Simple examples of non-representable functors

Question

I am looking for examples of non-representable functors, to see how the Yoneda lemma works in these cases.

Here is one: let $\mathbf{C}$ be the category of finite-dimensional Euclidean spaces, with morphisms being isometries (including into higher dimensions). Let $F$ be the functor $\mathbf{C}^{op}\to \mathbf{Set}$ that sends each space $C$ to the set of real-valued functions $C\to \mathbb{R}$ (not necessarily linear or even continuous), and sends each map $f:D \to C$ to the set function sending $g:C\to \mathbb{R}$ to $g \circ f$. We can think of $F$ as defining a presheaf of nonlinear or noncontinuous functions; which means (I think) that $F$ is not representable in $\mathbf{C}$. Yoneda's lemma says that a natural transformation from $Hom(-,C)$ to $F$ is uniquely specified by a section of this presheaf on $C$, i.e. by a specific function $g:C\to \mathbb{R}$.

In this case, $F(C)$ is explicitly a set of functions on the set underlying $C$. Are there any easy-to-understand examples where this is not the case?

Answer 1

You can build such presheaves by being a little bit silly about things. Observe that if your category $\mathbb{C}$ is the category of finite dimensional Euclidean spaces, $\mathbb{C}$ does not necessarily admit all colimits. In particular, consider the sequence $\phi_n:E^{n} \to E^{n+1}$ given by the mappings, for all $\mathbf{x} \in E^n$, $\phi_n(\mathbf{x}) := (\mathbf{x},0) \in E^{n+1}$ (so $\phi_n$ embeds $E^n$ as a subspace of $E^{n+1}$ by appending a zero at the end of each vector in $E^n$). It is easy to see that the colimit $$ E := \lim_{\substack{\longrightarrow \\ n \in \mathbb{N}}} (E^{n},\phi_n) $$ does not exist in $\mathbb{C}$ because $E$ must be infinite dimensional. However, note that the Yoneda embedding $\mathbf{y}:\mathbb{C} \to [\mathbb{C}^{\operatorname{op}},\mathbf{Set}]$ preserves colimits, and the functor category is cocomplete. In this category we have that the colimit $$ P(-) := \lim_{\substack{\longrightarrow \\ n \in \mathbb{N}}}(\mathbb{C}(-,E^n),\mathbb{C}(-,\phi_n)) $$ exists as a presheaf by the cocompleteness of $[\mathbb{C}^{\operatorname{op}},\mathbf{Set}]$. This presheaf, however, is not representable. If it were, it would necessarily need to be represented by the object $E$ described above, but since $E$ is not an object in your category, this functor is not representable. Essentially $P$ is telling you about one type of embedding of any finite dimensional space into some countably infinite Euclidean space, and since $\mathbb{C}$ does not admit infinite dimensional spaces as objects, there is no such space in $\mathbb{C}$ with which to represent this object.

Answer 2

Most $\mathbf{Set}$-valued functors aren't representable. You can more or less pick one at random and you'll be good. One approach then is to take a bunch of representable functors and combine them in some way. The result often won't be representable. As a particular case of that, a limit of representable functors being representable is equivalent to the corresponding limit existing in the embedded category. So take any category that doesn't have all limits and a limit it doesn't have gives rise to a limit presheaf that isn't representable.

More generally, all (1-categorical) universal properties correspond to some functor being representable. Again, this leads to plenty of non-representable functors. For example, in the category of vector spaces $\mathsf{Hom}(U\times -,W)$ will typically not be representable because $U\times -$ doesn't have a right adjoint, i.e. the category of vector spaces isn't cartesian closed.