How to define a map and questions about cones and functors.

Question

How do I define the bottom map of the following pushout diagram in this problem from Emily Reihl's Category Theory in context?

Also, how do I prove that a cone indexed by $J$ is the same thing as a functor from $\ J^{\lhd}$?

Answer 1

According to Category Theory in Context, the category $J^\triangleleft$ is made of the objects of $J$ (together with the arrows between them) and a new object $0$ together with exactly one arrow $0 \to j$ for every object $j \in J$. Observe that, for every arrow $j \to j'$ in $J$, it is the case that $(0 \to j \to j')=(0\to j')$ in $J^\triangleleft$.

As for every object $j$ in $J$ there is precisely one arrow $(j,0) \to (j,1)$ in the category $J \times 2$, in order to get $J^\triangleleft$ as vertex of that pushout, we want this pushout to identify the objects of the form $(j,0)$ of $J\times 2$ (for every object $j$ of $J$) with the unique object $0$ of the category $1$.

Hence the functor $i_0$ must send every object $j$ of $J$ to $(j,0)$ (and then every arrow of $J$ to the obvious one), so that indeed, for every $j$ in $J$, the objects $(j,0)$ of $J\times 2$ and $0$ of $1$ represent in $J^\triangleleft$ the same class. Therefore the "$J\times \{1\}$"-part of $J\times 2$ represents the "$J$"-part of $J^\triangleleft$, while the "$J\times \{0\}$" part of $J^\triangleleft$ represents the initial object $0$ of $J^\triangleleft$, as every object of the form $(j,0)$ in $J\times 2$ represents the same class in $J^\triangleleft$, that is, its initial object $0$. Indeed the arrow $(j,0)\to(j,1)$ in $J\times 2$ becomes in $J^\triangleleft$ the unique arrow $0=[(j,0)]\to [(j,1)]=j$.

So, what is the bottom functor $J\times 2 \to J$? It is the just the "quotient functor", the functor sending every object of $J\times 2$ to its equivalence class. In particular, as we said, it sends the "$J\times \{1\}$"-part of $J\times 2$ to the "$J$"-part of $J^\triangleleft$ and the "$J \times \{0\}$"-part of $J\times 2$ to the initial object $0$ of $J^\triangleleft$.

Finally, if we have a diagram $D \colon J \to C$, for some category $C$, a cone of $D$ can be seen as a constant functor $d \colon J \to C$ together with a natural transformation $\alpha \colon d \to D$. The corresponding functor $D'\colon J^\triangleleft \to C$, is the one sending $0$ to the image-object $c$ of the functor $d$, every object $j$ of $J$ to $Dj$ and every arrow $0 \to j$ to $\alpha_j \colon c \to Dj$.