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](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/pretopos)?



Do pretopoi have cohomology and homotopy groups?
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.)


