way monads are traditionally presented in category theory/categorical algebra (namely as monoid objects internal to a category of endofunctors), but is exactly the form commonly used for monads in computer science.
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.
In the general case that is distinct from , the functor category lacks a natural monoidal category structure (as opposed to the case of endofunctors ) so that the usual definition of monads as monoids cannot apply — but a suitable “skew” variant works:
The (skew-)composition of functors may be defined by first extending along and then composing with . The resulting composition product on is coherent but only laxly so, hence the need to appeal to skew-monoidal categories:
(ACU14, Thm. 3.4)
Suppose is such that exists (e.g. if is small and cocomplete). Then is skew-monoidal, with unit and product , and a relative monad is a monoid in .
When is a free completion of under colimits from some set of indexing types, then this skew-monoidal structure on is properly monoidal, since it is equivalent to the -colimit preserving functors , and the monoidal structure is just functor composition.
The above definition makes sense even more generally when is a profunctor , i.e., we require
a Kleisli extension natural in both
with essentially the same equations. This generalizes the previous definition by defining the profunctor to be .
(relative monads induced from actual monads) Any actual monad on induces a relative monad on , for any .
This is the content of ACU14, Prop. 2.3 (1)
A concrete instance of this case is spelled out in Exp. below.
A relative monad on the embedding is the same thing as an abstract clone. These are equivalent to finitary monads and single-sorted algebraic theories.
Fixing a category with finite products, to give a Freyd category is to give a strong relative monad on the Yoneda embedding .
The presheaf category-construction may be regarded as a relative pseudomonad on the inclusion . (See also Yoneda structures.)
A monad on with arities in is the same thing as a relative monad for the embedding . (Here is required to be a dense subcategory, so that to give a functor is to give a functor preserving -absolute colimits.)
(linear span)
We spell out the simple but maybe instructive example of the construction which sends a set to the vector space which it spans, i.e. to the -indexed direct sum of some ground field , regarded in -vector spaces.
In detail, for any ground field, consider:
Set;
(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 of base sets, there is a base change adjoint triple of functors
In particular, for the terminal singleton set, the left base change along the unique 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 -linear span,
carries the structure of a monad relative to the functor 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:
Abstract discussion via virtual equipments:
Nathanael Arkor, Dylan McDermott, The formal theory of relative monads, 2023. [arXiv:2302.14014]
Nathanael Arkor, Dylan McDermott, Relative monadicity, 2023. [arXiv:2305.10405]
Other references:
[doi:10.1007/s10485-023-09716-1, arXiv:2007.12982]
Last revised on May 28, 2023 at 16:42:08. See the history of this page for a list of all contributions to it.