# Why are (Pre)sheaves more important than Co(pre)sheaves?

I'm learning Sheaf Theory, and this is an issue that's been bothering me.

Fix a small category $$\mathcal{C}$$.

A $$\mathcal{V}$$-valued presheaf on the small category $$\mathcal{C}$$ is a functor $$F:\mathcal{C}^{\text{op}}\rightarrow \mathcal{V}$$. This determines a category $$[\mathcal{C}^{\text{op}},\mathcal{V}]$$ where objects are $$\mathcal{V}$$-valued presheaves, and morphisms are natural transformations between them.

If we impose certain gluing conditions (namely a Grothendieck Topology $$J$$), we can form the full subcategory $$\text{Sh}_{\mathcal{V}}(\mathcal{C},J)$$ of $$[\mathcal{C}^{\text{op}},\mathcal{V}]$$ called the category of $$\mathcal{V}$$-valued sheaves on the category $$\mathcal{C}$$.

There is also a dual notion called a copresheaf, which is a functor $$F:\mathcal{C}\rightarrow \mathcal{V}$$, determining a category $$[\mathcal{C},\mathcal{V}]$$ of copresheaves similar to earlier. Cosheaves form a category (which I'll denote $$\textbf{CoSh}_{\mathcal{V}}(\mathcal{C},J)$$) by a similar construction. (One can also view co(pre)sheaves $$\mathcal{C}$$ as (pre)sheaves on $$\mathcal{C}^{\text{op}}$$)

If $$\mathcal{V}$$ is an abelian category, then studying $$\text{Sh}_{\mathcal{V}}(\mathcal{C},J)$$ gives rise to Sheaf Cohomology. Also, if $$\mathcal{V} = \textbf{Set}$$, then $$\text{Sh}_{\textbf{Set}}(\mathcal{C},J)$$ is a very important example of a topos.

My question is why are (pre)sheaves more studied/important than co(pre)sheaves, when $$\mathcal{C}$$ and $$\mathcal{C}^{\text{op}}$$ are both small categories? Do (pre)sheaves give more important information about the category $$\mathcal{C}$$ (depending on what one is studying)?

• A simple answer is that the Yoneda embedding $C \to [C^{op}, \text{Set}]$ is covariant. There is a sort of dual to the Yoneda embedding $C \to [C, \text{Set}]^{op}$ but $[C, \text{Set}]^{op}$ isn't a very nice category for various reasons, e.g. it's not locally presentable. – Qiaochu Yuan Mar 30 at 22:53