# nLab unital magmoid

Contents

category theory

## Applications

#### Algebra

higher algebra

universal algebra

categorification

# Contents

## Idea

Just as a groupoid is the oidification of a group and a ringoid is the oidification of a ring, a unital magmoid should be the oidification of a unital magma.

## Definition

A unital magmoid $Q$ is a magmoid where every object $a \in Ob(Q)$ has an identity morphism $id_a: a \to a$, such that for any morphism $f:a \to b$, $f \circ id_a = f$, and for any morphism $g:c \to a$, $id_a \circ g = g$.

A unital magmoid is invertible if for every pair of objects $a,b \in Ob(Q)$ and for every morphism $f:a \to b$, there exists an inverse morphism $g:b \to a$ such that $f \circ g = id_b$ and $g \circ f = id_a$.