Zoran Skoda
essay geotop

UNDER CONSTRUCTION

This will grow into an essay, mainly for graduate students, outlining how the geometry and topology are organized in modern mathematics, especially as a result of the Grothendieck’s revolution in the architecture of mathematics.

Categorical preliminary: Yoneda lemma

Most mathematical structures are organized into categories. We take usual assumption that the morphisms between two given objects form a set, rather than a proper class (“local smallness”); while all objects (and hence all morphisms in a category) may form a proper class. If the objects also form a set, we say that the category is small. If CC is a category, c,dc,d objects in CC, the set of morphisms from cc to dd may be denoted by one of the 5 standard notations Hom(c,d)=Mor(c,d)=Hom C(c,d)=Mor C(c,d)=C(c,d)Hom(c,d)= Mor(c,d) = Hom_C(c,d) = Mor_C(c,d) = C(c,d). For example, TopTop is a category whose objects are topological spaces, morphisms are continuous maps and the composition is defined as usual. Useful notions are invariant under the change of object by an isomorphic object. Similarly, it is useful that the constructions changing structure are defined not only on objects but also on morphisms and that the constructions extend to a (covariant) functor, i.e. the construction commutes with taking the composition and sends identities to identities. we say that a functor F:CDF:C\to D is full (resp. faithful, fully faithful) if for each pair c,cOb(C)c,c'\in Ob(C), functor FF sends Hom C(c,c)Hom_C(c,c') into Hom D(F(c),F(c))Hom_D(F(c),F(c')) injectively (resp. surjectively, bijectively); and we say that FF is a essentially surjective if for every dOb(D)d\in Ob(D) there exists cOb(D)c\in Ob(D) and an invertible morphism f:F(c)df:F(c)\to d. Every equivalence is fully faithful and essentially surjective; if we assume the axiom of choice than the converse follows as well (this can be proven as an interesting exercise). Fully faithful functor is the proper analogue of an embedding in the world of categories (and often called that way); indeed, it is an equivalence with a full subcategory.

By inverting arrows and the order of composition in a category CC, one arrives at the dual or opposite category. It is convenient to look at contravariant functors from category CC as functors from the opposite category C opC^{op}; such functors, especially when the codomain is the category SetSet of sets and functions, or GrpGrp of groups and homomorphisms of groups are often called presheaves. In particular, a functor F:C opSetF:C^{op}\to Set is said to be a presheaf of sets on CC.

I assume that the reader is familiar with the notion of a natural transformation α:FG\alpha : F\to G between two functors F,G:CDF,G: C\to D, which is given by its components {α c} cC\{\alpha_c\}_{c\in C}, which are morphisms in DD such that for each morphism f:ccf: c\to c', G(f)α c=α cF(f):F(c)G(c)G(f)\circ \alpha_c = \alpha_{c'}\circ F(f):F(c)\to G(c') (what is usually drawn as a commutative square). Given a pair of small categories C,DC,D, the functors from CC to DD as objects together with natural transformations as morphisms, form a category Fun(C,D)=D CFun(C,D)=D^C.

An object cc in a category CC is (universal) initial (resp. terminal) object if for every other object xx in CC there exist a unique morphism cxc\to x (resp. xcx\to c). Category may either have an initial object or not, but if there are many initial objects then each two are isomorphic via a unique isomorphism.

Given a category CC, a diagram in CC is a functor d:DCd:D\to C whose domain is a small category DD. Given a diagram d:DCd:D\to C, a cone with vertex xx is a natural transformation α\alpha from a constant functor const x:DCconst_x:D\to C, zxz\mapsto x for all zOb(D)z\in Ob(D). A morphism from a cone α:const xd\alpha:const_x\to d to a cone β:const yd\beta:const_y\to d, is a map f:xyf: x\to y such that the transformation const f:const xconst yconst_f: const_x\to const_y whose every component is ff and for which βconst f=α\beta\circ const_f = \alpha. A terminal object in the category of cones over dd is called the limit cone or limit of the diagram dd; if it exists it is unique up to a unique isomorphism in the category of cones over dd. Dually, transformations to a vertex are called cocones; an initial object in the category of cocones is called the colimit cone, or simply colimit. By abuse of language one often says limit for a vertex of a limit. Special names to limits are attached if the source category of the diagram involved is discrete (product), directed (inverse directed limit), cofiltered (cofiltered limit) or if it consists of two objects a,ba,b and two distinct nontrivial arrows aba\to b (equalizer). Dually, one has colimits called coproducts, directed colimits, filtered colimits and coequalizers, respectively. Existence of arbitrary (small) (co)products and of (co)equalizers, together imply existence of arbitrary small (co)limits.

