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Claudio Pisani
Claudio Pisani
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Let $C$ be a category with finite hom-sets. Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition). For which categories $C$ does it followsfollow that $X \cong Y$? (Of course, it is true for posets).

A somewhat related question is the following.

Let $C$ be a symmetric monoidal closed category. Suppose that $X$ and $Y$ are objects in $C$ such that $[X,Z]\cong [Y,Z]$ for any Z (with no naturality condition). For which categories $C$ does it followsfollow that $X \cong Y$?

Let $C$ be a category with finite hom-sets. Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition). For which categories $C$ does it follows that $X \cong Y$? (Of course, it is true for posets).

A somewhat related question is the following.

Let $C$ be a symmetric monoidal closed category. Suppose that $X$ and $Y$ are objects in $C$ such that $[X,Z]\cong [Y,Z]$ for any Z (with no naturality condition). For which categories $C$ does it follows that $X \cong Y$?

Let $C$ be a category with finite hom-sets. Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition). For which categories $C$ does it follow that $X \cong Y$? (Of course, it is true for posets).

A somewhat related question is the following.

Let $C$ be a symmetric monoidal closed category. Suppose that $X$ and $Y$ are objects in $C$ such that $[X,Z]\cong [Y,Z]$ for any Z (with no naturality condition). For which categories $C$ does it follow that $X \cong Y$?

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Claudio Pisani
Claudio Pisani
  • 347
  • 1
  • 8

When is an object determined by the number of maps from the other objects?

Let $C$ be a category with finite hom-sets. Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition). For which categories $C$ does it follows that $X \cong Y$? (Of course, it is true for posets).

A somewhat related question is the following.

Let $C$ be a symmetric monoidal closed category. Suppose that $X$ and $Y$ are objects in $C$ such that $[X,Z]\cong [Y,Z]$ for any Z (with no naturality condition). For which categories $C$ does it follows that $X \cong Y$?