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Let $f: \mathcal{E} \rightarrow \mathcal{T}$ be a geometric morphism between two (Grothendieck) toposes (or maybe more generally a bounded geometric morphism between elementary toposes).

It is well known that if $f$ is an open or proper surjection then objects (and even locales) descend along $f$.

What I want to known is whether one also have descent for Higher categorical structures in $\mathcal{E}$ ?

To be precise, I am still speaking about ordinary $1$-toposes, but I want to consider for example the $2$-category of groupoid object, category object or toposes inside $\mathcal{E}$. I would like to know if for example groupoid objects/category objects in $\mathcal{E}$ satisfies a 2-categorical version of descent.

Or, even more optimistically do Grothendieck toposes in $\mathcal{E}$ satisfies a $2$-categorical version of descent (i.e. does open or proper surjections are effective descent morphism in the $(2,1)$-category of toposes.)

Maybe one need to add some hypothesis on $f$, for example being locally connected instead of just open , or tidy instead of proper ?

any results even for a restricted class of morphism will be of interest (for example, hyperconnected morphisms)

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    $\begingroup$ Just based on $n$-categorical voodoo, I would expect local connectedness and tidiness to be what you need for 2-descent, since they are the 2-categorical versions of openness and properness; and similarly for $n$-descent you would need their appropriate $n$-categorical versions. But that's just a guess. $\endgroup$ – Mike Shulman Mar 10 '16 at 20:17
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    $\begingroup$ @MikeShulman : that was also what I was thinking, but I just found strong evidence that open is in fact enough ! (see the edit) $\endgroup$ – Simon Henry Mar 11 '16 at 17:05
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I have been able to gather all the elements for the case of $2$-categorical descent along open surjections, so I will write it as an answer.

It is worth noting that the proof below follows the exact same path as the proof of descent for locales and objects along open surjections given in the Elephant. So what follow is probably optimal and obtaining statement for Higher categorical structures probably involves working with Higher toposes (with additional difficulties related to hypercompletness in the case of infnity categorical structures). Their is also a High chance that everything works exactly the same for proper surjections (and it suffices to proove the first theorem for proper surjections and everything will follow)

Theorem (Moerdijk) : Open surjections are effective descent morphism in the $2$-category of toposes.

Moerdijk's proof is in a paper called "Descent for toposes" in 1989 in Bulletin of the BMS, which is unfortunatly not very easy to find. It is a very nice proof that roughly goes along the same line as the modern proof that open/proper surjections are of effective descent for locales (the one in the elephant): it follow formally from:

1) the stability properties of open surjections

2) The fact that open surjection are (stably) the coequalizer of their kernel pair.

3) The fact that "equivalence relation" (groupoids) whose legs are open surjection have stable co-equalizer.

With the exception that for (3) he was only able to prove it for groupoids object whose action map $G_1 \rightarrow G_0 \times G_0$ is localic. But as toposes have localic diagonal it is enough for the proof of the descent theorem.

Proposition: Constructively, any topos $\mathcal{T}$ such that $p:\mathcal{T} \rightarrow *$ is open and $\Delta: \mathcal{T} \rightarrow \mathcal{T} \times \mathcal{T}$ is etale is isomorphic to the category of functor from a groupoid $G$ to sets.

Proof: Let $\mathcal{T}$ be such a topos, in particular $\mathcal{T}$ and its diagonal are open, hence $\mathcal{T}$ is atomic. As $\Delta$ is etale, there is an object $X$ in $\mathcal{T} \times \mathcal{T}$ such that $\mathcal{T} \simeq (\mathcal{T} \times \mathcal{T}) / X $. $X$ can be covered by objects of the form $a \otimes b$ where $a$ and $b$ are atoms of $\mathcal{T}$ (and $a \otimes b$ denotes the object $\pi_1^*(a) \times \pi_2^*(b)$). We fixe such an object $a \otimes b \rightarrow X$. As $a \otimes b$ is in $\mathcal{T} \times \mathcal{T}/X$ is corresponds to an object $c$ in $\mathcal{T}$. Moreover one has: $(\mathcal{T} \times \mathcal{T} )/( a \otimes b) \simeq \mathcal{T}/a \times \mathcal{T}/b \simeq \mathcal{T}/c$ and all those isomorphisms are compatible to the maps to $\mathcal{T} \times \mathcal{T}$. In particular, the map $\mathcal{T}/c \rightarrow \mathcal{T}/a \times \mathcal{T}/b$ is induced by morphisms from $c$ to $a$ and $b$, hence can be factored as the diagonal map $\mathcal{T}/c \rightarrow \mathcal{T}/c \times \mathcal{T}/c$ followed by an etale map $\mathcal{T}/c \times \mathcal{T}/c \rightarrow \mathcal{T}/a \times \mathcal{T}/b$. So we have an isomorphism which is factored as $f \circ g$ with $f$ etale, this implies that $g$ is an inclusion (indeed, internally in the target of $f$ it is a map from a point to a set). In our case, the diagonal map of $\mathcal{T}/c$ is an inclusion, by C2.4.14 this implies that $\mathcal{T}/c$ is localic and atomic hence discrete. As $\mathcal{T}$ can be covered by such toposes this impies the result.

From this, one easily deduces:

Corollary: Let $\mathcal{T}$ be a topos, then the $2$-category of groupoid object of $\mathcal{T}$ is equivalent to the full subcategory of the $2$-category of toposes over $\mathcal{T}$ whose map to $\mathcal{T}$ is open with an etale diagonal.

Corollary: Open surjections are of effective descent for the $2$-categories of groupoids objets.

Indeed, topos descend along open surjection, and the condition of the above corollary are detected by open surjection (C5.1.7 of the elephant) hence one has descent for toposes which are open with étale diagonal and those are the same as groupoids.

Proposition: Open surjections are of effective descent for categories.

Roughly, if you have an open surjection $f:\mathcal{E} \rightarrow \mathcal{T}$ and $\mathcal{C}$ a category object in $\mathcal{E}$ endowed with a descent data for $f$, then the descent data restrict to a descent data on the core groupoid of $\mathcal{C}$ hence one can apply descent to it and we obtain a groupoid $G_0$ in $\mathcal{T}$. Then for any pair of object $x,y \in G$ internally in $\mathcal{T}$ one has a pair of object $f^*x, f^* y$ in $f^* G$ and $f^* G$ is equivalent to the core of $C$, one easily see that the set of morphisms between $f^* x$ and $f^* y$ in $C$ comes with a descent data and the rest is just a boring manipulation of some $2$-categorical diagram. One can of course gives some more abstract proof of this relying in the end on the fact that the $2$-category of categories can be written explcitely as the category of model in groupoids of some "cartesian $2$-theory" (whatever that precisely means).

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