How is the product (and its unique map) related to terminal objects?
Call \(\mathbf{Cone}(X,Y)\) the /category of cones over X and Y in/ \(\mathcal{C}\), given two objects in \(\mathcal{C}\).
An object is pair of maps \(X \xleftarrow{f}C\xrightarrow{g}Y\) for some \(c \in \mathcal{C}\)
A morphism a from \(X \xleftarrow{f}C\xrightarrow{g}Y\) to \(X \xleftarrow{f'}C\xrightarrow{g'}Y\) is a morphism \(C \rightarrow C'\) in \(\mathcal{C}\) such that the following diagram commutes:
Suppose \(\mathcal{J}\) is the free category on the graph \(\boxed{1 \rightarrow 2 \leftarrow 3 \rightrightarrows 4 \rightarrow 5}\)
We may draw a diagram \(\mathcal{J}\xrightarrow{D}\mathcal{C}\) inside \(\mathcal{C}\) as below:
\(\boxed{D_1 \rightarrow D_2 \leftarrow D_3 \rightrightarrows D_4 \rightarrow D_5}\)
We can represent this diagram as a cone over the diagram by picking a \(C \in \mathcal{C}\) for which every pair of parallel paths that start from \(C\) are the same.
Terminal objects are imits where the indexing category is empty, \(\mathcal{J}=0\).
Products are limits where the indexing category consists of two objects with no arrows, \(\mathcal{J}=\boxed{\overset{v}\bullet\ \overset{w}\bullet}\).
A cone \((C,c_*)\) over a diagram \(\mathcal{J}\xrightarrow{D}\mathcal{C}\) and the category \(\mathbf{Cone}(D)\)
We require:
An object \(C \in Ob(\mathcal{C})\)
For each object \(j \in \mathcal{J}\), a morphism \(C \xrightarrow{c_j}D(j)\).
The following property must be satisfied:
\(\forall f \in \mathcal{J}(j,k):\) \(c_k=c_j;D(f)\)
A morphism of cones is a morphism \(C \xrightarrow{a} C'\) in \(\mathcal{C}\) such that, for all \(j \in \mathcal{J}\), we have \(c_j=a;c'_j\).
Cones and their morphisms form a category.
The limit of a diagram \(D\), \(lim(D)\)
The limit of \(D\), denoted is the terminal object in the category \(\mathbf{Cone}(D)\)
If \(lim(D)=(C,c_*)\) we refer to \(C\) as the limit object and the map \(c_j\) as the jth projection map
Check that the product \(X \xleftarrow{p_X} X \times Y \xrightarrow{p_Y} Y\) is terminal object in \(\mathbf{Cone}(X,Y)\)
The existence and uniqueness of the morphism to the product object in the product definition are the conditions of being a terminal object in \(\mathbf{Cone}(X,Y)\). The maps of the terminal object to \(X\) and \(Y\) are the projection maps of the product.