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Suppose I have a category $C$. Then I can construct its dual $C^\bot$, by way of a contravariant equvalence $C \to C^\bot$. Suppose that $C$ has an initial object $0 : C$ so that $0^\bot$ is terminal in $C^\bot$.
What is it called if I construct a new category $\bar{C}$ where there are inclusions $I : C \to \bar{C}$ and $I^\bot : C^\bot \to \bar{C}$ where the inclusions are disjoint except at $I(0)=I^\bot(0^\bot)$?
In case I have stated it in an unclear way, I think $I^\bot(0)$ should be terminal or initial in $\bar{C}$ if and only if $0$ is a zero object in $C$.

Here is the question: Is there a word for the property of an object $X$ like this, for which there are 2 functors which "split" the category $X$ lives in into a part in which $X$ is terminal, and a part where $X$ is initial?

Enid
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Aug 18 '24 at 01:46
  • Sounds a bit like the pushout in $\mathbf{Cat}$ of the two inclusion functors $0 : 1 \to C$ and $0^\bot : 1 \to C^\bot$, but I don't know if that satisfies your desired properties. – Naïm Camille Favier Aug 18 '24 at 08:02
  • Are you assuming that there exist a contravariant self-equivalence between C and its opposite? – fosco Aug 18 '24 at 15:55
  • There is always a contravariant equivalence between $C$ and its opposite, given by the identity functor $C^\bot \to C^\bot$ (or $C \to C$). – Naïm Camille Favier Aug 18 '24 at 16:30
  • Thank you, naïm. I will have to think it through but the pushout you are describing seems like a much cleaner way of defining what I have in mind. – Enid Aug 19 '24 at 20:57

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