monoidal category

monoidal category

A

monoidal category

is a

category

which has the

structure

of a monoid, that is, among the objects there is a

binary operation

which is

associative

and has an unique neutral or unit element. Specifically, a category

𝒞

is

monoidal

if

1.

there is a

bifunctor

⊗

:

𝒞

×

𝒞

→

𝒞

, where the images of object

(

A

,

B

)

and

morphism

(

f

,

g

)

are written

A

⊗

B

and

f

⊗

g

respectively,

2.

there is an

isomorphism

a

A

⁢

B

⁢

C

:

(

A

⊗

B

)

⊗

C

≅

A

⊗

(

B

⊗

C

)

, for arbitrary objects

A

,

B

,

C

in

𝒞

, such that

a

A

⁢

B

⁢

C

is natural in

A

,

B

and

C

. In other words,

–

a

-

B

⁢

C

:

(

-

⊗

B

)

⊗

C

⇒

-

⊗

(

B

⊗

C

)

is a

natural transformation

for arbitrary objects

B

,

C

in

𝒞

,

–

a

A

-

C

:

(

A

⊗

-

)

⊗

C

⇒

A

⊗

(

-

⊗

C

)

is a natural transformation for arbitrary objects

A

,

C

in

𝒞

,

–

a

A

⁢

B

-

:

(

A

⊗

B

)

⊗

-

⇒

A

⊗

(

B

⊗

-

)

is a natural transformation for arbitrary objects

A

,

B

in

𝒞

,

3.

there is an object

I

in

𝒞

called the

unit object

(or simply the

unit

),

4.

for any object

A

in

𝒞

, there are isomorphisms:

l

A

:

I

⊗

A

≅

A

and

r

A

:

A

⊗

I

≅

A

,

such that

l

A

and

r

A

are natural in

A

: both

l

:

I

⊗

-

⇒

-

and

r

:

-

⊗

I

⇒

-

are natural transformations

satisfying the following

commutative diagrams

:

•

unit coherence law

\xymatrix

⁢

@

+

=

2

⁢

c

⁢

m

⁢

(

A

⊗

I

)

⊗

B

⁢

\ar

⁢

[

r

⁢

r

]

a

A

⁢

I

⁢

B

⁢

\ar

⁢

[

d

⁢

r

]

r

A

⊗

1

B

⁢

&

⁢

&

⁢

A

⊗

(

I

⊗

B

)

⁢

\ar

⁢

[

d

⁢

l

]

1

A

⊗

r

B

⁢

&

⁢

A

⊗

B

⁢

&

•

associativity coherence law

\xymatrix

⁢

@

+

=

2

⁢

c

⁢

m

⁢

(

(

A

⊗

B

)

⊗

C

)

⊗

D

⁢

\ar

⁢

[

r

⁢

r

]

a

A

⊗

B

,

C

,

C

⁢

\ar

⁢

[

d

]

a

A

⁢

B

⁢

C

⊗

1

D

⁢

&

⁢

&

⁢

(

A

⊗

B

)

⊗

(

C

⊗

D

)

⁢

\ar

⁢

[

d

⁢

d

]

a

A

,

B

,

C

⊗

D

⁢

(

A

⊗

(

B

⊗

C

)

)

⊗

D

⁢

\ar

⁢

[

d

]

a

A

,

B

⊗

C

,

D

⁢

&

⁢

&

⁢

A

⊗

(

(

B

⊗

C

)

⊗

D

)

⁢

\ar

⁢

[

r

⁢

r

]

1

A

⊗

a

B

⁢

C

⁢

D

⁢

&

⁢

&

⁢

A

⊗

(

B

⊗

(

C

⊗

D

)

)

The bifunctor

⊗

is called the

tensor product

on

𝒞

, and the

natural isomorphisms

a

,

l

,

r

are called the

associativity isomorphism

, the

left unit isomorphism

, and the

right unit isomorphism

respectively.

Some examples of monoidal categories are

•

A prototype is the category of isomorphism classes of

vector spaces

over a field

𝕂

, herein the tensor product is the associative

operation

and the field

𝕂

itself is the unit element.

•

The

category of sets

is monoidal. The tensor product here is just the set-theoretic

cartesian product

, and any

singleton

can be used as the unit object.

•

The category of (left) modules over a ring

R

is monoidal. The tensor product is the usual tensor product (

http://planetmath.org/TensorProduct

) of modules, and

R

itself is the unit object.

•

The category of

bimodules

over a ring

R

is monoidal. The tensor product and the unit object are the same as in the previous example.

Monoidal categories play an important role in the topological quantum field theories (TQFT).

Title

monoidal category

Canonical

name

MonoidalCategory

Date of creation

2013-03-22 16:30:21

Last modified on

2013-03-22 16:30:21

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

14

Author

juanman (12619)

Entry type

Definition

Classification

msc 81-00

Classification

msc 18-00

Classification

msc 18D10

Synonym

monoid

Related topic

Category

Related topic

Algebroids

Related topic

Monoid

Related topic

StateOnTheTetrahedron

Defines

unit coherence

Defines

associativity coherence

Defines

tensor product

Defines

unit object

Defines

associativity isomorphism

Defines

left unit isomorphism

Defines

right unit isomorphism

Generated on Sat Feb 10 13:04:44 2018 by

LaTeXML