(i) Assuming that coproducts exist, prove commutativity: A ⊔B ≅B ⊔A . (ii) Assuming that coprodu

step 1 So the question is taken from determinants and matrices and we have to prove that the following

(i) Assuming that coproducts exist, prove commutativity: A...

(i) Assuming that coproducts exist, prove commutativity:
$$
A \sqcup B \cong B \sqcup A .
$$
(ii) Assuming that coproducts exist, prove associativity:
$$
A \sqcup(B \sqcup C) \cong(A \sqcup B) \sqcup C .
$$

Key Concepts

Isomorphism

An isomorphism in category theory is a morphism that has an inverse, meaning two objects are essentially the same in the context of the category. In the context of coproducts, showing two coproduct constructions are isomorphic confirms that their structure as categorical sums does not depend on the order or grouping of objects.

Associativity and Commutativity of Coproducts

These refer to the fact that, up to canonical isomorphism, the coproduct operation is independent of the order (commutativity) and grouping (associativity) of objects. Using the universal property, one can construct natural isomorphisms between different arrangements of coproducts, demonstrating that A ? B is isomorphic to B ? A, and A ? (B ? C) is isomorphic to (A ? B) ? C.

Coproduct

A coproduct in category theory is an object that serves as a 'sum' of other objects along with inclusion morphisms, satisfying a specific universal property. For any two objects, the coproduct is equipped with injections such that any pair of morphisms from these objects to another object factors uniquely through the coproduct.

Universal Property

The universal property is the defining characteristic of a coproduct, stating that for any object receiving morphisms from each summand, there exists a unique morphism from the coproduct that factors these morphisms. This property allows the construction of canonical morphisms used to prove isomorphisms, hence commutativity and associativity.

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