I've been reading some basic category theory, and am slightly confused about the definition of a cocone. I've been looking at the notes here – https://www.dpmms.cam.ac.uk/~jg352/pdf/CategoryTheoryNotes.pdf – specifically pages 15.
A cone with apex $A$ over diagram $D: J \to C$ is defined to be the
object $A$ with a set of maps $\mu_j: A \to D(j)$ s.t. for each
morphism $\alpha: j \to j'$ in $J$, the following diagram commutes: $$
\begin{array}{ccc} &A & \\ \swarrow&&\searrow\\
D(j)&\rightarrow&D(j') \end{array} $$
I'm fine with this, but I'm not quite sure how a cocone is defined.
My question is: which of the following two definitions is correct?
Defn 1: Just reverse all the arrows above, so under the map to the opposite category cocones go tocones and vice versa. Note that the
diagram in the opposite category then has to become a contravariant
functor (as arrows in the diagram are reversed).Defn 2: Same as the definition of the cone, but the object $A$ now lies "underneath" the diagram, and the triangle that has to commute is
$$ \begin{array}{ccc} D(j)&\rightarrow&D(j')\\ \searrow&&\swarrow\\
&A& \end{array} $$which isn't just the same as reversing all the arrows in the triangle
above.
The second definition seems to be what the notes suggest when they say a cocone is a "cone under D", but this seems less nice to me as it doesn't mesh nicely with the opposite category. But I don't see a nice way to define the dual of a cone without having to change the definition of the diagram in the opposite category (and make it into a contravariant functor).
I guess my question is equivalent to:
"do colimits always become limits when you go to the opposite category?".
If the first definition is correct then yes, if the second definition is correct then I don't think so in general.
Thanks for the help.
Best Answer
Regarding the duality between limits and colimits, what you're overlooking is that you can also take the opposite of the index category as well.
It may be helpful to observe that cones and cocones are morphisms in the functor category $C^J$. If $J$ is the index category, then there is the 'diagonal' embedding $ \Delta : C \to C^J $ that sends every object to the constant functor.
Then, for a diagram $J \to C$, a cone with apex $A$ is simply a natural transformation $\Delta(A) \to F$, whereas a cocone is a natural transformation $F \to \Delta(A)$.
And note that the $(C^J)^\circ \cong (C^\circ)^{J^\circ}$.
In fact, if $C$ has all $J$-indexed limits (respectively colimits), then $\lim$ (respectively $\mathop{\mathrm{colim}}$) is the right (respectively left) adjoint of $\Delta$.