• Source: Rig category

In category theory, a rig category (also known as bimonoidal category or 2-rig) is a category equipped with two monoidal structures, one distributing over the other.




Definition


A rig category is given by a category




C



{\displaystyle \mathbf {C} }

equipped with:

a symmetric monoidal structure



(

C

,
⊕
,
O
)


{\displaystyle (\mathbf {C} ,\oplus ,O)}


a monoidal structure



(

C

,
⊗
,
I
)


{\displaystyle (\mathbf {C} ,\otimes ,I)}


distributing natural isomorphisms:




δ

A
,
B
,
C


:
A
⊗
(
B
⊕
C
)
≃
(
A
⊗
B
)
⊕
(
A
⊗
C
)


{\displaystyle \delta _{A,B,C}:A\otimes (B\oplus C)\simeq (A\otimes B)\oplus (A\otimes C)}

and




δ

A
,
B
,
C

′

:
(
A
⊕
B
)
⊗
C
≃
(
A
⊗
C
)
⊕
(
B
⊗
C
)


{\displaystyle \delta '_{A,B,C}:(A\oplus B)\otimes C\simeq (A\otimes C)\oplus (B\otimes C)}


annihilating (or absorbing) natural isomorphisms:




a

A


:
O
⊗
A
≃
O


{\displaystyle a_{A}:O\otimes A\simeq O}

and




a

A

′

:
A
⊗
O
≃
O


{\displaystyle a'_{A}:A\otimes O\simeq O}


Those structures are required to satisfy a number of coherence conditions.




Examples


Set, the category of sets with the disjoint union as



⊕


{\displaystyle \oplus }

and the cartesian product as



⊗


{\displaystyle \otimes }

. Such categories where the multiplicative monoidal structure is the categorical product and the additive monoidal structure is the coproduct are called distributive categories.
Vect, the category of vector spaces over a field, with the direct sum as



⊕


{\displaystyle \oplus }

and the tensor product as



⊗


{\displaystyle \otimes }

.




Strictification


Requiring all isomorphisms involved in the definition of a rig category to be strict does not give a useful definition, as it implies an equality



A
⊕
B
=
B
⊕
A


{\displaystyle A\oplus B=B\oplus A}

which signals a degenerate structure. However it is possible to turn most of the isomorphisms involved into equalities.
A rig category is semi-strict if the two monoidal structures involved are strict, both of its annihilators are equalities and one of its distributors is an equality. Any rig category is equivalent to a semi-strict one.




References



Rig category at the nLab

Kata Kunci Pencarian:

• Cyberpunk 2077
• Second Life
• Rig category
• Rig (sailing)
• Lateen
• Square rig
• Bermuda rig
• Gaff rig
• Initial and terminal objects
• Rigveda
• Deepwater Horizon
• Barque