# ___ sum

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### 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}\coprod 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}\coprod X_{2}} and i 2 : X 2 → X 1 ∐ X 2 {\displaystyle i_{2}:X_{2}\to X_{1}\coprod 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}\coprod 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:

### Random information on the term “DIM”:

Dim is the fourth studio album by Japanese rock band the Gazette. It was released on July 15, 2009, in Japan. It includes the three lead up singles: “Guren”, “Leech”, and “Distress and Coma”. The album scored number two on the Oricon Daily Charts and number five on the Oricon Weekly Charts, selling 37,797 copies in its first week.

All lyrics are written by Ruki.

DVD (limited edition only)

The Limited Edition sold at Tower Records also came packaged with a Car Bumper Sticker, 5 Postcards and a Poster

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