# nLab associative quasigroupoid

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, an associative quasigroupoid should be the oidification of an associative quasigroup.

## Definition

An associative quasigroupoid is a magmoid $Q$ such that for every diagram

$a\underset{\quad f \quad}{\to}b\underset{\quad g \quad}{\to}c\underset{\quad h \quad}{\to}d \,,$
$h \circ (g \circ f) = (h \circ g) \circ f \,,$

for every span

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

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

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

in $Q$, there exists 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 19:19:34. See the history of this page for a list of all contributions to it.