---
tag: DAMK8XE1FR
kind: concept
name: Opposite category
msc: [18A05]
sources:
  - url: https://en.wikipedia.org/wiki/Opposite_category
same_as:
  wikidata: Q7098616
  wikipedia: Opposite category
---

## Summary

The opposite category $\mathcal{C}^{\mathrm{op}}$ of a [[category|A2DE66937D]] $\mathcal{C}$ has the same objects and the same morphisms, with every arrow reversed.
It turns [[monomorphisms|Y2NP1JFR86]] into [[epimorphisms|RVXDSY04KG]], products into coproducts, and limits into colimits.
It is the basis of [[duality in category theory|Z5EETTFWAN]].
It also lets one write a [[contravariant functor|RECR96V97B]] on $\mathcal{C}$ as an ordinary functor on $\mathcal{C}^{\mathrm{op}}$.

## Explanations

### L1 human

Take a drawing of a category and reverse the direction of every arrow.
The result is again a category, because chains of arrows can still be composed, in the reverse order.
For a [[preorder|0VABYWJWH9]] seen as a category, with an arrow $a \to b$ when $a \le b$, the opposite category is the reversed order $\ge$.
For the numbers $1 \le 2 \le 3$, the opposite category has arrows $3 \to 2 \to 1$.