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Suppose we have a small category $\mathcal{I}$, a diagram $D : \mathcal{I} \to \mathcal{C}$, and a functor $F : \mathcal{C} \to \mathcal{D}$. De know that the functor $\hat{F} : \text{Cone}_{\mathcal{C}}(D) \to \text{Cone}_\mathcal{D}(F \circ D)$ preserves the terminal object(s). In categories with zero objects (like Top), can we not use the $(—)^\text{op}$-functor for $F$ to show that the completeness of Top implies its cocompleteness..?

Martin Brandenburg
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Jos van Nieuwman
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    $\text{Top}$ doesn't have a zero object, and it is not true that if a category with a zero object is complete then it is cocomplete, although counterexamples are a bit tricky to construct IIRC. I don't understand what your proposed argument is at all; can you write it out in more detail? – Qiaochu Yuan Jan 29 '23 at 20:43
  • I was so foolish to think that for a category to have zero objects it must have both initial and terminal objects. Of course, these must be the same, so it would never work. I'm just trying to find a shortcut in showing that Top is cocomplete, whilst just having shown it is complete. (But as you can tell, I know very little about CT just yet.. :p) – Jos van Nieuwman Jan 29 '23 at 20:49
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    I'm not aware of any such shortcut and you don't need one anyway. You just need to show that it has coproducts and coequalizers and both of those are computed the same way as in $\text{Set}$, with more or less obvious topologies. – Qiaochu Yuan Jan 29 '23 at 21:08
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    Incidentally, a small category is complete iff it is cocomplete. (This result is not as interesting as it sounds.) – Zhen Lin Jan 29 '23 at 22:09
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    You may be interested in the concept of a topological category. Every such category is complete and cocomplete, by almost the same proof (one uses initial lifts, the other uses final lifts). And of course (Top) is the standard example. – Martin Brandenburg Jan 30 '23 at 00:07

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