The free strict monoidal category with k monoids

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 Δ0 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 Pj, 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 Pj to members of Qj.

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 Pj, then for f(i) ≤ e < f(i + 1), e is a member of Qj .

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].

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