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, differentials, (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...

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...