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Step 1: Define the objects and morphisms in the category.u000Au002D Objects: Pairs u005C((A, Au0027)u005C) where u005C(A

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Let the objects consist of pairs, ( $A, A^{\prime}$ ), where $A$ is a set and $A^{\prime}$ is a subset of $A$. Let a morphism from $\left(A, A^{\prime}\right)$ to $\left(B, B^{\prime}\right)$ consist of a mapping $\varphi$ from set $A$ to set $B$ such that, whenever $a^{\prime}$ is in $A^{\prime}, \varphi\left(a^{\prime}\right)$ is in $B^{\prime}$. Let composition of morphisms be composition of mappings. Prove that this is a category. Discuss monomorphisms, epimorphisms, direct products, and direct sums.

   Let the objects consist of pairs, ( $A, A^{\prime}$ ), where $A$ is a set and $A^{\prime}$ is a subset of $A$. Let a morphism from $\left(A, A^{\prime}\right)$ to $\left(B, B^{\prime}\right)$ consist of a mapping $\varphi$ from set $A$ to set $B$ such that, whenever $a^{\prime}$ is in $A^{\prime}, \varphi\left(a^{\prime}\right)$ is in $B^{\prime}$. Let composition of morphisms be composition of mappings. Prove that this is a category. Discuss monomorphisms, epimorphisms, direct products, and direct sums.

Mathematical physics

Robert Geroch 1st Edition

Chapter 2, Problem 3

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Step 1

- Objects: Pairs $(A, A')$ where $A$ is a set and $A'$ is a subset of $A$. - Morphisms: A morphism from $(A, A')$ to $(B, B')$ is a function $\varphi: A \to B$ such that if $a' \in A'$, then $\varphi(a') \in B'$. Show more…

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Let the objects consist of pairs, ( $A, A^{\prime}$ ), where $A$ is a set and $A^{\prime}$ is a subset of $A$. Let a morphism from $\left(A, A^{\prime}\right)$ to $\left(B, B^{\prime}\right)$ consist of a mapping $\varphi$ from set $A$ to set $B$ such that, whenever $a^{\prime}$ is in $A^{\prime}, \varphi\left(a^{\prime}\right)$ is in $B^{\prime}$. Let composition of morphisms be composition of mappings. Prove that this is a category. Discuss monomorphisms, epimorphisms, direct products, and direct sums.

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