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Grothendieck topoi have cohomology: the abelian category of abelian group objects in a topos has enough injectives, hence one can consider the right derived functors of the global sections functor from abelian group objects to abelian groups. (For details see Chapter 8 in Johnstone's book Topos theory.)

Also, Grothendieck topoi have homotopy groups. (I think a reference is Artin and Mazur's Etale homotopy.)

Question: Can some of these algebraic invariants (or other "topological properties") of Grothendieck topoi be generalized to pretopoi?

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    $\begingroup$ I don't know what a pretopos is, but if the axioms are strong enough to prove that the category of abelian groups in it is an abelian category, then you should be able to form Yoneda type $Ext$ groups, so that gives cohomology as $Ext^i(\mathbb{Z}, -)$. $\endgroup$ Jan 24 at 15:10
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    $\begingroup$ It is indeed the case that abelian groups in a pretopos form an abelian category. $\endgroup$ Jan 25 at 2:54

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There's a long story that can be told here but I will try to be brief. In one sense, the answer is yes – you can certainly define cohomology and homotopy groups and so on for pretoposes and have them coincide with the classical definitions for Grothendieck toposes – but in another sense the answer is no – because you are essentially just embedding the pretopos into a suitable Grothendieck topos and reducing to that case.

Let $\mathcal{E}$ be a pretopos. That means $\mathcal{E}$ is a category with a terminal object, pullbacks, finitary coproducts, and coequalisers of internal equivalence relations, such that finitary coproducts are disjoint and preserved by pullback, and coequalisers of internal equivalence relations are effective and preserved by pullback. In short, $\mathcal{E}$ satisfies the exactness part of the Giraud axioms, with finitary coproducts eplacing infinitary coproducts. That in itself should be a powerful reason to believe that any finitary constructions that can be carried out in a Grothendieck topos can also be carried out in $\mathcal{E}$ with the same results. Indeed:

Proposition. Assuming $\mathcal{E}$ is small, there is a fully faithful embedding of $\mathcal{E}$ into a Grothendieck topos where the embedding preserves finite limits, finitary coproducts, and coequalisers of internal equivalence relations.

Proof. Regard $\mathcal{E}$ as a site where the covering sieves are those that contain a sieve generated by a finite family that is jointly strongly epimorphic, and take the topos of sheaves on this site. ◼

(If $\mathcal{E}$ is not small then go up to a universe where it is, or find a subpretopos that is small and contains the objects and morphisms you care about.)

Concretely, the category $\textbf{Ab} (\mathcal{E})$ of internal abelian groups in $\mathcal{E}$ is an abelian category (but not necessarily AB4 or AB4*, let alone AB5). So you can go on to define the category $\textbf{Ch} (\mathcal{E})$ of chain complexes in $\textbf{Ab} (\mathcal{E})$ and then the (unbounded) derived category $\mathbf{D} (\mathcal{E})$. What you do not get is the existence of enough injectives in $\textbf{Ab} (\mathcal{E})$ itself. Nonetheless, the definition of derived functors as (absolute, or at least pointwise) Kan extensions makes sense, and some derived functors can be constructed without injective resolutions. For example, although $\mathbf{R} \textrm{Hom}_{\textbf{Ch} (\mathcal{E})} (A, -) : \mathbf{D} (\mathcal{E}) \to \mathbf{D} (\textbf{Ab})$ itself does not have an obvious construction, $H_0 \mathbf{R} \textrm{Hom}_{\textbf{Ch} (\mathcal{E})} (A, -) : \mathbf{D} (\mathcal{E}) \to \textbf{Ab}$ always exists: you can directly check that $\textrm{Hom}_{\mathbf{D} (\mathcal{E})} (A, -) : \mathbf{D} (\mathcal{E}) \to \textbf{Ab}$ works. Also, if $\mathcal{E}$ is small, then any functor $\textbf{Ch} (\mathcal{E}) \to \textbf{Ab}$ whatsoever admits a pointwise left Kan extension along $\textbf{Ch} (\mathcal{E}) \to \mathbf{D} (\mathcal{E})$... but it is unclear to me whether this is consistent with what $(\infty, 1)$-category theory would give.

