symmetric monoidal (∞,1)-category of spectra
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
monoid theory in algebra:
For $A$ and $B$ two commutative monoids, their tensor product $A \otimes B$ is a new commutative monoid such that a monoid homomorphism $A \otimes B \to C$ is equivalently a bilinear map out of $A$ and $B$.
Let CMon be the collection of commutative monoids, regarded as a multicategory whose multimorphisms are the multilinear maps $A_1, \cdots, A_n \to B$.
The tensor product $A, B \mapsto A \otimes B$ in this multicategory is the tensor product of commutative monoids.
Equivalently this means explicitly:
For $A, B$ two commutative monoids, their tensor product of commutative monoids is the commutative monoid $A \otimes B$ which is the quotient of the free commutative monoid on the product of their underlying sets $A \times B$ by the relations
$(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$
$(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$
$(0,b)\sim 0$
$(a,0)\sim 0$
for all $a, a_1, a_2 \in A$ and $b, b_1, b_2 \in B$.
By definition of the free construction and the quotient there is a canonical function of the underlying sets
(where $U \colon CMon \to Set$ is the forgetful functor).
On elements this sends $(a,b)$ to the equivalence class that it represents under the above equivalence relations.
The following relates the tensor product to bilinear functions. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of commutative monoids.
A function of underlying sets $f : A \times B \to C$ is a bilinear function precisely if it factors by the morphism of through a monoid homomorphism $\phi : A \otimes B \to C$ out of the tensor product:
Equipped with the tensor product $\otimes$ of def. and the exchange map $\sigma_{A, B}: A\otimes B \to B \otimes A$ generated by $\sigma_{A, B}(a, b) = (b, a)$, CMon becomes a symmetric monoidal category.
The unit object in $(CMon, \otimes)$ is the additive monoid of natural numbers $\mathbb{N}$.
To see that $\mathbb{N}$ is the unit object, consider for any commitative momoid $A$ the map
which sends for $n \in \mathbb{N}$
Due to the quotient relation defining the tensor product, the element on the left is also equal to
This shows that $A \otimes \mathbb{N} \to A$ is in fact an isomorphism.
Showing that $\sigma_{A, B}$ is natural in $A, B$ is trivial, so $\sigma$ is a braiding. $\sigma^2$ is identity, so it gives CMon a symmetric monoidal structure.
The tensor product of commutative monoids distributes over the biproduct of commutative monoids
Let $(A, \cdot)$ be a monoid in $(CMon, \otimes)$. The fact that the multiplication
is bilinear means by the above that for all $a_1, a_2, b \in A$ we have
This is precisely the distributivity law and absorption law? of the rig.
Last revised on July 18, 2023 at 09:20:30. See the history of this page for a list of all contributions to it.