nLab
universal element

Definition

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

$$\hat{\theta }:\mathrm{hom}(x,-)\to F$$\hat{\theta}: \hom(x, -) \to F

$${\hat{\theta }}_y(f:x\to y)\stackrel{\mathrm{def}}{=}F(f)(\theta )$$\hat{\theta}_y(f: x \to y) \stackrel{def}{=} F(f)(\theta)

and $\theta$ is universal if $\hat{\theta }$ 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

$$\mathrm{hom}(F(c),-)\cong \mathrm{hom}(c,G-),$$\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 $\wedge :\Omega \times \Omega \to \Omega$.

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 $C$ is a small category, then the colimit of the composite

$$C^{\mathrm{op}}\stackrel{(-\downarrow C)}{\to }\mathrm{Cat}\stackrel{\mathrm{Nerve}}{\to }{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$$C^{op} \stackrel{(- \downarrow C)}{\to} Cat \stackrel{Nerve}{\to} Set^{\Delta^{op}}

is equivalent to the nerve of $C$.

As a warm-up:

Lemma

The colimit of

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

is isomorphic to $C$.

Proof

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

$$(c\stackrel{f}{\to }d)\sim (c\prime \stackrel{g}{\to }c\stackrel{f}{\to }d)$$(c \stackrel{f}{\to} d) \sim (c' \stackrel{g}{\to} c \stackrel{f}{\to} d)

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

Now we prove the proposition on colimits of nerves.

Proof

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

$$C^{\mathrm{op}}\stackrel{(c\downarrow C)}{\to }\mathrm{Cat}\stackrel{\mathrm{nerve}}{\to }{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\stackrel{{\mathrm{ev}}_n}{\to }\mathrm{Set},$$C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{nerve}{\to} Set^{\Delta^{op}} \stackrel{ev_n}{\to} Set,

where ${\mathrm{ev}}_n$ is evaluation at an object $n$, agrees with $\mathrm{nerve}(C{)}_n$. This is

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

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

$$c\to (d_0\to \dots \to d_n)$$c \to (d_0 \to \ldots \to d_n)

is equivalent to

$$d_0\stackrel{1_{d_0}}{\to }(d_0\to \dots \to d_n)$$d_0 \stackrel{1_{d_0}}{\to} (d_0 \to \ldots \to d_n)

but the collection of such $d_0\to \dots \to d_n$ is the same as $\mathrm{nerve}(C{)}_n$. This completes the verification.