Contents
Context
Monoidal categories
monoidal categories
With symmetry
With duals for objects
With duals for morphisms
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Category theory
category theory
Concepts
Universal constructions
Theorems
Extensions
Applications
Contents
Idea
The category of functors on a monoidal category canonically inherits itself a monoidal category structure via a categorified convolution product. This holds generally in the context of enriched category theory. This was first observed by (Day 70) and accordingly these monoidal structures are called Day convolution products.
In more detail, just as there is convolution of functions whenever carries the structure of a group, or more generally just the structure of a monoid, so there is convolution of functors whenever the category carries the structure of a monoidal category.
This may be generalized by replacing Set with a more general cocomplete symmetric monoidal category . The technical condition is that is a monoidally cocomplete category: i.e. the tensor product preserves colimits in its two arguments separately; hence that the functors and preserve colimits. This occurs notably when is symmetric closed monoidal (so that these functors are left adjoints).
Definition
Let be a closed symmetric monoidal category with all small limits and colimits.
For a -enriched category, write for the -enriched functor category to , etc.
We discuss two equivalent ways of defining Day convolution
-
In terms of coends
-
In terms of profunctors
In terms of coends
Definition
Let be a small -enriched monoidal category.
Then the Day convolution tensor product on
is given by the following coend
We observe now that Day convolution is equivalently a left Kan extension. This will be key for understanding monoids and modules with respect to Day convolution.
Definition
Let be a small -monoidal category. Its external tensor product is
given by
i.e.
Proposition
The Day convolution product (def. ) of two functors is equivalently the left Kan extension of their external tensor product (def. ) along the tensor product : there is a natural isomorphism
Hence the adjunction unit is a natural transformation of the form
This perspective is highlighted in (MMSS 00, p. 60).
Proof
By prop. we may compute the left Kan extension as the following coend:
Proposition implies the following fact, which is the key for the identification of βfunctors with smash productβ.
Corollary
The operation of Day convolution (def. ) is universally characterized by the property that there are natural isomorphisms
where is the external product of def. .
In terms of profunctors
The Day convolution can also be expressed in terms of profunctors. The tensor product induces a representable profunctor . The above definition can be interpreted to say that if are regarded as profunctors , where is the unit -category, then is the composite of profunctors
A more βglobalβ way to say the same thing is to consider the βevaluationβ functor to be a profunctor . Then the profunctor composite
is a functor , which by exponential transpose gives a functor ; this is the Day convolution product.
The above description in terms of profunctors makes it clear that the construction only depends on the representable profunctor induced by , i.e. on the underlying promonoidal category of . In the original article (Day 70), a stronger form of the convolution is discussed, in which is assumed only to be a promonoidal category.
Before continuing our discussion, we comment a bit on a convention adopted in (Dayβs thesis). To define promonoidal structures, Day used functors of the form , whereas the nLab convention is that a profunctor βΈ is a functor . Following modern usage and (Corner 2016), instead of defining Day convolution for -enriched functors, we do so for -presheaves.
Let be a BΓ©nabou cosmos, be a small -enriched category equipped with the structure of a promonoidal category, and write for (this is called the Einstein notation for profunctors; see (Loregian 2019, Definition 5.1.10)).
This is claimed without proof in (Day 70).
Properties
Closed monoidal structure
Proposition
For a small monoidal -enriched category, the Day convolution product of def. makes
a monoidal category with tensor unit co-represented by the tensor unit of .
This may be deduced fairly abstractly from the above description of Day convolution in terms of profunctors, using the associativity of the promonoidal structure on .
Proposition
For a small monoidal -enriched category, the monoidal category with Day convolution from def. is a closed monoidal category. Its internal hom is given by the end
or equivalently by the end
Proof
First note that the equivalence between the two formulas follows from the Yoneda lemma. (We mention them both, even though the second is undoubtedly simpler, because the more general case of a promonoidal this simplification is unavailable.)
In analogy to the cartesian closed monoidal structure on presheaves we see that if the internal hom in exists at all, (with right adjoint to ) then by the enriched Yoneda lemma and by the end-expression for the hom-objects in the enriched functor category it has to be given by
This exists, by the assumption that is small and that has all small limits. Now to check that this really gives a right adjoint:
While most of this page discusses the covariant Day convolution, the contravariant one interacts conveniently with the Yoneda embedding.
