# What's a (infinity-) semi-stack?

A stack is an object that mixes the notions of (algebraic) space and group. The key insight of stack theory is that most things you would want to do with spaces you can do with stacks: namely, you have a symmetric monoidal category of quasicoherent sheaves, you have forms and differentials, you have pullbacks, (several) notions of locality, an étale theory, etc.

There is a generalization to infinity-stacks, which mix the notions of algebraic space and infinity-groupoid (equivalently, $(\infty, 0)$-category, or topological space).

I want a generalization of this to a notion of "semi-stack", which mixes the concepts of algebraic space and semigroup -- and more generally, in the derived context, a notion of $(\infty, 1)$-stacks which interpolates between spaces and $(\infty, 1)$-categories. Probably because I'm not familiar with higher algebra, I haven't been able to find a definition. Here are some properties I would like from the category of "derived semistacks", $Stck_{(\infty,1)}$, roughly in decreasing order of importance (non-derived versions to be supplied by the reader):

(1) Given any derived semistack $X\in Stck_{(\infty, 1)}$ I want defined a symmetric monoidal dg category (at least in characteristic zero, with some infinity-categorical modifications in characteristic p) of (quasi-)coherent sheaves $QCoh(X)$.

(2) It should be true that any suitably finite $(\infty, 1)$-category $c$ has a corresponding object $c\times \text{pt}\in Stck_{(\infty,1)}.$ The category $QCoh(c\times \text{pt})$ is the symmetric monoidal representation category $Rep_k(c)$. (Tensor product of representation of categories is defined object-wise, like for quivers.)

(3) The category $Stck_{(\infty, 1)}$ is fibered over the category of small categories (i.e. $Hom(X, Y)$ is a small, "discrete" category for X, Y derived semistacks, with composition given by functors in a functorial way).

(4) The category $Stck_{(\infty)}$ (viewed as fibered in $(\infty)$-groupoids) is fully faithfully contained in $Stck_{(\infty,1)}$. Moreover, I want a "stackification" functor $X\mapsto X^\times$ from $Stck_{(\infty,1)}\to Stck_{\infty}$ which satisfies the following adjunction property: $Hom(X, Y)^\times = Hom(X, Y^\times)$, for $X\in Stck_\infty$ and $Y\in Stck_{\infty, 1}$. Here $Hom(X, Y)^\times$ is the $(\infty, 0)$-category underlying $Hom(X, Y)$ (consisting of the same objects and all invertible morphisms).

(5) Let $\overline{G}_m$ be the semigroup scheme given by $\mathbb{A}^1$ with product given by $x\times y:\, = xy$, extending the group scheme $G_m = \mathbb{A}^1\setminus \{0\}$. I want an object $\text{pt}/\overline{G}_m$ (one point with "endomorphisms $\overline{G}_m$") such that $(\text{pt}/\overline{G}_m)^\times = BG_m$ is the classifying space of the multiplicative group, and such that for an ordinary scheme (or stack) $X$, we have $$Hom(X, \text{pt}/\overline{G}_m)$$ given by the category with objects line bundles $L$ over $X$ and morphisms $$\pi_0Hom(L, M) \cong Hom_{Coh}(L, M)$$ ("morphisms between line bundles which are allowed to go to zero".)

Does anyone know of such a construction? Would it be sufficient to consider infinity-categories (in Lurie's simplicial set formulation) over, say, the fppf site on a point?

It would be nice if this category (perhaps under some additional conditions, like smoothness of some sort) were to exhibit some of the good behavior of stacks, viz. good notions of locality, an étale homology theory, other motivic invariants, etc...

• "I want a generalization of this to a notion of semi-stack, which mixes the concepts of algebraic space and semigroup": it might be useful to explain what is your motivation for wanting such a generalization. It would be great if you could also explain why your own proposed answer isn't satisfactory: considering simplicial sheaves over the (say) fppf site, which satisfy a local inner-Kan condition sounds like a very reasonable thing to do. Which ones of the desiderata (1), (2), (3), (4), (5) are you having difficulty checking? – André Henriques Dec 21 '17 at 22:56
• The motivation is essentially #5. I don't know enough of this theory to check it, and don't know if the fppf local simplicial construction is "correct" (in particular, whether it gives the right answer for (5)) -- it would be great if it were! – Dmitry Vaintrob Dec 21 '17 at 23:00
• More generally (following some ideas of Roman Bezrukavnikov and David Kazhdan) I want to study the category of Z_p-points of "semistacks" similar to $\text{pt}/\overline{G}_m$, whose representation theory should have nice properties. – Dmitry Vaintrob Dec 21 '17 at 23:23
• I always find units very confusing, but aren't you looking for a stacky analog of monoids (like (5)) rather than semigroups? (that's also what Scott's answer provides) – David Ben-Zvi Dec 22 '17 at 16:07

If we remove the $\infty$ from your question, the idea you want already has the name "stack" or "stack of categories", and it's been considered for about 50 years. See for example Giraud's Non-abelian cohomology book (from 1971). Stacks are just fibered categories that satisfy good descent properties with respect to some chosen topology. Standard examples fibered over schemes include the stack $QCoh$ of all quasi-coherent sheaves and the stack $Ell^{isog}$ of elliptic curves, where morphisms are pullback squares composed with isogenies. Your example of line bundles, where morphisms are allowed to be non-invertible, is a substack of $QCoh$.
Notational suggestion: As David Ben-Zvi mentioned in the comments, some people are introducing new terms to distinguish between stacks in categories and stacks in groupoids. In fact, many people have now done so, and these new terms generally don't agree with each other. It might be less confusing to add numerical prefixes describing the type of fiber, following the practice of higher categories (and the part of this question that doesn't use "semi-stack"). That is, a $(1,0)$-stack is a stack in groupoids, a $(0,0)$-stack is a sheaf of sets (or setoids), a $(1,1)$-stack is a stack in categories, an $(\infty,0)$-stack is what is commonly called an infinity-stack, and an $(\infty,1)$-stack is what the question seems to be seeking.
• Such a pile gives an ordinary stack, by looking at the maximal subgroupoid, together with an algebra object in self-spans, given by the Hom spaces. As such these things are fairly straightforward to capture in an $\infty$-setting --- e.g. the book of Gaitsgory-Rozenblyum discusses "stacky monoids" --Segal objects in derived stacks – David Ben-Zvi Dec 22 '17 at 16:12