symmetric monoidal (∞,1)-category of spectra
Classically, a monoid is a set $M$ equipped with a binary operation $\mu: M \times M \to M$ and special element $1 \in M$ such that $1$ and $x \cdot y = \mu(x,y)$ satisfy the usual axioms of an associative product with unit, namely the associative law:
and the left and right unit laws:
More generally, we can define a monoid (sometimes called monoid object) in any monoidal category $(C,\otimes,I)$. 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.
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).
The data of a monoid may be written in string diagrams as:
Thanks to the distinctive shapes, one can usually omit the labels:
The axioms $\mu \cdot (\eta \otimes M) = 1_M = \mu \cdot (M \otimes \eta)$ and $\mu \cdot (M \otimes \mu) = \mu \cdot (\mu \otimes M)$ then appear as:
Equivalently, and more efficiently, we may say that a (classical) monoid is the hom-set of a category with a single object, equipped with the structure of its unit element and composition.
More tersely, one may say that a monoid is a category with a single object, or more precisely (to get the proper morphisms and $2$-morphisms) a pointed category with a single object. But taking this too literally may create conflicts in notation. To avoid this, for a given monoid $M$, we write $\mathbf{B}M$ for the corresponding category with single object $\bullet$ and with $M$ as its hom-set: the delooping of $M$, so that $M = Hom_{\mathbf{B}M}(\bullet, \bullet)$. This realizes every monoid as a monoid of endomorphisms.
Similarly, a monoid in $(C,\otimes,I)$ may be defined as the hom-object of a $C$-enriched category with a single object, equipped with its composition and identity-assigning morphisms; and so on, as in the classical (i.e. $\mathbf{Set}$-enriched) case.
For more on this see also group.
The notion of associative monoids discussed above are controled by the associative operad. More generally in higher algebra, for $\mathcal{O}$ any operad or (infinity,1)-operad, one can consider $\mathcal{O}$-monoids. (Lurie, def. 2.4.2.1)
These are closely related to (infinity,1)-algebras over an (infinity,1)-operad with respect to $\mathcal{O}$ (Lurie, prop. 2.4.2.5).
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.
It can be important to distinguish between a $k$-tuply monoidal structure and the corresponding $k$-tuply degenerate category, even though there is a map identifying them. The issue appears here for instance when discussing the universal $G$-bundle in its groupoid incarnation. This is
(where $\mathbf{E}G = G//G$ is the action groupoid of $G$ acting on itself). On the left we crucially have $G$ as a monoidal 0-category, on the right as a once-degenerate 1-category. Without this notation we cannot even write down the universal $G$-bundle!
Or take the important difference between group representations and group 2-algebras, the former being functors $\mathbf{B}G \to Vect$, the latter functors $G \to Vect$. Both these are very important.
Or take an abelian group $A$ and a codomain like $2Vect$. Then there are 3 different things we can sensibly consider, namely 2-functors
and
All of these concepts are different, and useful. The first one is an object in the group 3-algebra of $A$. The second is a pseudo-representation of the group $A$. The third is a representations of the 2-group $\mathbf{B}A$. We have notation to distinguish this, and we should use it.
Finally, writing $\mathbf{B}G$ for the 1-object $n$-groupoid version of an $n$-monoid $G$ makes notation behave nicely with respect to nerves, because then realization bars $|\cdot|$ simply commute with the $B$s in the game: $|\mathbf{B}G| = B|G|$.
This behavior under nerves shows also that, generally, writing $\mathbf{B}G$ gives the right intuition for what an expression means. For instance, what’s the “geometric” reason that a group representation is an arrow $\rho : \mathbf{B}G \to Vect$? It’s because this is, literally, equivalently thought of as the corresponding classifying map of the vector bundle on $\mathbf{B}G$ which is $\rho$-associated to the universal $G$-bundle:
the $\rho$-associated vector bundle to the universal $G$-bundle is, in its groupoid incarnations,
where $V$ is the vector space that $\rho$ is representing on, and this is classified by the representation $\rho : \mathbf{B}G \to Vect$ in that this is the pullback of the universal $Vect$-bundle
In summary, it is important to make people understand that groups can be identified with one-object groupoids. But next it is important to make clear that not everything that can be identified should be, for instance concerning the crucial difference between the category in which $G$ lives and the 2-category in which $\mathbf{B}G$ lives.
monoid, internal monoid/monoid object,