# Presheaves saturated by monomorphisms

This is Corollary 1.3.10, p12. Note we are fixing a small Eilenberg Zilber category. The terminology are all given in the text. Tell me if the link is not accessible.

1.3.10: If a class of prehseaves over $A$ is saturated by monomorphisms and contains all representable presheaves, then it contains all presheaves over $A$.

The proof is by simple induction. But I do not understand two parts of the proof:

1. Why is $Sk_0(Y)$ a sum of representable presheaf (which I assume means presheaves of the form $h_a$ from Yonnea embedding $A \rightarrow \hat{A}$).
2. The axioms only guarantee that $\bigsqcup h_a \in \mathcal{C}$ but not $\bigsqcup \partial h_a \in \mathcal{C}$.

## 1 Answer

1. I would explain this via the fact that $sk_nF$ is constructed by restricting $F$ to the full subcategory of $A$ on the objects of length at most $n$, then left Kan extending (applying the left adjoint of restriction) to get back to $A$. Then the claim follows from the fact that the full subcategory of length-$0$ objects is discrete, since every presheaf on a discrete category is a coproduct of representables. If you don't like this argument, the point is that $sk_0F$ has as sections the non-degenerate sections of $F$ over length $0$ objects together with their degeneracies.

2. $\partial h_a$ is its own $n-1$-skeleton, so the inductive hypothesis covers it.

• Sorry, I am still unclear of 1. It would really help if you spell out details. I don't understand how every presheaf over a discrete category is a coproduct of representables. I also dont' know what left Kan extending means. Are there any introductory references for these? – CL. Sep 5 '18 at 9:42