• Source: Mac Lane coherence theorem

In category theory, a branch of mathematics, Mac Lane's coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. But regarding a result about certain commutative diagrams, Kelly is states as follows: "no longer be
seen as constituting the essence of a coherence theorem". More precisely (cf. #Counter-example), it states every formal diagram commutes, where "formal diagram" is an analog of well-formed formulae and terms in proof theory.
The theorem can be stated as a strictification result; namely, every monoidal category is monoidally equivalent to a strict monoidal category.




Counter-example


It is not reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell.
Let






S
e
t



0


⊂


S
e
t




{\displaystyle {\mathsf {Set}}_{0}\subset {\mathsf {Set}}}

be a skeleton of the category of sets and D a unique countable set in it; note



D
×
D
=
D


{\displaystyle D\times D=D}

by uniqueness. Let



p
:
D
=
D
×
D
→
D


{\displaystyle p:D=D\times D\to D}

be the projection onto the first factor. For any functions



f
,
g
:
D
→
D


{\displaystyle f,g:D\to D}

, we have



f
∘
p
=
p
∘
(
f
×
g
)


{\displaystyle f\circ p=p\circ (f\times g)}

. Now, suppose the natural isomorphisms



α
:
X
×
(
Y
×
Z
)
≃
(
X
×
Y
)
×
Z


{\displaystyle \alpha :X\times (Y\times Z)\simeq (X\times Y)\times Z}

are the identity; in particular, that is the case for



X
=
Y
=
Z
=
D


{\displaystyle X=Y=Z=D}

. Then for any



f
,
g
,
h
:
D
→
D


{\displaystyle f,g,h:D\to D}

, since



α


{\displaystyle \alpha }

is the identity and is natural,




f
∘
p
=
p
∘
(
f
×
(
g
×
h
)
)
=
p
∘
α
∘
(
f
×
(
g
×
h
)
)
=
p
∘
(
(
f
×
g
)
×
h
)
∘
α
=
(
f
×
g
)
∘
p


{\displaystyle f\circ p=p\circ (f\times (g\times h))=p\circ \alpha \circ (f\times (g\times h))=p\circ ((f\times g)\times h)\circ \alpha =(f\times g)\circ p}

.
Since



p


{\displaystyle p}

is an epimorphism, this implies



f
=
f
×
g


{\displaystyle f=f\times g}

. Similarly, using the projection onto the second factor, we get



g
=
f
×
g


{\displaystyle g=f\times g}

and so



f
=
g


{\displaystyle f=g}

, which is absurd.




Proof






= Coherence condition

=
In monoidal category



C


{\displaystyle C}

, the following two conditions are called coherence conditions:

Let a bifunctor



⊗
:

C

×

C

→

C



{\displaystyle \otimes \colon \mathbf {C} \times \mathbf {C} \to \mathbf {C} }

called the tensor product, a natural isomorphism




α

A
,
B
,
C




{\displaystyle \alpha _{A,B,C}}

, called the associator:





α

A
,
B
,
C


:
(
A
⊗
B
)
⊗
C
→
A
⊗
(
B
⊗
C
)


{\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)}


Also, let



I


{\displaystyle I}

an identity object and



I


{\displaystyle I}

has a left identity, a natural isomorphism




λ

A




{\displaystyle \lambda _{A}}

called the left unitor:





λ

A


:
I
⊗
A
→
A


{\displaystyle \lambda _{A}:I\otimes A\rightarrow A}


as well as, let



I


{\displaystyle I}

has a right identity, a natural isomorphism




ρ

A




{\displaystyle \rho _{A}}

called the right unitor:





ρ

A


:
A
⊗
I
→
A


{\displaystyle \rho _{A}:A\otimes I\rightarrow A}

.




= Pentagon identity and triangle identity

=




See also


Coherency (homotopy theory)
Monoidal category
Symmetric monoidal category
Coherence condition




Notes






References






Further reading






External links


Armstrong, John (29 June 2007). "Mac Lane's Coherence Theorem". The Unapologetic Mathematician.
Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. "18.769, Spring 2009, Graduate Topics in Lie Theory: Tensor Categories §.Lecture 3". MIT Open Course Ware.
"coherence theorem for monoidal categories". ncatlab.org.
"Mac Lane's proof of the coherence theorem for monoidal categories". ncatlab.org.
"coherence and strictification". ncatlab.org.
"coherence and strictification for monoidal categories". ncatlab.org.

Kata Kunci Pencarian:

• Mac Lane coherence theorem
• Coherency (homotopy theory)
• Saunders Mac Lane
• Strictification
• Coherence condition
• Permutoassociahedron
• Monoidal category
• Weak Hopf algebra
• Rig category
• Timeline of category theory and related mathematics