Let cc be an object in a category CC; then we define the corresponding representable presheaf of sets h c=Hom(,c):C opSeth_c = Hom(-,c):C^{op}\to Set by

  • on objects: h c(d)=Hom(d,c)h_c(d) = Hom(d,c)

  • on morphisms: for f:ddf:d\to d', define h c(f)(g)=Hom(f,c)(g)=gf:dc h_c(f)(g) = Hom(f,c)(g) = g\circ f:d\to c for every g:dcg:d'\to c.

A representable presheaf of sets on CC is a presheaf of sets FF together with natural isomorphism of functors (i.e. invertible natural transformation) α:Fh c\alpha: F\cong h_c for some cOb(C)c\in Ob(C).

Yoneda lemma is a very simple but important theorem in modern mathematics and it comes in two versions, strong and weak. Strong version says that the natural transformations from any functor FF to a representable functor h ch_c (cc a fixed object in CC) are in a natural bijection with the set F(c)F(c), i.e. there is a bijection F(c)Nat(h c,F)F(c) \cong Nat(h_c,F) which is natural in CC (i.e. it is a component of natural transformation at cc). It is easy to prove this as an exercise. In particular, take F=h eF = h_e for some other object eCe\in C; then Nat(h e,h c)Hom(e,c)Nat(h_e,h_c) \cong Hom(e,c); therefore one obtains the weak Yoneda lemma: the functor h:ch ch: c\mapsto h_c is fully faithful (covariant) functor CFun(C op,Set)C\to Fun(C^{op},Set). The category of presheaves of sets Fun(C op,Set)Fun(C^{op},Set) is also denoted by C ^C^\hat{}. The weak Yoneda lemma, thus, means that every small category CC embeds as a full and faithful subcategory of the category of presheaves of sets C ^C^\hat{}, this embedding h:ch ch:c\mapsto h_c is called the Yoneda embedding CC ^C\hookrightarrow C^{\hat{}}. Being full and faithful means that we did not loose any information; the morphisms between the image objects look just as before. But there is an advantage: we are in a better category of presheaves of sets. The category of presheaves is better because it is complete and cocomplete, i.e. it has all small categorical limits and colimits (because the category SetSet has it, and we compute objectwise for presheaves) over small diagrams; one has more ‘space’ to do various constructions in the bigger category C ^=Set C opC^\hat{}=Set^{C^{op}}. Moreover, one can check directly that the Yoneda embedding CSet C opC\hookrightarrow Set^{C^{op}} preserves limits, but not necessarily the colimits. There are refinements of Yoneda embedding CIndCC sm ^C ^C\hookrightarrow Ind C\hookrightarrow C^{\hat{}}_{sm}\hookrightarrow C^{\hat{}}, where IndCInd C is the full subcategory whose objects are the ind-objects in CC, i.e. those presheaves C opSetC^{op}\to Set which are colimits of small filtered diagrams of representables and C ^C^{\hat{}} the full subcategory whose objects are the small presheaves, i.e. those presheaves which are colimits of small diagrams of representables.

