# nLab magmoid

Contents

### Context

category theory

#### 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 magmoid should be the oidification of a magma.

## Definition

### With a set of morphisms

Let $Q$ be a quiver with a collection of objects $Ob(Q)$, a set of morphisms $Mor(Q)$, a function $s: Mor(Q) \to Ob(Q)$ called the source or domain and a function $t: Mor(Q) \to Ob(Q)$ called the target or codomain. A magmoid is a quiver $Q$ with a partial binary operation

$\left(-\right)\circ\left(-\right): Mor(Q) \times Mor(Q) \to Mor(Q) + 1$

with coproduct injections $inj:Mor(Q) \to Mor(Q) + 1$ and $pt: 1 \to Mor(Q) + 1$, such that for $f, g \in Mor(Q)$, there is a function $h$ in the inverse image of $inj(h) = f \circ g$ if $s(f) = t(g)$, and $f \circ g = pt(\bullet)$ otherwise.

### With a family of sets of morphisms

Let $Q$ be a quiver with a collection of objects $Ob(Q)$ and a Set-valued functor $Mor: Ob(Q) \times Ob(Q) \to Set$ for all objects $a, b \in Ob(Q)$. A magmoid is a quiver $Q$ with a binary operation

$\left(-\right)\circ\left(-\right): Mor(b,c) \times Mor(a,b) \to Mor(a,c)$

for all $a,b,c \in Ob(Q)$.

### Weak and strict magmoids

A weak magmoid is a magmoid $Q$ whose collection of objects $Ob(Q)$ form a groupoid, while a strict magmoid is a magmoid $Q$ whose collection of objects $Ob(Q)$ form a set.

### Enriched magmoids

Let $V$ be a monoidal category (or a monoidal (infinity,1)-category) and let $Q$ be a $V$-enriched quiver, with a collection of objects $Ob(Q)$ and a $V$-valued functor $Mor: Ob(Q) \times Ob(Q) \to V$ for all objects $a, b \in Ob(Q)$. A $V$-enriched magmoid is a $V$-enriched quiver $Q$ with a binary operation

$\left(-\right)\circ\left(-\right): Mor(b,c) \times Mor(a,b) \to Mor(a,c)$

for all $a,b,c \in Ob(Q)$.

## Examples

A transitive relation is a magmoid enriched in truth values, or a magmoid $M$ where there is at most one morphism from every object $a$ to another object $b$ in $M$.

## References

The terminology seems to have been introduced in:

Further discussion:

• Miklos Bartha, p. 101 in: An algebraic model of synchronous systems, Information and Computation 97 1 (1992) 97-131

• Dan Jonsson, Def. 2.2 in: On Group-Like Magmoids [arXiv:1902.06109]

• Fiona Torzewska, §2.1 of: Topological quantum field theories and homotopy cobordisms [arXiv:2208.14504]

Last revised on December 9, 2023 at 17:00:53. See the history of this page for a list of all contributions to it.