As far as I understand, cone is a pair of some object $X \in Obj(\mathcal{C})$ (which can be viewed as a $\Delta_X$ – constant functor from some other category to the $\mathcal{C}$) and a set of morphisms in $\mathcal{C}$ obeying a known requirement, which I ignore here.
So, by the definition cone is $(X \in Obj(\mathcal{C}), M \subset Ars(\mathcal{C}))$ (once again, commutation requirement ignored).
Now my textbook says that comma category $(\Delta | F)$, where $F$ is a functor mapping somewhat category $\mathcal{C'}$ to the $\mathcal{C}$, produces a category of cones. More than that, initial object in a $(\Delta | F)$ is a limit to the $F$. I can't make it out:
Objects of comma category are triplets, each ships single morphism rather than a set of those. But cone requires a set of morphisms. Therefore, to gather a cone (by given definition) one has to select a subcategory of $(\Delta|F)$ and "merge" each morphism of the selected triplet into a resulting set of morphisms.
Thus, I am confused with what is called a "cone" in the category theory? And how does it relate to the category of cones (special case of comma category)?
Best Answer
Writing $(\Delta \mid F)$ is a slight abuse of notation. Remember that $\Delta$ is a functor $\mathcal C \to \mathcal C^{\mathcal C'}$, while $F$ is a functor $\mathcal C' \to \mathcal C$.
To form a comma category $(G \mid H)$, the functors $G$ and $H$ need to have the same codomain.
So, we actually consider $F$ as a functor $\bf1 \to \mathcal C ^{\mathcal C'}$. Let's write it as $\hat F :\bf 1 \to \mathcal C ^{\mathcal C'}$, where $\hat F \star = F$ (and $\star$ is the unique object of $\bf 1$).
Now we see that the objects of $(\Delta \mid \hat F)$ are triples $(A, \star, h)$, where $A$ is an object of $\mathcal C$ and $h$ is a morphism $\Delta A \to \hat F\star = F$. So $h$ is a natural transformation between the functors $\Delta A$ and $F$.
It is not hard to verify that this is the same thing as a cone from $A$ to $F$.
Edit: By “cone to $F$”, I mean an object $A$ of $\mathcal C$, and a family of morphisms $\gamma_X :A \to F X$, indexed by objects $X$ of $\mathcal C’$, such that for every $f : X \to Y$, we have $\gamma_Y = Ff \circ \gamma_X$.
A morphism of cones $(A, \{\gamma_X\}_X) \to (B, \{\delta_X\}_X)$ is a morphism $f : A \to B$ such that for every X we have $\gamma_X = \delta_Y \circ f$.
So given a category $\mathcal C$ and a functor $F : \mathcal C’ \to \mathcal C$ we can form the category of cones to $F$ in $\mathcal C$. This is isomorphic to the comma category $(\Delta \mid \hat F)$.