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SUM.
Last seen on: Daily Crossword Club Crossword Sunday, 23 April 2023
Random information on the term “SUM”:
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the “least specific” object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.
Let C {\displaystyle C} be a category and let X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} be objects of C . {\displaystyle C.} An object is called the coproduct of X 1 {\displaystyle X_{1}} and X 2 , {\displaystyle X_{2},} written X 1 ⊔ X 2 , {\displaystyle X_{1}\sqcup X_{2},} or X 1 ⊕ X 2 , {\displaystyle X_{1}\oplus X_{2},} or sometimes simply X 1 + X 2 , {\displaystyle X_{1}+X_{2},} if there exist morphisms i 1 : X 1 → X 1 ⊔ X 2 {\displaystyle i_{1}:X_{1}\to X_{1}\sqcup X_{2}} and i 2 : X 2 → X 1 ⊔ X 2 {\displaystyle i_{2}:X_{2}\to X_{1}\sqcup X_{2}} satisfying the following universal property: for any object Y {\displaystyle Y} and any morphisms f 1 : X 1 → Y {\displaystyle f_{1}:X_{1}\to Y} and f 2 : X 2 → Y , {\displaystyle f_{2}:X_{2}\to Y,} there exists a unique morphism f : X 1 ⊔ X 2 → Y {\displaystyle f:X_{1}\sqcup X_{2}\to Y} such that f 1 = f ∘ i 1 {\displaystyle f_{1}=f\circ i_{1}} and f 2 = f ∘ i 2 . {\displaystyle f_{2}=f\circ i_{2}.} That is, the following diagram commutes: