This paper demystifies the notion of the smashing spectrum of a stable presentably symmetric monoidal $\infty$-category, defined as a locale whose opens correspond to smashing localizations. Previously, this concept was studied in tensor-triangular geometry in the compactly generated rigid setting. Our main result identifies the smashing spectrum functor as the right adjoint to the spectral sheaves functor, providing in particular an external characterization that avoids explicit reference to objects, ideals, or localizations. The sheaves--spectrum adjunction formalizes the intuition that the smashing spectrum constitutes the best approximation of a given $\infty$-category by $\infty$-categories of sheaves. We establish an unstable generalization of this result by identifying the correct unstable analog of the smashing spectrum, which parametrizes smashing colocalizations instead.
As an application, we give a categorical presentation of Clausen--Scholze's categorified locales.
We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and equip it with a natural paracyclic structure (``rotational symmetry''). This fulfills a part of the general program to provide a rigorous account of perverse schobers which are (thus far conjectural) categorifications of perverse sheaves. Namely, in terms of our previous identification of perverse sheaves on Riemann surfaces with Milnor sheaves, the relative …
This second part comes to the construction of the spectrum associated to a situation of multi-adjunction. Exploiting a geometric understanding of its multi-versal property, the spectrum of an object is obtained as the spaces of local units equipped with a topology provided by orthogonality aspects. After recalling Diers original construction, this paper introduces new material. First we explain how the situation of multi-adjunction can be corrected in a situation of adjunction between categories of modeled spaces as in the topos-theoretic approach. Then we come to the 2-functorial aspects of t…
Diers developed a general theory of right multi-adjoint functors leading to a purely categorical, point-set construction of spectra. Situations of multiversal properties return sets of canonical solutions rather than a unique one. In the case of a right multi-adjoint, each object deploys a canonical cone of local units jointly assuming the role of the unit of an adjunction. This first part revolves around the theory of multi-adjoint and recalls or precises results that will be used later on for geometric purpose. We also study the weaker notion of local adjoint, proving Beck-Chevalley conditio…
This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure s…
There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is naturally isomorphic to the identity, such that F restricts to the Zariski spectrum functor Spec on the category of commutative rings (in a compatible way with the natural isomorphism). Both of these no-go re…
In this paper we prove that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one. We also discuss dimension spectrums of these categories.
In the (covariant) topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to each unital C*-algebra, A, a topos T(A) of sheaves on a locale and a commutative C*-algebra, a, within that topos. The Gelfand spectrum of a is a locale S in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale S, by noting that the construction of the Gelfand spectrum in a general topos can be described using geometric logic. As a consequence, the spectrum, seen as a bundle, is computed fibrewise. As a by-product of th…
The Grothendieck monoid of an exact category is a monoid version of the Grothendieck group. We use it to classify Serre subcategories of an exact category and to reconstruct the topology of a noetherian scheme. We first construct bijections between (i) the set of Serre subcategories of an exact category, (ii) the set of faces of its Grothendieck monoid, and (iii) the monoid spectrum of its Grothendieck monoid. By using (ii), we classify Serre subcategories of exact categories related to a finite dimensional algebra and a smooth projective curve. In particular, we determine the Grothendieck mon…
Optics and lenses are abstract categorical gadgets that model systems with bidirectional data flow. In this paper we observe that the denotational definition of optics - identifying two optics as equivalent by observing their behaviour from the outside - is not suitable for operational, software oriented approaches where optics are not merely observed, but built with their internal setups in mind. We identify operational differences between denotationally isomorphic categories of cartesian optics and lenses: their different composition rule and corresponding space-time tradeoffs, positioning t…
Given a tensor triangulated category we investigate the geometry of the Balmer spectrum as a locally ringed space. Specifically we construct functors assigning to every object in the category a corresponding sheaf and a notion of support based upon these sheaves. We compare this support to the usual support in tt-geometry and show that under reasonable conditions they agree on compact objects. We show that when tt-categories satisfy a scheme-like property then the sheaf associated to an object is quasi-coherent, and that in the presence of an appropriate t-structure and affine assumption, this…
