Examination through isomorphism

Iso­mor­phism is the cor­rect no­tion of equal­ity be­tween ob­jects in a cat­e­gory. From the cat­e­gory-the­o­retic point of view, if you want to dis­t­in­guish be­tween two ob­jects which are iso­mor­phic but not equal, it means that the mor­phisms in your cat­e­gory don’t pre­serve what­ever as­pect of the ob­jects al­lows you to make this dis­tinc­tion, and hence the cat­e­gory doesn’t re­ally cap­ture what you want to be work­ing with. If you want to talk about it cat­e­gor­i­cally, you should con­sider a cat­e­gory with mor­phisms that pre­serve all of the struc­ture you care about, in­clud­ing what­ever al­lowed the dis­tinc­tion to be made.

For ex­am­ple (this ex­am­ple is due to Qiaochu Yuan), con­sider the cat­e­gory with ob­jects met­ric spaces, and mor­phisms con­tin­u­ous maps. The di­ame­ter of a met­ric space \((X,d)\), which is the max­i­mum value of \(d(x,y)\) for \(x,y \in X\), is a fea­ture of met­ric spaces which is not in­var­i­ant un­der iso­mor­phism in this cat­e­gory; for ex­am­ple, the sub­sets \([0,1]\) and \([0,2]\) of \(\mathbb{R}\), equipped with the usual met­rics in­her­ited from \(\mathbb{R}\), are iso­mor­phic in this cat­e­gory. There is a con­tin­u­ous map \(f : [0,1] \to [0,2]\) and a con­tin­u­ous map \(g : [0,2] \to [0,1]\) such that \(fg\) and \(gf\) are iden­tities. For ex­am­ple, one could take \(f\) to be mul­ti­pli­ca­tion by \(2\), and \(g\) to be di­vi­sion by \(2\). How­ever, the di­ame­ter of \([0,1]\) is \(1\), and the di­ame­ter of \([0,2]\) is \(2\). There­fore, in­so­far as “di­ame­ter” is a prop­erty of met­ric spaces, the ob­jects of these cat­e­gories are not met­ric spaces. The cor­rect name for them is “metriz­able spaces”, since this cat­e­gory is equiv­a­lent to the cat­e­gory whose ob­jects are topolog­i­cal spaces whose topol­ogy is in­duced by some met­ric and whose mor­phisms are con­tin­u­ous maps.

For a less re­al­is­tic (but more ob­vi­ous) ex­am­ple, con­sider the cat­e­gory of groups and ar­bi­trary func­tions be­tween their un­der­ly­ing sets. The ob­jects of this cat­e­gory are, sup­pos­edly, groups, but prop­er­ties of groups, such as “sim­ple”, do not re­spect iso­mor­phism in this cat­e­gory.

Another ex­am­ple of this is the product of, say, sets. It de­ter­mines a func­tor \(\text{Set}\times\text{Set}\to\text{Set}\). We would like to say that this is as­so­ci­a­tive, but this is false; a typ­i­cal el­e­ment of \(A \times (B \times C)\) looks like \((a,(b,c))\), while a typ­i­cal el­e­ment of \((A \times B) \times C\) looks like \(((a,b),c)\). Since these sets have differ­ent el­e­ments, they are not equal. How­ever, they are iso­mor­phic. In fact, the two func­tors \(\text{Set}\times\text{Set}\times\text{Set}\to\text{Set}\) given by \((A,B,C) \mapsto A \times (B \times C)\) and \((A,B,C) \mapsto (A \times B) \times C\) are iso­mor­phic in the cat­e­gory of func­tors \(\text{Set}\times\text{Set}\times\text{Set}\to\text{Set}\). That is, they are nat­u­rally iso­mor­phic. link to a “nat­u­ral iso­mor­phism” page


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.