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An initial object of a category $\mathcal{C}$ is an object $I$ in $\mathcal{C}$ such that for every object $X$ in $\mathcal{C}$, there exists precisely one arrow $I → X$.

Let $\mathcal{X}$ and $\mathcal{A}$ be categories and let $U:\mathcal{X} \rightarrow \mathcal{A}$ be a functor. Let $A$ be an object of $\mathcal{A}$ and define $(A\downarrow U)$ to be the category with

  • Objects: pairs $\langle X, h: A \rightarrow UX \rangle$ where $X$ is an object of $\mathcal{X}$ and $h$ is an arrow of $\mathcal{A}$
  • Arrows: $f: \langle X, h: A \rightarrow UX \rangle \rightarrow \langle X', h': A \rightarrow UX' \rangle$ given by arrows $f: X \rightarrow X'$ of $\mathcal{X}$ such that $(Uf)\circ h = h'$ in $\mathcal{A}$.

Question: If there is no initial object in the category $\mathcal{X}$ can there still be an initial object in the category defined above?

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    $\begingroup$ Yes, for the case that $\mathcal{A}$ is the category of one object and exactly one morphism. I'm sure there are also more interesting examples out there. $\endgroup$ – Shaun Ault Apr 10 '12 at 12:15
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Certainly $A\downarrow U$ can have an initial object without $\cal X$ having one. For example, take ${\cal A} = {\cal X}$ and let $U:{\cal X}\to{\cal A}$ be the identity functor. Then $(A,id_A)$ is the initial object.

On the other hand, even if $\cal X$ has an initial object, that cannot guarantee the existence of an initial object in $A\downarrow U$, for every object $A$, because this would imply that $U$ has a left adjoint.

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I don't understand Shaun's example. Nevertheless, it is easy to think of other counterexamples. Take $\mathcal X = \mathcal A$ the category with two objects, two identity morphisms and no other maps, and let $U$ be the identity functor. Then $\mathcal X$ has no initial object, but the comma category is the one-object one-morphism category, which does have one.

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