I'm learning about representable functors from [Vistoli notes](http://homepage.sns.it/vistoli/descent.pdf) thanks to [Charles Siegel's answer](https://mathoverflow.net/questions/10314/every-scheme-as-a-sheaf-references/10316#10316).


I see that any category $\mathcal C$ can be embedded into $\text{Hom}\\,(\mathcal C^{op}, \mathcal Set)$ by means of Yoneda embedding. I wonder if there are examples where the latter category would be interesting in itself, other then for these purposes?



Yoneda embedding target
Lots! Categories of that form (when C is small) are often called "presheaf categories". Many interesting categories are presheaf categories, such as simplicial sets, cubical sets, symmetric sets, etc. In particular, any presheaf category is a topos, and many interesting toposes are presheaf categories. The category of G-sets for any discrete group G is another nice example, since G can be regarded as a groupoid, hence as a category. Presheaves on a topological space are also interesting, if only as a means to the construction of sheaves. And *simplicial* presheaves on a category C (which are the same as presheaves on $C\times \Delta$) are sometimes easier to work with (once you put a nice model structure on them) than simplicial sheaves.


Many other interesting categories are full subcategories of some presheaf category; in fact a category is a full subcategory of a presheaf category as soon as it has a small dense subcategory. Thus, in particular, any accessible category is a subcategory of a presheaf category. This includes almost any "algebraic" category, such as groups, rings, fields, Lie algebras, etc.


