# Subobjects in the category of presheaves?

Suppose $$\mathcal{C}$$ is a locally small category, and $$X$$ be an element of $$\mathcal{C}.$$ A sub-object of $$X$$ is an isomorphism class of monomorphisms in to $$X.$$ Now suppose we embedd $$X$$ in $$[\mathcal{C}^{op}, \mathbf{Set}]$$ using Yoneda $$X\mapsto y(X)=\text{Hom}(-,X)$$. I would like to understand sub-objects (or sub-functors) of the functor $$y(X).$$ Without much of luck, I am struggling with this for sometime. Can anyone give me a hint on this problem?

• Subobjects of representable functors are the same thing as sieves on the representing objects, i.e. collections of maps into the representing object that are stable under precomposition with maps in $\mathcal C$. – asdq Apr 14 at 4:14
• @asdq: How would you prove this? – Bumblebee Apr 14 at 4:15
Subobjects of representable functors are the same thing as sieves on the representing objects, i.e. collections of maps into the representing object that are stable under precomposition with maps in $$\mathcal C$$. The correspondence is established by sending a subfunctor $$R\hookrightarrow y(X)$$ to the collection $$\bigcup_{Y\in \mathcal C} R(Y)$$ and conversely by sending a sieve $$S$$ on $$X$$ to the functor $$R$$ that maps an object $$Y$$ to the set of maps $$Y\to X$$ that are contained in $$S$$ and a map $$Z\to Y$$ to the map $$R(Y)\to R(Z)$$ given by precomposition.
• Thank you very much. I can see that for each $Y$ there is an injective map $R(Y)\to \operatorname{Hom}(Y, X).$ But still I don't see why $\bigcup_{Y\in \mathcal C} R(Y)$ form a sieve over $X.$ – Bumblebee Apr 14 at 4:34
• This is because $R$ is a subfunctor of $y(X)$, which implies that for any map $Z\to Y$ the map $y(X)(Y)\to y(X)(Z)$ restricts to a map $R(Y)\to R(Z)$, which precisely means that $\bigcup_{Y\in \mathcal C} R(Y)$ is stable under precomposition with the map $Z\to Y$. – asdq Apr 14 at 4:40