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The properties of the category of monoids , especially with respect to colimits, are markedly different according to whether or not the tensor product of preserves colimits in each variable. (This is automatically the case if is closed.)
Most “algebraic” situations have this property, but others do not. For instance, the category of monads on a fixed category is , where is the category of endofunctors of with composition as its monoidal structure. This monoidal product preserves colimits in one variable (since colimits in are computed pointwise), but not in the other (since most endofunctors do not preserve colimits). So far, the material on this page focuses on the case where does preserve colimits in both variables, although some of the references at the end discuss the more general case.
This appears in (Porst, page 7).
Free and relative free monoids
Let be a monoidal category with countable coproducts that are preserved by the tensor product. Then the forgetful functor has a left adjoint . On an object the underlying object of is
in , with the monoidal structure given by tensor product/juxtaposition.
A morphism in with components is entirely fixed by its component on , because by the homomorphism property and the special free nature of the product in
it follows that
Conversely, every choice for extends to a morphism in this way.
Free algebras of the form are called tensor algebras, at least for Vect and similar.
The elements of the free algebra are somtimes called lists, at least for Set and similar.
Of noncommutative monoids
If is cocomplete and its tensor product preserves colimits on both sides, then the category of monoids has all pushouts
along morphisms , for a morphism in and the free monoid functor from above.
Moreover, these pushouts in are computed in as the colimit over a sequence
of objects , which are each given by pushouts in inductively as follows.
Assume has been defined. Write for the poset of subsets of the -element set (this is the poset of paths along the edges of an -dimensional cube). Define a diagram
by setting on subsets
and by assigning to a morphism the morphism which is the tensor product of identities on , identities on and the given morphism .
Write for the same diagram minus the terminal object .
Now take to be the pushout
where the top morphism is the canonical one induced by the commutativity of the diagram , and where the left morphism is defined in terms of components of the colimit for a proper subset by the tensor product morphisms of the form
This gives the underlying object of the monoid . Take the monoid structure on it as follows. The unit of is the composite
with the unit of . The product we take to be the image in the colimit of compatible morphisms defined by induction on as follows. we observe that we have a pushout diagram
where is the colimit as in the above at stage .
There is a morphism from the bottom left object to given by the induction assumption. Moreover we have a morphism from the top right object to obtained by first multiplying the two adjacent factors of and then applying the morphism . These are compatible and hence give the desired morphism .
This construction is spelled out for instance in the proof of SchwedeShipley, lemma 6.2
First we need to discuss that this definition is actually consistent, in that the morphism is well defined and the monoid structure on is well defined.
That is a morphism of monoids follows then essentially by the definition of the monoid structure on .
Finally we need to check the universal property of the cocone obtained this way:
Of commutative monoids
Suppose that is
Then for and two morphisms in the category of commutative monoids in , the underlying object in of the pushout in coincides with that of the pushout in the category Mod of -modules
Here and are regarded as equipped with the canonical -module structure induced by the morphisms and , respectively.
This appears for instance as Johnstone, page 478, cor. 1.1.9.
This appears for instance as (Johnstone, C1.1 lemma 1.1.8).
Structure induced from monoidal functors
If is a lax monoidal functor, then it induces canonically a functor between categories of monoids
This is one good way to remember the difference between lax and colax monoidal functors.
If is a monoidal model category, then may inherit itself the structure of a model category. See model structure on monoids in a monoidal model category.
A general discussion of categories of monoids in symmetric monoidal categories is in
- Hans Porst, On Categories of Monoids, Comonoids and bimonoids (pdf)
Free monoid constructions are discussed in
Eduardo Dubuc, Free monoids Algebra J. 29, 208–228 (1974)
Max Kelly, A unified treatment of transfinite constructions for free algebras, free monoids,colimits, associated sheaves, and so on Bull. Austral. Math. Soc. 22(1), 1–83 (1980)
Stephen Lack, Note on the construction of free monoids Appl Categor Struct (2010) 18:17–29
The detailed discussion of pushouts along free monoid morphisms is in the proof of lemma 6.2 of
Some remarks on commutative monoids are in section C1.1 of