Denní konvoluce - Day convolution
V matematice, konkrétně v teorie kategorií , Denní konvoluce je operace na funktory které lze považovat za kategorizováno verze konvoluce funkce . Poprvé byl představen Brianem Dayem v roce 1970 [1] obohacený kategorie funktorů . Denní konvoluce funguje jako tenzorový produkt pro a monoidní kategorie struktura kategorie funktorů [ C , PROTI ] { displaystyle [ mathbf {C}, V]} PROTI { displaystyle V}
Definice Nechat ( C , ⊗ C ) { displaystyle ( mathbf {C}, otimes _ {c})} ( PROTI , ⊗ ) { displaystyle (V, otimes)} F , G : C → PROTI { displaystyle F, G colon mathbf {C} na V} coend .[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 } Li ⊗ C { displaystyle otimes _ {c}} ⊗ d { displaystyle otimes _ {d}}
( 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}, -) [5 bodů ] 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} další _ {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}, -) [5 bodů] cong {} & F otimes _ {d} (G otimes _ {d} H) end {zarovnáno}}} Reference ^ Day, Brian (1970). "O uzavřených kategoriích funktorů". Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics . 139 : 1–38. ^ Loregian, Fosco (2015). „Toto je (spolu) konec, můj jediný (spolu) přítel“. str. 51. arXiv :1501.02503 math.CT ]. externí odkazy
![{ displaystyle [ mathbf {C}, V]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cefe6959c0bae6a141dc8ba8444b96548c91fcd)
![{ 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}, -) [5 bodů ] 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} další _ {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}, -) [5 bodů] cong {} & F otimes _ {d} (G otimes _ {d} H) end {zarovnáno}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dfe1b50869adba9c90fbf51aa653f905779f14b)