nLab
universal element

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Definition

A universal element of a functor F:CSetF: C \to Set is an element θF(x)\theta \in F(x), where xx is some object of CC, which exhibits representability of FF via the Yoneda lemma. That is, any element θF(x)\theta \in F(x) induces, in natural bijective fashion, a natural transformation

θ^:hom(x,)F\hat{\theta}: \hom(x, -) \to F

[]

θ^ y(f:xy)=defF(f)(θ)\hat{\theta}_y(f: x \to y) \stackrel{def}{=} F(f)(\theta)

and θ\theta is universal if θ^\hat{\theta} 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

hom(F(c),)hom(c,G),\hom(F(c), -) \cong \hom(c, G-),

the Brown representability theorem, and there are many others. A few more examples are discussed below.

Examples

Internal logic in toposes

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 :Ω×ΩΩ\wedge: \Omega \times \Omega \to \Omega.

Colimits of nerves

A question was brought to the nForum on colimits of nerves, conjecturing the following which we state as a proposition (proof below):

Proposition

If CC is a small category, then the colimit of the composite

C op(C)CatNerveSet Δ opC^{op} \stackrel{(- \downarrow C)}{\to} Cat \stackrel{Nerve}{\to} Set^{\Delta^{op}}

is equivalent to the nerve of CC.

As a warm-up:

Lemma

The colimit of

C op(C)CatC^{op} \stackrel{(- \downarrow C)}{\to} Cat

is isomorphic to CC.

Proof

The objects of colim c:C op(cC)colim_{c: C^{op}} (c \downarrow C) are equivalence classes of arrows or generalized elements cdc \to d where the equivalence \sim is generated by

(cfd)(cgcfd)(c \stackrel{f}{\to} d) \sim (c' \stackrel{g}{\to} c \stackrel{f}{\to} d)

and it is immediate that every generalized element g:cdg: c \to d is equivalent to the universal generalized element 1 d:dd1_d: d \to d; in spirit, this is the Yoneda lemma in disguise.

Now we prove the proposition on colimits of nerves.

Proof

Since colimits in Set Δ opSet^{\Delta^{op}} (sSet) are computed pointwise, we just have to show the colimit of

C op(cC)CatnerveSet Δ opev nSet,C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{nerve}{\to} Set^{\Delta^{op}} \stackrel{ev_n}{\to} Set,

where ev nev_n is evaluation at an object nn, agrees with nerve(C) nnerve(C)_n. This is

C op(cC)Cathom([n],)SetC^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{\hom([n], -)}{\to} Set

Now an nn-simplex in the comma category (cC)(c \downarrow C), which is an element of this composite, is the same as an (n+1)(n+1)-simplex beginning with the vertex cc, and the colimit (in SetSet) consists of equivalence classes of (n+1)(n+1)-simplices where a simplex beginning with cc is deemed equivalent to a simplex beginning with cc' obtained by pulling back along any g:ccg: c' \to c. And again, it is a triviality that each (n+1)(n+1)-simplex

c(d 0d n)c \to (d_0 \to \ldots \to d_n)

is equivalent to

d 01 d 0(d 0d n)d_0 \stackrel{1_{d_0}}{\to} (d_0 \to \ldots \to d_n)

but the collection of such d 0d nd_0 \to \ldots \to d_n is the same as nerve(C) nnerve(C)_n. This completes the verification.

Revised on September 4, 2010 22:01:31 by Toby Bartels (173.190.156.19)