category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
symmetric monoidal (∞,1)-category of spectra
For $C$ a monoidal category, the category of monoids $Mon(C)$ in the $C$ is the category whose
morphisms are morphisms in $C$ of the underlying objects that respect the monoid structure.
Similarly for the category of commutative monoids $CMon(C)$, if $C$ is symmetric monoidal.
If Assoc denotes the associative operad in $C$, then $Mon(C) = Alg_C Assoc$ is the category of algebras over an operad for $Assoc$.
Every category of monoids comes with a forgetful functor $U \colon Mon(C) \to C$ which is faithful and conservative. In many cases it is monadic.
The properties of the category of monoids $Mon (C)$, especially with respect to colimits, are markedly different according to whether or not the tensor product of $C$ preserves colimits in each variable. (This is automatically the case if $C$ is closed.)
Most “algebraic” situations have this property, but others do not. For instance, the category of monads on a fixed category $A$ is $Mon (C)$, where $C= [A,A]$ is the category of endofunctors of $A$ with composition as its monoidal structure. This monoidal product preserves colimits in one variable (since colimits in $[A,A]$ 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 $\otimes$ does preserve colimits in both variables, although some of the references at the end discuss the more general case.
Let $C$ be a closed symmetric monoidal category with countable coproducts which is locally presentable.
Then
$U : Mon(C) \to C$ is a finitary monadic functor.
If $C$ is a $\lambda$-locally presentable category then so is $Mon(C)$.
This appears in (Porst, page 7).
Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the forgetful functor $U_C$ has a left adjoint $F_C : C \to Mon(C)$. On an object $X \in C$ the underlying object of $F_C X$ is
in $C$, with the monoidal structure given by tensor product/juxtaposition.
A morphism $f : F_C X \to A$ in $Mon(C)$ with components $f_k : X^{\otimes k} \to U_C A$ is entirely fixed by its component $\tilde f = f_1 : X \to U_C A$ on $X$, because by the homomorphism property and the special free nature of the product in $F_C X$
it follows that
Conversely, every choice for $f_1$ extends to a morphism $f$ in $Mon(C)$ this way.
Free algebras of the form $F(A)$ are called tensor algebras, at least for $C =$ Vect and similar.
The elements of the free algebra $F(A)$ are somtimes called lists, at least for $C =$ Set and similar.
We discuss forming pushouts in a category of monoids. The case
has a simple description. The case
is more incolved.
Suppose that $C$ is
with reflexive coequalizers
that are preserved by the tensoring functors $A \otimes (-): C \to C$ for all objcts $A$ in $C$.
Then for $f : A\to B$ and $g : A \to C$ two morphisms in the category $CMon(C)$ of commutative monoids in $C$, the underlying object in $C$ of the pushout in $CMon(C)$ coincides with that of the pushout in the category $A$Mod of $A$-modules
Here $B$ and $C$ are regarded as equipped with the canonical $A$-module structure induced by the morphisms $f$ and $g$, respectively.
This appears for instance as (Johnstone, page 478, cor. 1.1.9).
If $C$ is cocomplete and its tensor product preserves colimits on both sides, then the category $Mon(C)$ of monoids has all pushouts
along morphisms $F(f) : F(K) \to F(L)$, for $f : K \to L$ a morphism in $C$ and $F : C \to Mon(C)$ the free monoid functor from above.
Moreover, these pushouts in $Mon(C)$ are computed in $C$ as the colimit over a sequence
of objects $(P_n)_{n \in \mathbb{N}}$, which are each given by pushouts in $C$ inductively as follows.
Assume $P_{n-1}$ has been defined. Write $Sub(\mathbf{n})$ for the poset of subsets of the $n$-element set $\mathbf{n}$ (this is the poset of paths along the edges of an $n$-dimensional cube). Define a diagram
by setting on subsets $S \subset \mathbf{n}$
where
and by assigning to a morphism $S_1 \subset S_2$ the morphism which is the tensor product of identities on $X$, identities on $L$ and the given morphism $f : K \to L$.
Write $K^-$ for the same diagram minus the terminal object $S = \mathbf{n}$.
Now take $P_n$ to be the pushout
where the top morphism is the canonical one induced by the commutativity of the diagram $K$, and where the left morphism is defined in terms of components $K^-(S)$ of the colimit for $S \subset \mathbf{n}$ a proper subset by the tensor product morphisms of the form
This gives the underlying object of the monoid $P$. Take the monoid structure on it as follows. The unit of $P$ is the composite
with the unit of $X$. The product we take to be the image in the colimit of compatible morphisms $P_k \otimes P_k \to P_{k + l}$ defined by induction on $lk + l$ as follows. we observe that we have a pushout diagram
where $Q_n := (\lim_{\to} K)_n$ is the colimit as in the above at stage $n$.
There is a morphism from the bottom left object to $P_{k+l}$ given by the induction assumption. Moreover we have a morphism from the top right object to $P_{k+1}$ obtained by first multiplying the two adjacent factors of $X$ and then applying the morphism $(X \otimes L)^{\otimes k+l} \otimes X \to P_{k+l}$. These are compatible and hence give the desired morphism $P_k \otimes P_k \to P_{k+l}$.
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 $\lim_\to K^- \to P_{n-1}$ is well defined and the monoid structure on $P$ is well defined.
(…)
That $X \to P$ is a morphism of monoids follows then essentially by the definition of the monoid structure on $P$.
Finally we need to check the universal property of the cocone $P$ obtained this way:
(…)
For $C$ a closed symmetric monoidal category the forgetful functor
from commutative monoids to $C$ created filtered colimits.
This appears for instance as (Johnstone, C1.1 lemma 1.1.8).
If $F : C\to D$ 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 $C$ is a monoidal model category, then $Mon(C)$ may inherit itself the structure of a model category. See model structure on monoids in a monoidal model category.
Some categories are implicitly enriched category|? over commutative monoids, in particular semiadditive categories. Also Ab-enriched categories (and hence in particualr abelian categories) of course have an underlying $CMon$-enrichment.
A general discussion of categories of monoids in symmetric monoidal categories is in
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
Discussion of the closed monoidal category structure on a category of algebras of a commutative algebraic theory is in