# Analog of a discrete fibration for sheafs

category-theoryfibration

Given a discrete fibration $$P : \text{Dis}(C^{\text{op}})$$ you can map to a copresheaf.

$$[[P]](x) = \Sigma s, C(P(s), x)$$

What is the equivalent if you want to map to a cosheaf on a site $$(C, J)$$?

I'm not so good with sheafs so I find this confusing.

I have a feeling you want a subset of discrete fibrations such that

$$\forall s, \text{id}_{P(s)} \in J(P(s))$$

You're thinking of discrete fibrations as mapping identity maps to identity maps.

I am not quite sure what you want for mapping to a sheaf either. I think you want something like:

$$[[P]](x) = \Sigma s, \{ f : C(P(s), x) | f \in J(s) \}$$

But this seems off to me.

There should also be an analogous operation for "sheafification" mapping discrete fibrations to the subset of fibrations.

$$\def\C{\mathcal{C}}\def\DFib{\text{DFib}}$$I will try to sketch a possible course of action for addressing this problem. The story can be told from different perspectives at different levels of generality (I forgot most of them, so I was hoping for some other user to answer before me...).

The moral of this story is that

Under the equivalence $$\Gamma : [\C^\text{op}, {\sf Set}] \cong \DFib(\C) : \Delta$$ sheaves correspond to those discrete fibrations that are stacks, i.e. such that two sets (the set of "descent data" and the set of "effective descent data") are in bijection.

In a sense, the conditions that define discrete fibration "satisfying descent" (i.e. those for which the above bijection exists) are just the sheaf conditions passed under the mirror of the equivalence above: a presheaf $$F$$ has an associated discrete fibration $$\Gamma F$$ (its category of elements, where an object is a pair $$(U, s\in FU)$$) over $$\C$$, and $$F$$ is a sheaf if and only if for every $$U$$ and every cover of $$U$$, $$F$$ fits into an equaliser diagram $$\textstyle FU \to \prod_i FU_i \rightrightarrows \prod_{i,j} FU_{ij}$$ so $$\Gamma F$$ will "satisfy descent" if it satisfies a similar property. In order to find which property is this, you have to understand what $$FU, FU_i$$ and $$FU_{ij}$$ are in terms of $$\Gamma F$$ (and the parallel maps induced by restriction of sections, of course).