A universal element of a functor is an element , where is some object of , which exhibits representability of via the Yoneda lemma. That is, any element induces, in natural bijective fashion, a natural transformation
and is universal if is an isomorphism.
Thus, universal elements are part and parcel of any discussion involving representability. Well-known examples include adjoint functors, where one has representability
the Brown representability theorem, and there are many others. A few more examples are discussed below.
Quite often, logical constructions that work for arbitrary toposes can be deduced by arguing from universal elements. Some simple examples follow.
Consider first the construction of internal conjunction .
is equivalent to the nerve of .
As a warm-up:
The colimit of
is isomorphic to .
The objects of are equivalence classes of arrows or generalized elements where the equivalence is generated by
and it is immediate that every generalized element is equivalent to the universal generalized element ; in spirit, this is the Yoneda lemma in disguise.
Now we prove the proposition on colimits of nerves.
where is evaluation at an object , agrees with . This is
Now an -simplex in the comma category , which is an element of this composite, is the same as an -simplex beginning with the vertex , and the colimit (in ) consists of equivalence classes of -simplices where a simplex beginning with is deemed equivalent to a simplex beginning with obtained by pulling back along any . And again, it is a triviality that each -simplex
is equivalent to
but the collection of such is the same as . This completes the verification.