Proposition
The Yoneda embedding constitutes a strong monoidal functor .
Proof
The proof that the tensor unit is respected is dual to the argument in prop. . To see that the tensor product is respected, apply the co-Yoneda lemma twice to get the following natural isomorphism
Monoids with respect to Day convolution
Given any monoidal category then one may consider monoid objects and module objects inside it.
Proposition
For a small (symmetric) monoidal -enriched category, then (commutative) monoid objects in the Day convolution monoidal category of prop. are equivalent to (symmetric) lax monoidal functors :
In functional programming, these monoids give rise to the notion of Applicative.
Moreover, module objects over these monoid objects are equivalent to the corresponding modules over monoidal functors.
This is stated in some form in (Day 70, example 3.2.2). It was highlighted again in (MMSS 00, prop. 22.1). See also MO discussion here.
Proof
A lax monoidal functor is given by natural transformations
satisfying compatibility conditions. Under the natural isomorphism of corollary these are identified with natural transformations
satisfying analogous conditions. This is just the structure of a monoid object on under .
Similarly for module objects and modules over monoidal functors.
(MMSS 00, section 22).
Modules with respect to Day convolution
Definition
For a small monoidal -enriched category, and for a monoid object with respect to Day convolution over , write
for the full subcategory of the category of modules over on those that are free modules and moreover free on objects in (under the Yoneda embedding). Hence the objects of are those of and the hom-objects are
Proposition
For a small -enriched category, and for a monoid object with respect to Day convolution over , then there is an equivalence of categories
between the category of right modules over and the enriched functor category out of the opposite category of that of free -modules from def. .
(MMSS 00, theorem 2.2)
Proof idea
Use the identification from prop. of with a lax monoidal functor and of any -module object as a functor with the structure of a module over a monoidal functor, given by natural transformations
Notice that these transformations have just the same structure as those of the enriched functoriality of of the form
Hence we may unify these two kinds of transformations into a single kind of the form
and subject to certain identifications.
Now observe that the hom-objects of (def. ) have just this structure:
We claim that under this identification, composition in is given by
where
-
the first morphism is, in the integrand, the tensor product of
-
forming the tensor product of hom-objects of with the identity of
-
the monoidal functor incarnation of the monoid structure on ;
-
the second morphism is, in the integrand, given by composition in ;
-
the last morphism is the morphism induced on coends by regarding extranaturality in and separately as a special case of extranaturality in (and then renaming).
It is fairly straightforward to see that, under the above identifications, functoriality under this composition is equivalently functoriality in together with the action property over .
Examples
Example
Let be a discrete category over a set, which is hence a monoid (for instance a group) with product .
Then the Day convolution product is
Notice that if we regard the presheaves and here, assuming they take values in finite sets, as categorifications of -valued functions , where is the cardinality operation on finite sets, then this reproduces precisely the ordinary convolution product of these -valued functions
This uses in particular that for every object the functor
is in this sense the Kronecker delta-function on the set supported at . Precisely because by assumption has only identity morphisms.
Further examples:
References
The concept originates in
- Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 (pdf),
as well as in Dayβs thesis
- Brian Day, Construction of Biclosed Categories, PhD thesis. School of Mathematics of the University of New South Wales, September 1970. Link.
(Note that some unproven statements in (Dayβs report) are proved in (Dayβs thesis) and vice versa.)
The universal property of the Day convolution, in the sense of free monoidal cocompletion, is discussed in
- Geun Bin Im and G. M. Kelly, A universal property of the convolution monoidal structure, J. Pure Appl. Algebra 43 (1986), no. 1, 75-88.
General discussion includes
The application of Day convolution to the construction of symmetric smash products of spectra for highly structured spectra is due to
and for excisive functors due to
- Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)
(see also at functors with smash product).
Day convolution for monoidal bicategories is developed in
- Alexander Corner?, Day convolution for monoidal bicategories, School of Mathematics and Statistics of the University of Sheffield. Available through the White Rose theses database.
Day convolution for (β,1)-categories is discussed in
Other references cited in this page:
On -colimits and Day convolution in the context of enriched -categories: