# nLab loopoid

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 loopoid should be the oidification of a loop.

## Definition

A loopoid $Q$ 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 $g:c \to a$, $id_a \circ g = g$, where every span

$\array{ && s \\ & {}^{f}\swarrow && \searrow^{g} \\ x &&&& y }$

in $Q$ has morphisms $i:x\to y$ and $j:y \to x$ such that $i \circ f = g$ and $j \circ g = f$, and where every cospan

$\array{ && a &&&& b \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& c &&&& }$

in $Q$ has morphisms $d:a\to b$ and $e:b \to a$ such that $g \circ d = f$ and $f \circ e = g$.

## Examples

Last revised on May 23, 2021 at 18:55:53. See the history of this page for a list of all contributions to it.