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Post Closed as "Duplicate" by Johannes Hahn, Gabriel C. Drummond-Cole, John Pardon, John Wiltshire-Gordon, მამუკა ჯიბლაძე
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Rafael
Rafael
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Let $\mathcal{I}$ be a small category and $\mathcal{A}$ an abelian category. If $\mathcal{A}$ is complete (that is, the product of any set of objects exists) and has enough injectives, how can I prove that the functor category $\mathcal{A}^{\mathcal{I}}$ has enough injectives?

I know that if $R$ is a right adjoint functor to an exact functor $L$ then $R$ preserves injective objects. This may be used to solve the problem; there is a functor $R:\mathcal{A}^{\mathcal{I}}\rightarrow\mathcal{A}$$R:\mathcal{A}\rightarrow\mathcal{A}^{\mathcal{I}}$ satisfying the proposition above? If yes, how can I prove it?

Let $\mathcal{I}$ be a small category and $\mathcal{A}$ an abelian category. If $\mathcal{A}$ is complete (that is, the product of any set of objects exists) and has enough injectives, how can I prove that the functor category $\mathcal{A}^{\mathcal{I}}$ has enough injectives?

I know that if $R$ is a right adjoint functor to an exact functor $L$ then $R$ preserves injective objects. This may be used to solve the problem; there is a functor $R:\mathcal{A}^{\mathcal{I}}\rightarrow\mathcal{A}$ satisfying the proposition above? If yes, how can I prove it?

Let $\mathcal{I}$ be a small category and $\mathcal{A}$ an abelian category. If $\mathcal{A}$ is complete (that is, the product of any set of objects exists) and has enough injectives, how can I prove that the functor category $\mathcal{A}^{\mathcal{I}}$ has enough injectives?

I know that if $R$ is a right adjoint functor to an exact functor $L$ then $R$ preserves injective objects. This may be used to solve the problem; there is a functor $R:\mathcal{A}\rightarrow\mathcal{A}^{\mathcal{I}}$ satisfying the proposition above? If yes, how can I prove it?

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Rafael
Rafael
  • 183
  • 1
  • 6

About functor categories and enough injectives

Let $\mathcal{I}$ be a small category and $\mathcal{A}$ an abelian category. If $\mathcal{A}$ is complete (that is, the product of any set of objects exists) and has enough injectives, how can I prove that the functor category $\mathcal{A}^{\mathcal{I}}$ has enough injectives?

I know that if $R$ is a right adjoint functor to an exact functor $L$ then $R$ preserves injective objects. This may be used to solve the problem; there is a functor $R:\mathcal{A}^{\mathcal{I}}\rightarrow\mathcal{A}$ satisfying the proposition above? If yes, how can I prove it?