Generalized element

In cat­e­gory the­ory, a gen­er­al­ized el­e­ment of an ob­ject \(X\) of a cat­e­gory is any mor­phism \(x : A \to X\) with codomain \(X\). In this situ­a­tion, \(A\) is called the shape, or do­main of defi­ni­tion, of the el­e­ment \(x\). We’ll un­pack this.

Gen­er­al­ized el­e­ments gen­er­al­ize elements

We’ll need a set with a sin­gle el­e­ment: for con­crete­ness, let us de­note it \(I\), and say that its sin­gle el­e­ment is \(*\). That is, let \(I = \{*\}\). For a given set \(X\), there is a nat­u­ral cor­re­spon­dence be­tween the fol­low­ing no­tions: an el­e­ment of \(X\), and a func­tion from the set \(I\) to the set \(X\). On the one hand, if you have an el­e­ment \(x\) of \(X\), you can define a func­tion from \(I\) to \(X\) by set­ting \(f(i) = x\) for any \(i \in I\); that is, by tak­ing \(f\) to be the con­stant func­tion with value \(x\). On the other hand, if you have a func­tion \(f : I \to X\), then since \(*\) is an el­e­ment of \(I\), \(f(*)\) is an el­e­ment of \(X\). So in the cat­e­gory of sets, gen­er­al­ized el­e­ments of a set \(X\) that have shape \(I\), which are by defi­ni­tion maps \(I \to X\), are the same thing (at least up to iso­mor­phism, which as usual is all we care about).

Gen­er­al­ized el­e­ments in sets

In the cat­e­gory of sets, if a set \(A\) has \(n\) el­e­ments, a gen­er­al­ized el­e­ment of shape \(A\) of a set \(X\) is an \(n\)-tu­ple of el­e­ments of \(X\). is there more to say here? or less?

Some­times there is no `best shape’

Based on the case of sets, you might ini­tially think that it suffices to con­sider gen­er­al­ized el­e­ments whose shape is the ter­mi­nal ob­ject add link \(1\). How­ever, in the cat­e­gory of groups, since the ter­mi­nal ob­ject is also ini­tial ex­plain this some­where, each ob­ject has a unique gen­er­al­ized el­e­ment of shape \(1\). How­ever, in this case, there is a sin­gle shape that suffices, namely the in­te­gers \(\mathbb{Z}\). A gen­er­al­ized el­e­ment of shape \(\mathbb{Z}\) of an abelian group \(A\) is just an or­di­nary el­e­ment of \(A\).

How­ever, some­times there is no sin­gle ob­ject whose gen­er­al­ized el­e­ments can dis­t­in­guish ev­ery­thing up to iso­mor­phism. For ex­am­ple, con­sider \(\text{Set} \times \text{Set}\) link to a page about the product of two cat­e­gories. If we use gen­er­al­ized el­e­ments of shape \((X,Y)\), then they won’t be able to dis­t­in­guish be­tween the ob­jects \((2^A, 2^{X + B})\) and \((2^{Y + A}, 2^{B})\), up to iso­mor­phism, since maps from \((X,Y)\) into the first are the same as el­e­ments of \((2^A)^X\times(2^{X+B})^Y \cong 2^{X\times A + Y \times (X + B)} \cong 2^{X \times A + Y \times B + X \times Y}\), and maps from \((X,Y)\) into the sec­ond are the same as el­e­ments of \((2^{Y+A})^X \times (2^B)^Y \cong 2^{X\times(Y+A) + Y \times B} \cong 2^{X \times A + Y \times B + X \times Y}\). Th­ese ob­jects will them­selves be non-iso­mor­phic as long as at least one of \(X\) and \(Y\) is not the empty set; if both are, then clearly the func­tor still fails to dis­t­in­guish ob­jects up to iso­mor­phism. (More tech­ni­cally, it does not re­flect iso­mor­phisms. ex­plain or avoid this ter­minol­ogy) In­tu­itively, be­cause ob­jects of this cat­e­gory con­tain the data of two sets, the in­for­ma­tion can­not be cap­tured by a sin­gle hom­set. This in­tu­ition is con­sis­tent with the fact that it can be cap­tured with two: the gen­er­al­ized el­e­ments of shapes \((0,1)\) and \((1,0)\) to­gether de­ter­mine ev­ery ob­ject up to iso­mor­phism.

Mor­phisms are func­tions on gen­er­al­ized elements

If \(x\) is an \(A\)-shaped el­e­ment of \(X\), and \(f\) is a mor­phism from \(X\) to \(Y\), then \(f(x) := f\circ x\) is an \(A\)-shaped el­e­ment of \(Y\). The Yoneda lemma cre­ate Yoneda lemma page states that ev­ery func­tion on gen­er­al­ized el­e­ments which com­mutes with repa­ram­e­ter­i­za­tion, i.e. \(f(xu) = f(x) u\), is ac­tu­ally given by a mor­phism in the cat­e­gory.


  • Mathematics

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