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 $(B,m^{\prime },a^{\prime })$, by Theorem 2.7.2 (http://planetmath.org/27sigmatypes#Thmprethm1), the type of paths$(A,m,a){=}_{\mathrm{𝖲𝖾𝗆𝗂𝗀𝗋𝗈𝗎𝗉}}(B,m^{\prime },a^{\prime })$is equal to the type of pairs

$p_1$$:A{=}_{𝒰}B\mathit{\hspace{1em}\hspace{1em}}\text{and}$

$p_2$$:{\mathrm{𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍}}^{\mathrm{𝖲𝖾𝗆𝗂𝗀𝗋𝗈𝗎𝗉𝖲𝗍𝗋}}(p_1,(m,a))=(m^{\prime },a^{\prime }).$

By univalence, $p_1$ is $\mathrm{𝗎𝖺}(e)$ for some equivalence $e$. ByTheorem 2.7.2 (http://planetmath.org/27sigmatypes#Thmprethm1), function extensionality, and the above analysis oftransport in the type family $\mathrm{𝖲𝖾𝗆𝗂𝗀𝗋𝗈𝗎𝗉𝖲𝗍𝗋}$, $p_2$ is equivalent to a pairof proofs, the first of which shows that

$$\prod _{y_1,y_2:B}e(m(e^{-1}(y_1),e^{-1}(y_2)))=m^{\prime }(y_1,y_2)$$

and the second of which shows that $a^{\prime }$ is equal to the inducedassociativity proof constructed from $a$ in2.14.3 (http://planetmath.org/2141liftingequivalences#S0.E1). But by cancellation of inverses2.14.2 (http://planetmath.org/2141liftingequivalences#S0.E2X) is equivalent to

$$\prod _{x_1,x_2:A}e(m(x_1,x_2))=m^{\prime }(e(x_2),e(x_2)).$$

This says that $e$ commutes with the binary operation, in the sensethat it takes multiplication in $A$ (i.e. $m$) to multiplication in $B$(i.e. $m^{\prime }$). A similar rearrangement is possible for the equation relating$a$ and $a^{\prime }$. Thus, an equality of semigroups consists exactly of anequivalence on the carrier types that commutes with the semigroupstructure.

For general types, the proof of associativity is thought of as part ofthe structure of a semigroup. However, if we restrict to set-like types(again, see http://planetmath.org/node/87576Chapter 3), theequation relating $a$ and $a^{\prime }$ is trivially true. Moreover, in thiscase, an equivalence between sets is exactly a bijection. Thus, we havearrived at a standard definition of a semigroup isomorphism: abijection on the carrier sets that preserves the multiplicationoperation. It is also possible to use the category-theoretic definitionof isomorphism, by defining a semigroup homomorphism to be a mapthat preserves the multiplication, and arrive at the conclusion that equality ofsemigroups is the same as two mutually inverse homomorphisms; but wewill not show the details here; see §9.8 (http://planetmath.org/98thestructureidentityprinciple).

The conclusion is that, thanks to univalence, semigroups are equalprecisely when they are isomorphic as algebraic structures. As we will see in §9.8 (http://planetmath.org/98thestructureidentityprinciple), theconclusion applies more generally: in homotopy type theory, all constructions ofmathematical structures automatically respect isomorphisms, without anytedious proofs or abuse of notation.