In set theory, starting from a model $V$ of $ZFC$, a forcing notion $\mathbb{P}$, and a generic filter $G \subset \mathbb{P}$ over $V$, we can find a generic extension which is a model of $ZFC$ and is the smallest model, having the same ordinals as $V$ such that $V[G] \supseteq V$ and $G \in V[G].$
My question is: what is the corresponding construction in terms of toposes, if we start with an arbitrary topos $T$.
Giving references is appreciated.
The informal analogue, is simply the notion of topos of sheaves.
If I work in a "ground" topos (whose object I call set), then a "forcing extention" would be just a Grothendieck topos, that is a topos of sheaves on a small site.
If you want to adopt an external point of view and start from an elementary topos $\mathcal{E}$ , then a forcing extension of $\mathcal{E}$ is a topos $\mathcal{F}$ that can be obtained as the category of $\mathcal{E}$-valued sheaves on an internal site in $\mathcal{E}$, where internal site means " a category object in $\mathcal{E}$ endowed with a "topology".
The simplest way to define the word "topology" here is to say that it is a Lawvere-Tierney operator in the topos of $\mathcal{E}$-valued presheaves on the category object. But one can also define it in a more Grothendieckian style using collection of subobjects of power objects satisfying the internal version of the axioms of a topology.
It is a well known theorem of topos theory that the topos that can be obtained from $\mathcal{E}$ this way are exactly the topos endowed with a bounded geometric morphism $\mathcal{F} \to \mathcal{E}$. (see section B3.3 of P.T.Johnstone Sketches of an elephant).
The best way to get a feeling of why this is a good analogy is to look at the topos theoretic proof of the independence of the continuum hypothesis in MacLane and Moerdijk "Sheaves in geometry and logic" (section VI.2).
However, it is not a perfect analogy: First as, pointed out by Andreas Blass in comment, the standard set theoretic forcing corresponds only the case of a double negation topology on a poset. Though it can be shown that any sheaves topos admit a cover by one of this form, so this is not a strong restriction.
But there is a more subtle difference: Informally, in set theory people construct a a model that contains a "generic filter", in topos theory we construct the model that contains "the universal (generic) filter" (in the sense of classifying toposes). The point here is that the toposes obtained this way are not well-pointed in general, so they can not directly corresponds to a model of ZFC.
If you want a more precise analogy you need to combine the construction of the topos of sheaves with a construction that reproduce a model of ZFC out of a topos. For this I recommend to look at Mike Shulman's paper that give a very good exposition to the topic.
