Cofibre

Let $\mathcal{C}$ be a category with a Terminal Object $1:\mathcal{C}$. The cofibre of a morphism $f:X\to Y$ is the Pushout

Intuitively, this adjoints a new point to $Y$, and identifies the entire Image of $f$ to that point.

In terms of Cocartesian Fibrations

We can define the notion of a cofibre generally in any Cocartesian Fibration that has a Cofibrewise Terminal Object.

Properties

• The cofibre of the unique map $0\to X$ is simply the Coproduct $1+X$.

• The cofibre of the identity map $mathrm{id}{ifx..lse\par i}:X\to X$ gives a sort of "cone" in $X$.

• The cofibre of a Global Element $x:1\to X$ is Isomorphic to $X$.