nLab cofiltered category

Idea

The notion of a cofiltered category is dual to that of a filtered category. We refer to the latter page for the general theory; here we simply spell out a few things explicitly.

Details

Definition

A category $\mathcal{C}$ is cofiltered if its opposite category $\mathcal{C}^{op}$ is filtered.

Remark

In other words, a cofiltered category is one in which every finite diagram in $\mathcal{C}$ has a cone.

Remark

Explicitly, a cofiltered category $\mathcal{C}$ is one for which the following hold.

1. $\mathcal{C}$ has at least one object.
2. For every pair of objects $c_{1}$ and $c_{2}$ of $\mathcal{C}$, there is an object $c_{3}$ of $\mathcal{C}$ such that there exists an arrow $c_{3} \rightarrow c_{1}$ and there exists an arrow $c_{3} \rightarrow c_{2}$.
3. For every pair of objects $c_{1}$ and $c_{2}$ of $\mathcal{C}$, and every pair of arrows $f, g: c_{1} \rightarrow c_{2}$, there is an arrow $h: c \rightarrow c_{1}$ such that $f \circ h = g \circ h$.

Remark

In the final condition of Remark , note that $h$ is not required to satisfy any uniqueness condition with regard to the stated property. In particular, it is not necessarily an equaliser of $f$ and $g$.

Related concepts

• pro-object