as the unit. The category of pointed spaces (restricted to compactly generated spaces for example) is monoidal with the smash product serving as the product and the pointed 0-sphere (a two-point discrete space) serving as the unit. The category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit. Just like for any category E, the full subcategory spanned by any given object is a monoid, it is the case that for any 2-category E, and any object C in Ob(E), the full 2-subcategory of E spanned by {C} is a monoidal category. In the case E = Cat, we get the endofunctors example above. Bounded-above meet semilattices are strict symmetric monoidal categories: the product is meet and the identity is the top element. Any ordinary monoid is a small monoidal category with object set , only identities for morphisms, as tensorproduct and as its identity object. Conversely, the set of isomorphism classes (if such a thing makes sense) of a monoidal category is a monoid w.r.t. the tensor product. Any commutative monoid can be realized as a monoidal category with a single object. Recall that a category with a single object is the same thing as an ordinary monoid. By an Eckmann-Hilton argument, adding another monoidal product on requires the product to be commutative. Monoidal preorders Monoidal preorders, also known as "preordered monoids", are special cases of monoidal categories. This sort of structure comes up in the theory of string rewriting systems, but it is plentiful in pure mathematics as well. For example, the set of natural numbers has both a monoid structure (using + and 0) and a preorder structure (using ≤), which together form a monoidal preorder, basically because and implies . We now present the general case. It's well known that a preorder can be considered as a category C, such that for every two objects , there exists at most one morphism in C. If there happens to be a morphism from c to c' , we could write , but in the current section we find it more convenient to express this fact in arrow form . Because there is at most one such morphism, we never have to give it a name, such as . The reflexivity and transitivity properties of an order are respectively accounted