With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
The category of presheaves over the opposite of 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 convolutions products.
More in 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 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).
For monoidal categories
For a small monoidal category and for two presheaves on , their Day convolution product is the presheaf given by the coend
Let be a good symmetric monoidal category for -enriched category theory (in particular having all small colimits, e.g. a Benabou cosmos). For a -enriched category, write for the -enriched functor category to , etc.
For a small monoidal -enriched category, the Day convolution product on is the functor
given by the coend
We may think of this equivalently as a Kan extension:
For a monoidal -enriched category, its external tensor product is the -functor
The Day convolution product, def. 1, is equivalently given by left Kan extension (along the tensor product ) of the result of the external tensor product, def. 2: there is a natural isomorphism
This perspective is highlighted in (MMSS 00, p. 60).
The general formula for pointwise Kan extension via coends (here) says that left Kan extension of any along some is given by
In our case
and hence the general formula here becomes
Day convolution , def. 1, is universally characterized by the property that there are natural isomorphisms
where is the external product of def. 2.
for the -Yoneda embedding, so that for any object, is the corepresented functor .
For a small monoidal -enriched category, the Day convolution product of def. 1 makes
a closed monoidal category with tensor unit co-represented by the tensor unit of .
To see that is the tensor unit for , use the Fubini theorem for coends and then twice the co-Yoneda lemma to get for any that
In the original article (Day 70), a stronger form of the convolution is discussed, in which is assumed only to be a promonoidal category.
Let be a Benabou cosmos, and a small -enriched category.
There is an equivalence of categories between the category of pro-monoidal structures on with strong pro-monoidal functors between them and the category of biclosed monoidal structures on with strong monoidal functors between them.
This is claimed without proof in (Day 70).
Let be the Yoneda embedding.
With the tensor unit of , then the presheaf that it represents is a tensor unit for the Day convolution product.
Using the co-Yoneda lemma on the two coends we have
For a small monoidal category, regard the category of presheaves as a monoidal category with tensor product the Day convolution product and unit the unit of under the Yoneda embedding .
is a closed monoidal category;
the Yoneda embedding constitutes a strong monoidal functor .
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 Yoneda lemma it has to be given by
Monoids with respect to Day convolution
Given any monoidal category then one may consider monoid objects and module objects inside it.
For a small (symmetric) monoidal -enriched category, then (commutative) monoid objects in the Day convolution monoidal category of prop. 2 are equivalent to (symmetric) lax monoidal functors :
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.
A lax monoidal functor is given by natural transformations
satisfying compatibility conditions. Under the natural isomorphism of corollary 1 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
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. Hence the objects of are those of and the hom-objects are
For a small monoidal -enriched category, and for a monoid object with respect to Day convolution over , then there is an equivalence of categories
between the category of modules over and the enriched functor category out of the opposite category of that of free -modules from def. 3.
(MMSS 00, theorem 2.2)
Use the identification from prop. 5 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
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 transformarmations in a single kind of the form
and subject to certain identifications.
By comparison with def. 3, this is just the form of the functoriality of an enriched functor over .
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.
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)
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 (∞,1)-categories is discussed in