# 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.

Parents:

• Mathematics

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