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I am looking for solution verification for a rather basic category theory problem.

Observe a category where:

  • objects are triples $(V, v_1, v_2)$ where $V$ is a vector space and $v_1, v_2 \in V$ are distinct vectors
  • morphisms $(U, u_1, u_2) \rightarrow (V, v_1, v_2)$ are linear maps $L : U \rightarrow V$ such that $L(u_1)=v_1$ and $L(u_2)=v_2$.

The task is to find the initial and terminal objects in this category.

I worked out that $(V, v_1, v_2)$ is initial iff $V$ is two-dimensional and $v_1, v_2$ are linearly independent. This works since a linear map $L: (V, v_1, v_2) \rightarrow (X, x_1, x_2)$ given by its values on basis $v_1, v_2$ is unique.

I think that there is no terminal object in this category. Consider an arbitrary object $(V, v_1, v_2)$. If $v_2 = \alpha v_1$ then there are no morphisms $(U, u_1, \beta u_1) \rightarrow (V, v_1, \alpha v_1)$ if $\beta \neq \alpha$, since $L(\beta u_1) = \beta L(u_1) = \beta v_1 \neq \alpha v_1$ (note that $v_1\neq 0$ since $v_1 \neq \alpha v_1$ by object definition). If though $v_1, v_2$ are linearly independent, then again a morphism $(U, u_1, \beta u_1) \rightarrow (V, v_1, v_2)$ would immediately imply $L(\beta u_1) = \beta v_1 = v_2$, which is not the case.

I am not sure about my solution only because the problem states "to find" the object and not to prove the contrary... Also I'd be interested in other approaches, if any.

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1 Answer 1

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The forgetful functor is representable in this category. Therefore the underlying set of the terminal object, if it exists, must be a 1-element set. This contradicts the requirement that the two chosen elements be distinct.

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