internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
monoid theory in algebra:
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 lax monoidal functor for more details.
For special properties of categories of monoids, see category of monoids.
A monoid in a monoidal category of modules $R Mod$ (over any ground ring $R$ and equipped with the tensor product of modules) is an associative unital algebra over $R$.
As the special case of Exp. for $R = \mathbb{Z}$ the integers:
A monoid object in the monoidal category Ab of abelian groups with the tensor product of abelian groups, is a ring.
As the special case of Exp. for $R = k$ a field:
A monoid object in the category Vect of vector spaces (over any ground field $k$) with the tensor product of vector spaces is an associative unital algebra over $k$.
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.
(co)monad name | underlying endofunctor | (co)monad structure induced by |
---|---|---|
reader monad | $W \to (\text{-})$ on cartesian types | unique comonoid structure on $W$ |
coreader comonad | $W \times (\text{-})$ on cartesian types | unique comonoid structure on $W$ |
writer monad | $A \otimes (\text{-})$ on monoidal types | chosen monoid structure on $A$ |
cowriter comonad | $\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})}$ on monoidal types | chosen monoid structure on $A$ chosen comonoid structure on $A$ |
Frobenius (co)writer | $\array{A \to (\text{-}) \\ A \otimes (\text{-})}$ on monoidal types | chosen Frobenius monoid structure |
Original references (including the case of a commutative monoids in a symmetric monoidal category, but see there for more):
Jean Bénabou, Algèbre élémentaire dans les catégories (1964), C. R. Acad. Sci. Paris 258 (1964) pp.771-774, gallica
Saunders MacLane, Sections III.6 & VII.3 in: Categories for the Working Mathematician, Springer (1979, 2nd ed.) [doi:10.1007/978-1-4757-4721-8]
Hans-Joachim Baues, Mamuka Jibladze, Andy Tonks, Cohomology of monoids in monoidal categories, in: Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) 137-166 [doi:10.1090/conm/202, preprint:pdf]
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)
Discussion for commutative monoids in a symmetric monoidal category including proof that/when the category of module objects is itself closed symmetric monoidal:
See also:
Florian Marty, Sections 1.2, 1.3 in: Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Toulouse (2009) [theses:2009TOU30071, pdf]
Martin Brandenburg, Section 4.1 of: Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)
Lecture notes:
In cartesian monoidal categories:
and here formalized as mathematical structures in proof assistants:
in a context of plain Agda:
Last revised on September 6, 2023 at 08:09:14. See the history of this page for a list of all contributions to it.