symmetric monoidal (∞,1)-category of spectra
categorification
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.
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
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.
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
for all $a,b,c \in Ob(Q)$.
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.
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
for all $a,b,c \in Ob(Q)$.
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 $M$.
The terminology seems to have been introduced in:
A. Arnold, M. Dauchet, Théorie des magmoïdes (I), RAIRO – Informatique Théorique 12 3 (1978) 235-257 [numdam:ITA_1978__12_3_235_0, eudml:92079]
A. Arnold, M. Dauchet, Théorie des magmoïdes (II), RAIRO – Informatique Théorique 13 2 (1979) 135-154 [numdam:ITA_1979__13_2_135_0]
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.