The category theoretical definition of stacks (as given for instance in Giraud: Cohomologie non-abélienne) allow for arbitrary categories as targets (the stack condition only involves isomorphisms however). A natural example is the category of quasi-coherent sheaves (which has the category of vector bundles as a subcategory). However, when you are talking about algebraic stacks (which are category theoretic stacks fulfilling extra conditions) they only involve isomorphisms. Note that given any stack restricting to isomorphisms gives a stack in groupoids. This is what one does when one considers the algebraic stack of vector bundles: Start with the stack of vector bundles with arbitrary morphisms. This is not an algebraic stack but restricting to isomorphisms gives one.
General stacks (with non-isomorphisms) are used extensively as they encode the idea of descent. This is somewhat orthogonal to algebraic stacks which try to encode the idea of a moduli problem.
Addendum: All morphisms in a descent datum are isomorphisms (this actually
follows and does not have to be assumed). However full descent means
that you can descend objects (a descent datum of objects comes from an object downstairs) but also arbitrary morphisms (a descent datum of morphisms comes from a morphisms downstairs). These two properties together can be formulated as an equivalence of categories between the category of obejcts downstairs and the category of descent data.
Addendum 1: Charles poses an interesting question. One answer can be based on the fact there seems to also be a philosophical difference between general stacks and algebraic stacks. General stacks are based on the idea that we have some objects and relations between them, the morphisms, that can be glued together over some kind of covering. Hence, usually the objects themselves are the things of main interest and the gluing condition is just an extra (though very important) condition on such objects.
Algebraic stacks on the other hand are things that themselves are glued. The relevant idea is that groupoids are a natural generalisation of equivalence relations. (One can more or less arrive at the idea of a groupoid by thinking of bereasoned equivalence relations, elements do not just happen to be equivalent but there are specific, in general several, reasons for them to be equivalent.)
Having said that, one could start with the fact that an algebraic is the stack associated to a smooth algebraic groupoid (i.e., source and target maps are smooth). This gives a candidate generalisation by just looking at smooth algebraic categories instead. However, no natural examples that are not groupoids comes (at least) to my mind. I think the reason might be the above philosophical distinction.
While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still an especially important place in our heart. Especially when we say algebraic topology instead of homotopy theory. This says that while all of category theory and all of homological algebra belongs to the study of $(\infty, 1)$-categories, this is not where our aim is.
The roots of our subject lie in the study of nice spaces like manifolds. Important questions are:
- Can we classify manifolds up to some equivalence relation?
- Can we understand maps between manifolds?
The coarsest useful equivalence relation for the classification of manifolds is bordism and this is also the basis of most other classification results (those using surgery theory). Computing bordism groups was an important topic in earlier algebraic topology and was done succesfully for some flavors rather early ($\Omega_O$, $\Omega_U$, $\Omega_{SO}$,...). But one of the most important variants, both theoretically and from the viewpoint of clasification of manifolds, is framed bordism. By an old theorem by Pontryagin, the framed bordism groups are isomorphic to the stable homotopy groups of spheres, connecting it to the second question.
One can say that much of algebraic topology was invented or can be used to study the stable homotopy groups of spheres. One of the most recent spectacular advances in algebraic topology was the solution of (most of) the Kervaire invariant 1 problem by Hill, Hopkins and Ravenel about framed manifolds/stable homotopy groups of spheres. They used a tremendous amount of stuff to solve this classical problem: equivariant topology, chromatic homotopy theory, spectral sequences, orthogonal spectra, abstract homotopy theory, ...
Likewise topological modular forms $tmf$ have important applications to the stable homotopy groups of spheres and also to string bordism. And to really understand $tmf$, you have to study some spectral algebraic geometry.
I do not want to say that all of algebraic topology still directly aims at classical questions. As soon as we see an interesting structure, we also study it for its own sake; new phenomena need explanations and developing abstract frameworks is also fun. But like in the relationship between mathematics and physics, sharpening our tools and exploring by pure curiosity can be quite useful for the classical questions. When people replaced older, in some aspects more clumsy models of spectra by symmetric and orthogonal spectra, they probably didn't have in mind any direct applications to framed manifolds. But what Hill, Hopkins and Ravenel did would have been much harder without these tools in their hands.
Best Answer
The work on analytic geometry is all joint with Dustin Clausen!
Your main question seems a little vague to me, but let me try to get at it by answering the subquestions. See also the discussion at the nCatCafe. Also, as David Corfield comments, much of this had been observed long before: https://nforum.ncatlab.org/discussion/5473/etale-site/?Focus=43431#Comment_43431
Yes, I think cohesion does not work in algebraic or $p$-adic contexts. The issue is that schemes or rigid-analytic spaces are just not locally contractible.
Cohesion does not seem to have been applied in algebraic or $p$-adic contexts. However, I realized recently (before this nCatCafe discussion), in my project with Laurent Fargues on the geometrization of the local Langlands correspondence, that the existence of the left adjoint to pullback ("relative homology") is a really useful structure in the pro-etale setting. I'm still somewhat confused about many things, but to some extent it can be used as a replacement to the functor $f_!$ of compactly supported cohomology, and has the advantage that its definition is completely canonical and it exists and has good properties even without any assumptions on $f$ (like being of finite dimension), at least after passing to "solid $\ell$-adic sheaves". So it may be that the existence of this left adjoint, which I believe is a main part of cohesion, may play some important role.
As I already hinted in 2, this relative notion of cohesiveness may be a convenient notion. In brief, there are no sites relevant in algebraic geometry that are cohesive over sets, but there are such sites that are (essentially) cohesive over condensed sets; for example, the big pro-etale site on all schemes over a separably closed field $k$. So in this way the approach relative to condensed sets has benefits.
All of these questions sidestep the question of why condensed sets are not cohesive over sets, when cohesion is meant to model "toposes of spaces" and condensed sets are meant to be "the topos of spaces". I think the issue here is simply that for Lawvere a "space" was always built from locally contractible pieces, while work in algebraic geometry has taught us that schemes are just not built in this way. But things are OK if instead of "locally contractible"(="locally contractible onto a point") one says "locally contractible onto a profinite set", and this leads to the idea of cohesion relative to the topos of condensed sets.
Let me use this opportunity to point out that this dichotomy between locally contractible things as in the familar geometry over $\mathbb R$ and profinite things as codified in condensed sets is one of the key things that Dustin and I had to overcome in our work on analytic geometry. To prove our results on liquid $\mathbb R$-vector spaces we have to resolve real vector spaces by locally profinite sets!