nLab
cofiltered category

Contents

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

Remark

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

Remark

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

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

Remark

In the final condition of Remark , note that hh is not required to satisfy any uniqueness condition with regard to the stated property. In particular, it is not necessarily an equaliser of ff and gg.

Last revised on April 21, 2020 at 02:22:10. See the history of this page for a list of all contributions to it.