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.

I think mostly I'm looking for a name I can google to lookup more information.

## Best Answer

$\def\C{\mathcal{C}}\def\DFib{\text{DFib}}$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

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).