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I have come across the assertion that a terminal object $T$ in a category $\mathcal{C}$ can be understood as the limit of an empty diagram, i.e. a diagram $D:\mathcal{J}\to\mathcal{C}$ whose index category $\mathcal{J}$ is the empty category.

Typically, in such a formulation, the limit object $T$ is understood the image of a constant functor $\Delta_T:\mathcal{J}\to\mathcal{C}$ that maps every object of $\mathcal{J}$ to $T$, and every arrow of $\mathcal{J}$ to $1_T$.

But if $\mathcal{J}$ is empty, then the object part of $\Delta T$ is the empty function, so its image is $\varnothing$. This means that $T$ cannot be in the image of this function.

Is this a case of "special dispensation"? In other words, is the vertex of the cone in this case simply defined to be the object $T$?

kjo
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1 Answers1

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A cone for $D$ is defined to be an object $c \in \mathcal C$ together with a natural transformation $\Delta(c) \Rightarrow D$. It is not defined to be the image of a constant functor; as you say, this would not make sense when $\mathcal J$ is empty. In other words, a cone always specifies the apex $c$ as part of the data, rather than inferring it from the functor. (If you have come across a reference that suggests that the object is inferred from the functor, I think it has been written confusingly.)

varkor
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