(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
In the context of dg-geometry an ∞-stack $X$ is called perfect if its (∞,1)-category $QC(X)$ of quasicoherent ∞-stacks (of modules over the structure sheaf $\mathcal{O}(X)$) is generated from compact objects/dualizable objects: modules that are locally perfect chain complexes.
Let $k$ be a field of characteristic 0. Let $T$ be the Lawvere theory of commutative associative algebras over $k$. When this is regarded as an (∞,1)-algebraic theory, the $T$-$\infty$-algebras are modeled (by the monoidal Dold-Kan correspondence equivalently) by the
model structure on dg-algebras (over $k$, in non-positive degree, with positively graded differential).
The higher geometry/derived geometry over formal duals of these algebras is sometimes called dg-geometry: a general space in this context is given by an ∞-stack over a full sub-(∞,1)-site
of the opposite (∞,1)-category of these $\infty$-algebras.
The (∞,2)-presheaf of quasicoherent ∞-stacks is
given by
where on the right we take the $(\infty,1)$-category of $\infty$-modules over the $\infty$-algebra $A$, regarded as an unbounded dg-algebra.
For $X \in Sh_{(\infty,1)}(C)$ an ∞-stack in dg-geometry, write
for the $(\infty,1)$-category of quasicoherent $\infty$-stacks on $X$.
By the co-Yoneda lemma we may express every $X \in Sh_{(\infty,1)}(C)$ as an (∞,1)-colimit of representables
We have then
This appears as (Ben-ZviFrancisNadler, section 3.1).
(Ben-ZviFrancisNadler, section 3.1).
Let $A \in T Alg_\infty$ . An $A$-module is a perfect module if it lies in the smallest sub-(∞,1)-category of $A Mod$ containing $A$ and closed under finite (∞,1)-colimits and retracts.
For a ∞-stack $X \in Sh_{(\infty,1)}(C)$, the $\infty$-category $Perf(X)$ is the full sub-$(\infty,1)$-category of $QC(X)$ consisting of those modules that are prefect over every affine $U\to X$.
This appears as (Ben-ZviFrancisNadler, definition 3.1).
A ∞-stack $X \in Sh_{(\infty,1)}(C)$ is called a perfect stack if
it has affine diagonal $X \to X \times X$;
and $QC(X)$ is the (∞,1)-category of ind-objects
of the full sub-(∞,1)-category $Perf(X) \subset QC(X)$ of perfect complexes of modules on $X$.
A morphism $X \rightarrow Y$ is said to be perfect morphism if its fibers $X \times_Y U$ over affines $U \rightarrow Y$ are perfect.
A stable (∞,1)-category $C$ is compactly generated if it has a small set $\{c_i\}_{i \in I}$ of compact object that are generators in the sense that if for $N \in C$ we have that $C(c_i, N)$ is equivalent to the zero morphism, then $N$ is the zero object.
For a ∞-stack $X \in Sh_{(\infty,1)}(C)$ with affine diagonal, the following are equivalent:
$X$ is perfect
$QC(X)$ is
compactly generated,
and its compact and dualizable objects coincide.
The assigmnent
of the $(\infty,2)$-algebras $QC(X)$ of quasicoherent ∞-stacks to perfect $$\infty$-stacks $X$ constitutes a geometric ∞-function theory: this assignment commutes with (∞,1)-pullbacks and admits a ggood pull-push theory of integral transforms on sheaves.
Perfect stacks cover a broad array of spaces of interest, with notable exceptions being the (constant ∞-stack on a) classifying space $\mathcal{B}G$ of a topological group $G$ such as the circle $S^1 \simeq \mathcal{B} \mathbb{Z}$ or the classifying spaces of most algebraic groups in non-zero characteristic. This is because if $X$ is perfect, then the global sections functor $\Gamma$ must preserve colimits, which fails when the global sections $\Gamma(X, \mathcal{O}_X)$ of the structure sheaf is ‘too large’, as in the previous cases.
But the following are examples of perfect $\infty$-stacks
quasi-compact derived schemes with affine diagonal;
the total space of a quasi-projective morphism over a perfect base;
a quasi-projective derived scheme;
the quotient $X/G$ of a quasi-projective derived scheme $X$ by a linear action of an affine group (for $k$ of characteristic 0).
The concept of a perfect stack in the context of dg-geometry is considered in