symmetric monoidal (∞,1)-category of spectra
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 group 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 May 21, 2021 at 18:34:16. See the history of this page for a list of all contributions to it.