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Generalizing the classical notion of monoid, one can define a monoid (or monoid object) in any monoidal category $(C,\otimes,I)$. Classical monoids are of course just monoids in Set with the cartesian product.
By the microcosm principle, in order to define monoid objects in $C$, $C$ itself must be a “categorified monoid” in some way. The natural requirement is that it be a monoidal category. In fact, it suffices if $C$ is a multicategory. Contrast this with a group object, which can only be defined in a cartesian monoidal category (or a cartesian multicategory).
Namely, a monoid in $C$ is an object $M$ equipped with a multiplication $\mu: M \otimes M \to M$ and a unit $\eta: I \to M$ satisfying the associative law:
and the left and right unit laws:
Here $\alpha$ is the associator in $C$, while $\lambda$ and $\rho$ are the left and right unitors.
The analogue of a monoid homomorphism, called a morphism of monoids, is a morphism, $\f: M \to M'$ between two monoid objects, satisfying the equations;
$f \circ \mu = \mu' \circ (f \otimes f)$
$f \circ \eta = \eta'$
corresponding to the commutative diagrams;
Just as the category of regular monoids is equivalent to the category of locally small (i.e. Set-enriched) categories with one object, the category of monoids in $C$ (with the obvious morphisms) is equivalent to the category of $C$-enriched categories with one object.
Monoid structure is preserved by lax monoidal functors. Comonoid structure by oplax monoidal functors. See there for more.
For special properties of categories of monoids, see category of monoids.
These are examples of monoids internal to monoidal categories. More generally, given any bicategory $B$ and a chosen object $a$, the hom-category $B(a,a)$ has the structure of a monoidal category. So, the concept of monoid makes sense in any bicategory $B$: we define a monoid in $B$ to be a monoid in $B(a,a)$ for some object $a \in B$. This often called a monad in $B$. The reason is that a monad in Cat is the same as monad on a category.
A monoid in a bicategory $B$ may also be described as the hom-object of a $B$-enriched category with a single object.
Saunders MacLane, Section III.6 in: Categories for the Working Mathematician, Springer (1971)
Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 7 in: Internal object actions, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)
Florian Marty, Dections 1.2, 1.3 in: Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Thesis, 2009 (web)
Martin Brandenburg, Section 4.1 of: Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)
See also MO/180673.
Last revised on March 22, 2021 at 04:22:06. See the history of this page for a list of all contributions to it.