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In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970cite-ref-1[1] in the general context of enriched functor categories.

Day convolution gives a symmetric monoidal structure on H o m ( C , D ) {\displaystyle \mathrm {Hom} (\mathbf {C} ,\mathbf {D} )} for two symmetric monoidal categories C , D {\displaystyle \mathbf {C} ,\mathbf {D} } .

Another related version is that Day convolution acts as a tensor product for a monoidal category structure on the category of functors [ C , V ] {\displaystyle [\mathbf {C} ,V]} over some monoidal category V {\displaystyle V} .

• Definition

• First version

• Enriched version

• References

• External links

Given F , G : : C → → D {\displaystyle F,G\colon \mathbf {C} \to \mathbf {D} } for two symmetric monoidal C , D {\displaystyle \mathbf {C} ,\mathbf {D} } , we define their Day convolution as follows.

It is the left kan extension along C × × C → → ⊗ ⊗ C {\displaystyle \mathbf {C} \times \mathbf {C} \to ^{\otimes }\mathbf {C} } of the composition C × × C → → F , G D × × D → → ⊗ ⊗ D {\displaystyle \mathbf {C} \times \mathbf {C} \to ^{F,G}\mathbf {D} \times \mathbf {D} \to ^{\otimes }\mathbf {D} }

Thus evaluated on an object O ∈ ∈ C {\displaystyle O\in \mathbf {C} } , intuitively we get a colimit in D {\displaystyle \mathbf {D} } of F ( x ) ⊗ ⊗ G ( y ) {\displaystyle F(x)\otimes G(y)} along approximations of O ∈ ∈ C {\displaystyle O\in \mathbf {C} } as a pure tensor x ⊗ ⊗ y {\displaystyle x\otimes y}

Left kan extensions are computed via coends, which leads to the version below.

Let ( C , ⊗ ⊗ c ) {\displaystyle (\mathbf {C} ,\otimes _{c})} be a monoidal category enriched over a symmetric monoidal closed category ( V , ⊗ ⊗ ) {\displaystyle (V,\otimes )} . Given two functors F , G : : C → → V {\displaystyle F,G\colon \mathbf {C} \to V} , we define their Day convolution as the following coend.cite-ref-2[2]

F ⊗ ⊗ d G = ∫ ∫ x , y ∈ ∈ C C ( x ⊗ ⊗ c y , − − ) ⊗ ⊗ F x ⊗ ⊗ G y {\displaystyle F\otimes _{d}G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes _{c}y,-)\otimes Fx\otimes Gy}

If ⊗ ⊗ c {\displaystyle \otimes _{c}} is symmetric, then ⊗ ⊗ d {\displaystyle \otimes _{d}} is also symmetric. We can show this defines an associative monoidal product:

( F ⊗ ⊗ d G ) ⊗ ⊗ d H ≅ ≅ ∫ ∫ c 1 , c 2 ( F ⊗ ⊗ d G ) c 1 ⊗ ⊗ H c 2 ⊗ ⊗ C ( c 1 ⊗ ⊗ c c 2 , − − ) ≅ ≅ ∫ ∫ c 1 , c 2 ( ∫ ∫ c 3 , c 4 F c 3 ⊗ ⊗ G c 4 ⊗ ⊗ C ( c 3 ⊗ ⊗ c c 4 , c 1 ) ) ⊗ ⊗ H c 2 ⊗ ⊗ C ( c 1 ⊗ ⊗ c c 2 , − − ) ≅ ≅ ∫ ∫ c 1 , c 2 , c 3 , c 4 F c 3 ⊗ ⊗ G c 4 ⊗ ⊗ H c 2 ⊗ ⊗ C ( c 3 ⊗ ⊗ c c 4 , c 1 ) ⊗ ⊗ C ( c 1 ⊗ ⊗ c c 2 , − − ) ≅ ≅ ∫ ∫ c 1 , c 2 , c 3 , c 4 F c 3 ⊗ ⊗ G c 4 ⊗ ⊗ H c 2 ⊗ ⊗ C ( c 3 ⊗ ⊗ c c 4 ⊗ ⊗ c c 2 , − − ) ≅ ≅ ∫ ∫ c 1 , c 2 , c 3 , c 4 F c 3 ⊗ ⊗ G c 4 ⊗ ⊗ H c 2 ⊗ ⊗ C ( c 2 ⊗ ⊗ c c 4 , c 1 ) ⊗ ⊗ C ( c 3 ⊗ ⊗ c c 1 , − − ) ≅ ≅ ∫ ∫ c 1 , c 3 F c 3 ⊗ ⊗ ( G ⊗ ⊗ d H ) c 1 ⊗ ⊗ C ( c 3 ⊗ ⊗ c c 1 , − − ) ≅ ≅ F ⊗ ⊗ d ( G ⊗ ⊗ d H ) {\displaystyle {\begin{aligned}&(F\otimes _{d}G)\otimes _{d}H\\[5pt]\cong {}&\int ^{c_{1},c_{2}}(F\otimes _{d}G)c_{1}\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2}}\left(\int ^{c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\right)\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{2}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&\int ^{c_{1},c_{3}}Fc_{3}\otimes (G\otimes _{d}H)c_{1}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&F\otimes _{d}(G\otimes _{d}H)\end{aligned}}}

cite-note-11. ↑ citerefday1970Day, Brian (1970). "On closed categories of functors". Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics. 139: 1–38.

cite-note-22. ↑ citerefloregian2021Loregian, Fosco (2021). (Co)end Calculus. p. 51. arXiv:1501.02503. doi:10.1017/9781108778657. ISBN 9781108778657. S2CID 237839003.

• Day convolution at the nLab