# Discrete Category on Set is Discrete Category

Jump to navigation
Jump to search

Work In ProgressIn particular: Theorem can't be formulated atm, what is up now must move to Definition:Set Category (not to be confused with Definition:Category of Sets)You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

## Theorem

Let $S$ be a set.

Let $\map {\mathbf {Dis} } S$ be the discrete category on $S$.

Then $\map {\mathbf {Dis} } S$ determines a unique (up to isomorphism discrete category $\map {\mathbf {Dis} } S$ whose objects precisely comprise $S$.

This article is complete as far as it goes, but it could do with expansion.In particular: That is, $\mathbf {Dis}: \mathbf {Set} \to \mathbf {Cat}$ is a functor (with left inverse $\mathbf {ob}$)You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Proof

This needs considerable tedious hard slog to complete it.In particular: Trivial.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

$\blacksquare$

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.4.12$