It is well known that the augmented simplex category is the free strict monoidal category with a monoid: the monoidal product is concatenation, and the monoid is the terminal object (called [0] or Δ^{0} by geometers). Perhaps less well known is the free strict monoidal category with *k* monoids (where *k* is any cardinal, possibly infinite):

- The objects are pairs (
*n*,*P*) where*n*is a natural number (possibly zero) and*P*is a partition of the set { 0, …,*n*– 1 } into*k*disjoint subsets*P*_{j}, 0 ≤*j*<*k*. - The morphisms (
*n*,*P*) → (*m*,*Q*) are monotone maps { 0, …,*n*– 1 } → { 0, …,*m*– 1 } that respect the partitions, i.e. send members of*P*_{j}to members of*Q*_{j}.

In other words, it is the category of finite linear orders equipped *k*-colouring of its elements. From this point of view, the monoidal product is best interpreted as concatenation of (coloured) linear orders. The *k* monoids are the *k* different objects of the form (1, *P*). Amusingly, the monoidal product on the set of objects makes it the free monoid on *k* generators.

The opposite category, i.e. the free strict monoidal category with *k* comonoids, can be described concretely as follows:

- The objects are pairs (
*n*,*P*) as before. - The morphisms (
*m*,*Q*) → (*n*,*P*) are monotone maps*f*: { 0, …,*m*} → { 0, …,*n*} that satisfy the following axioms:*f*(0) = 0 and*f*(*m*) =*n*.- If
*i*is a member of*P*_{j}, then for*f*(*i*) ≤*e*<*f*(*i*+ 1),*e*is a member of*Q*_{j}.

The above should be interpreted as the category of finite intervals equipped with a *k*-colouring of its *edges*. From this point of view, the monoidal product is the wedge sum. The special case where *k* = 2 is especially useful in relation to the problem of inverting morphisms in a category: for this, see [Dwyer and Kan, *Calculating simplicial localizations*].