2.14.2 Equality of semigroups

2.14.2 Equality of semigroups

Using the

equations

for path spaces discussed in the previous

sections

,
we can investigate when two

semigroups

are equal. Given semigroups

(

A

,

m

,

a

)

and

(

B

,

m

′

,

a

′

)

, by

Theorem

2.7.2 (

http://planetmath.org/27sigmatypes#Thmprethm1

), the type of paths

(

A

,

m

,

a

)

=

𝖲𝖾𝗆𝗂𝗀𝗋𝗈𝗎𝗉

(

B

,

m

′

,

a

′

)

is equal to the type of pairs

p

1

:

A

=

𝒰

B

and

p

2

:

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖲𝖾𝗆𝗂𝗀𝗋𝗈𝗎𝗉𝖲𝗍𝗋

⁢

(

p

1

,

(

m

,

a

)

)

=

(

m

′

,

a

′

)

.

By univalence,

p

1

is

𝗎𝖺

⁢

(

e

)

for some equivalence

e

. By
Theorem 2.7.2 (

http://planetmath.org/27sigmatypes#Thmprethm1

), function extensionality, and the above

analysis

of
transport in the type family

𝖲𝖾𝗆𝗂𝗀𝗋𝗈𝗎𝗉𝖲𝗍𝗋

,

p

2

is

equivalent

to a pair
of proofs, the first of which shows that

∏

y

1

,

y

2

:

B

e

⁢

(

m

⁢

(

e

-

1

⁢

(

y

1

)

,

e

-

1

⁢

(

y

2

)

)

)

=

m

′

⁢

(

y

1

,

y

2

)

and the second of which shows that

a

′

is equal to the induced

associativity

proof constructed from

a

in
2.14.3 (

http://planetmath.org/2141liftingequivalences#S0.E1

). But by cancellation of

inverses

2.14.2 (

http://planetmath.org/2141liftingequivalences#S0.E2X

) is equivalent to

∏

x

1

,

x

2

:

A

e

⁢

(

m

⁢

(

x

1

,

x

2

)

)

=

m

′

⁢

(

e

⁢

(

x

2

)

,

e

⁢

(

x

2

)

)

.

This says that

e

commutes with the

binary operation

, in the sense
that it takes

multiplication

in

A

(i.e.

m

) to multiplication in

B

(i.e.

m

′

). A similar rearrangement is possible for the equation relating

a

and

a

′

. Thus, an equality of semigroups consists exactly of an
equivalence on the

carrier

types that commutes with the semigroup

structure

.

For general types, the proof of associativity is thought of as part of
the structure of a semigroup. However, if we restrict to set-like types
(again, see

http://planetmath.org/node/87576

Chapter 3), the
equation relating

a

and

a

′

is trivially true. Moreover, in this
case, an equivalence between sets is exactly a

bijection

. Thus, we have
arrived at a standard definition of a

semigroup

isomorphism

: a
bijection on the carrier sets that preserves the multiplication

operation

. It is also possible to use the category-theoretic definition
of isomorphism, by defining a

semigroup homomorphism

to be a map
that preserves the multiplication, and arrive at the

conclusion

that equality of
semigroups is the same as two mutually inverse

homomorphisms

; but we
will not show the details here; see §9.8 (

http://planetmath.org/98thestructureidentityprinciple

).

The conclusion is that, thanks to univalence, semigroups are equal
precisely when they are isomorphic as

algebraic structures

. As we will see in §9.8 (

http://planetmath.org/98thestructureidentityprinciple

), the
conclusion applies more generally: in

homotopy type theory

, all constructions of
mathematical structures automatically respect isomorphisms, without any
tedious proofs or abuse of notation.

Title

2.14.2 Equality of semigroups

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