Generalization of cones and cocones

Question

I'm a real newb to category theory and I don't know if it's the right place to ask this trivial question but:

I've learned about limits and colimits (as being special objects in the category of cones/cocones), and I wonder whether there existed some kind of generalization/mix between cones and cocones.

In other words, let's say I have a diagram $J$ with some additional + or - attribute on each node (maybe this can be formalized with a functor $J \rightarrow 2$ where 2 is the discrete two-element category). I'd like to define a generalized cone as a cone where the arrows between the apex $c$ and objects of the base of the diagram can be in both directions(for + objects in $J$, the arrows would go from $c$ to $F(c)$ where $F: J \rightarrow C$, and for - objects in $J$, from $F(c)$ to $c$), plus the resulting commutation diagrams. I tried to define it as a natural transformation (between the functor that sends diagrams objects to $c$ if +, or $F(c)$ otherwise and the functor that does the inverse), but it did not seem to work well with morphisms in $J$ (which did not map to morphisms in the base of the cone).

So my question is: does the category of such generalized cones have a name, and have interesting properties (initial and terminal objects)? I could not find a clear answer on this on the Internet, that did not involve other constructions I'm not familiar with (cocomma categories, for instance).

(I've seen What are generalized cocones/colimits really called? but it seems to me that it is still always going from the base to the apex in that case)

Answer 1

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor, with $\pm:\mathcal{C}\to 2$ a functor assigning $+'s$ and $-'s$ as appropriate. Suppose we have an object $D\in{\bf Ob}_\mathcal{D}$ together with arrows $$\{\pi_C:D\to F(C)\}_{C\in\pm^{-1}(0)}$$ $$\{\pi'_C:F(C')\to D\}_{C\in\pm^{-1}(1)}$$ which commute with arrows in the image of $F$ as appropriate. Let $\mathcal{C}^+$ and $\mathcal{C}^-$ be the full subcategories of $\mathcal{C}$ on $\pm^{-1}(0)$ and $\pm^{-1}(1)$, respectively, and let$$F^+:\mathcal{C}^+\to\mathcal{D}$$ $$F^-:\mathcal{C}^-\to\mathcal{D}$$ be the functors obtained by restricting $F$ as appropriate. Then $\pi$ is a cone to $F^+$ and $\pi'$ is a cocone from $F^-$, so the category you want to consider here is really just $${\sf Cone}(F^+)+{\sf Cocone}(F^-),$$ the coproduct of the cone and cocone categories to the appropriate subfunctors of $F$.

This explains the lack of good 'limit defining' properties for this construction on the overall functor -- we never have terminal or initial objects in this category since it's disconnected unless ${\sf Cone}(F^+)$ or ${\sf Cocone}(F^-)$ is trivial, i.e. unless we label everything with a $+$ or a $-$, in which case it obviously collapses back to a cone or cocone. This construction is consequently unlikely to pop up anywhere in the published literature, and likely remains unnamed.

Alternatively, you could be considering ${\sf Cone}(F^+)\times{\sf Cocone}(F^-)$ which can have limits and colimits without one factor being trivial, but a terminal/initial object would be a terminal/initial pair consisting of a cone and a cocone, respectively, and initial cones/terminal cocones aren't generally what's encountered in practice (I'm not sure why, though).