Similarly (in some sense...), the category $\textbf{Kan} (\mathcal{E})$ of internal Kan complexes in $\mathcal{E}$ is a category of fibrant objects (in the sense of Brown) where the fibrations are the internal Kan fibrations and the weak equivalences are the internal weak homotopy equivalences. Thus the homotopy category $\mathbf{H} (\mathcal{E})$, obtained by localising $\textbf{Kan} (\mathcal{E})$ with respect to internal weak homotopy equivalences, is reasonable in the sense that there is a nice-ish formula for its hom-sets. The category $\textbf{Set}_\textrm{fin}$ of finite sets is the initial pretopos, so we get an induced functor $L : \mathbf{H} (\textbf{Set}_\textrm{fin}) \to \mathbf{H} (\mathcal{E})$. The homotopy type of $\mathcal{E}$ is "morally" a representing object for the functor $\textrm{Hom}_{\mathbf{H} (\mathcal{E})} (1, L {-}) : \mathbf{H} (\textbf{Set}_\textrm{fin}) \to \textbf{Set}$, where $1$ is the terminal object, but in practice this functor is rarely representable (even if $\mathcal{E}$ is a Grothendieck topos) so we are forced to make various tweaks like replacing $\textbf{Set}_\textrm{fin}$ with $\textbf{Set}_{< \kappa}$ (if $\mathcal{E}$ has coproducts of families of size $< \kappa$), or allowing more generalised notions of representability (e.g. pro-representability), or both.

(Actually, this phenomenon can already be seen at homotopy level 0 for Grothendieck toposes, so in some sense the difficulty does not (only) come from trying to work in higher homotopy levels or with general pretoposes instead of Grothendieck toposes.)

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    $\begingroup$ RHom always exist: consider the dg category of cochain complexes, take the Drinfeld dg quotient by acyclic complexes and RHom will be the Hom of the latter. Taking (the spaces of the) connective cover of such Rhom will always coincide with the mapping space of the $(\infty,1)$-category obtained by inverting quasi-isomorphisms in the 1-category of cochain complexes. This is a story you can tell with the derived category of any abelian category (no need of projectives/injectives). Smilarly, $Kan(E)$ has an $(\infty,1)$-theoretic localization which is perfectly understandable. $\endgroup$ Jan 25 at 20:24
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    $\begingroup$ About Kan extensions and $(\infty,1)$-categories: if $E$ is small, than the $\infty$-category $D(E)$ obtained by inverting quasi-isomorphisms is small. For any $(\infty,1)$-category $C$ with small colimits, we can left Kan extend along any functor between small $(\infty,1)$-category $A\to B$ any functor $A\to C$. Take $A=Ch(E)$ and $D(E)$. If $C$ is a $1$-category, all this will factor through the homotopy categories of $A$ and $B$ for free. $\endgroup$ Jan 25 at 20:32
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    $\begingroup$ If we restrict with absolute Kan extensions (as resolutions give us), this gives immediate compatibilities between derived functors defined via $(\infty,1)$-categories and those defined via $1$-categories. $\endgroup$ Jan 25 at 20:32
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Already elementary toposes can fail to have enough injectives: Let $M$ be the model of ZF by Andreas Blass in his 1979 paper Injectivity, Projectivity, and the Axiom of Choice. The category of $M$-sets and $M$-maps is an elementary topos in which the only injective abelian group is the zero group. Hence this pretopos fails to have enough injectives.

However, this observation only rules out one particular approach to cohomology. Donu already mentioned one more in the comments and there are several others.

My current favorite to constructivize and predicativize cohomology is to use of a mix of the approaches of Emily Riehl (employing pointwise Kan extensions) and the Stacks Project. This framework is sufficiently flexible to not demand derived functors to be defined everywhere. Rather, they will be defined just for those objects for which we happen to have a suitable resolution. Classically, by using injectives, we can pretend that any object has such a resolution, but these are irrelevant for computing purposes. My thesis is that for any particular object for which we can actually compute the value of a particular derived functor of interest, we also have a more effective resolution available.

Filling out the details of this program is work in progress. It seems to work for quasicoherent sheaves on quasicompact separated schemes (using Čech resolutions) and for arbitrary sheaves on spectral spaces (using a variant of flabby resolutions).

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    $\begingroup$ In case you are thinking of flabby resolutions as a substitute for injective ones: I'm in the middle of a revision of this paper on injective and flabby objects in toposes. The main new result will be that there are enough flabby modules in any elementary topos with an NNO. However, in the absence of Zorn's Lemma, these (still) fail to have the exactness properties which sheaf cohomology requires. A better variant of the notion of flabby objects will be subject to a forthcoming paper. $\endgroup$ Jan 24 at 16:07

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