Codiscrete Category

definition

The codiscrete (indiscrete, chaotic) category on a set $X$ has objects $X$ and exactly one morphism $x\to y$ for every pair $(x,y)$. It is a Groupoid and the Codiscrete Preorder viewed as a category. The construction $\mathbf{Set}\to \mathbf{Cat}$ is right adjoint to the objects functor $\mathrm{Ob}:\mathbf{Cat}\to \mathbf{Set}$ (7 Sketches Example 3.74: “codiscrete things are right adjoints”). Contrast: Discrete Category.

Indiscrete categories are equivalent to a point (CTfS Example 4.3.4.3). For a nonempty set $S$, the indiscrete category $K_S$ is isomorphic to the terminal category $1$ only if $|S|=1$, but it is always equivalent to $1$: pick any $s_0\in S$; the unique isomorphisms $s\to s_0$ assemble into a natural isomorphism ${\mathrm{id}}_{K_S}\cong (\text{constant at }{s}_0)$. Every object of $K_S$ is both initial and terminal (CTfS Exercise 4.5.3.13). The functor $\mathrm{Ind}:\mathbf{Set}\to \mathbf{Cat}$ also exhibits a full subcategory ${\mathcal{C}}_{\mathrm{Ob}=X}$ as the fiber product $\mathcal{C}{\times }_{\mathrm{Ind}(\mathrm{Ob}\mathcal{C})}\mathrm{Ind}(X)$ (CTfS Example 4.6.3.4).

Sources: 7 Sketches Example 3.74; CTfS Examples 4.3.4.3, 4.6.3.4, 5.1.1.7.