Higher categories

While in a category (synonym:1-category) there are objects and morphisms, one can consider nn-categories where one has objects, morphisms, 2-morphisms which are morphisms among morphisms and so on till nn-morphisms, together with structure of various compositions. kk-morphisms are also called kk-cells, and objects 00-cells. In higher category theory a main problem is that the natural examples involve compositions which are not associative but associative up to higher morphisms, which themselves satisfy additional “coherence” relations. Particularly important are (n,1)(n,1)-categories, which are nn-categories such that all higher kk-morphisms for k>1k\gt 1 are invertible. It may look suprising, but natural, that various approaches to higher categories lead to equivalent theories. For (,1)(\infty,1)-categories in addition to models involving lots of explicit structure of cells, compositions, coherences and so on, there are low-structure models involving simplicial sets and required properties instead of additional structures. This is partly based on Grothendieck’s homotopy conjecture which states that the homotopy types of CW-complexes correspond to the homotopy types of \infty-groupoids, i.e. (,1)(\infty,1)-categories whose 1-morphisms are also invertible. This can more combinatorially be described using simplicial sets, which are presheaves of sets on the simplex category Δ\Delta whose objects are nonempty finite totally ordered sets 0<1<<n0\lt 1\lt\ldots \lt n and morphisms are the monotone maps. The \infinity-groupoids correspond those simplicial sets which satisfy so called Kan conditions; the (,1)(\infinity,1)-categories correspond to the simplicial sets satisfying just the subset of so called inner Kan conditions.

There are Yoneda embeddings in higher category theory. The functors and presheaves in higher category theory respect associativity up to coherent higher cells.

Sheaves, covers and Grothendieck topologies

Sheaf theory and its higher analogues (stacks and higher stacks; all together called descent theory) are about passage between the local and the global; about gluing global objects, sections etc. from local objects and sections; and about defining global spaces from local models.

Let a set XX be equipped with topology τ𝒫(X)\tau\subset \mathcal{P}(X). The topology is the set of objects of a category Ouv X=Ouv X,τOuv_X = Ouv_{X,\tau} where the morphisms UVU\to V are the inclusions of τ\tau-open sets; in other words the morphism set Hom(U,V)Hom(U,V) is either empty or a singleton. A presheaf of sets F:Ouv X,τ opSetF: Ouv_{X,\tau}^{op}\to Set is said to be a presheaf of sets on the topological space (X,τ)(X,\tau). Given the inclusion i UV:UVi_{U V}: U\subset V the map F(i UV):F(V)F(U)F(i_{U V}):F(V)\to F(U) is called a restriction map r UVr_{U V} from VV to UU; for each sF(U)s\in F(U) one also denotes the restriction r UV(s)=s| UF(U)r_{U V}(s) = s|_U\in F(U). A presheaf FF on (X,τ)(X,\tau) is a monopresheaf or separated presheaf if for every family {U α} αA\{U_\alpha\}_{\alpha\in A} of open sets two elements s,tF( αU α)s,t\in F(\cup_\alpha U_\alpha) are different whenever s| U α=t| U αs|_{U_\alpha} = t|_{U_\alpha} for all αA\alpha \in A. The presheaf FF is an epipresheaf if for every family {U α} αA\{U_\alpha\}_{\alpha\in A} of open sets and for every family s αF(U α)s_\alpha \in F(U_\alpha) such that s α| U αU β=s β| U αU βs_\alpha|_{U_\alpha\cap U_\beta} = s_\beta|_{U_\alpha\cap U_\beta} for all pairs (α,β)(\alpha,\beta), there is at least one sF( αU α)s\in F(\cup_\alpha U_\alpha) such that s α=s| U αs_\alpha = s|_{U_\alpha}. A presheaf is said to be a sheaf if it is a separated epipresheaf. Sheaves form a full subcategory of the category of sheaves (where the morphisms are natural transformations) which has a left adjoint functor, called the sheafification (or the associated sheaf) functor. Notice that, in the above epipresheaf and monopresheaf conditions, {U α} α\{U_\alpha\}_\alpha is a cover of αU α\cup_\alpha U_\alpha in the classical sense.

There is an equivalence between the sheaves of sets on (X,τ)(X,\tau) and surjective local homeomorphisms (Y,τ Y)(X,τ)(Y,\tau_Y)\to (X,\tau) which comes as a restriction of an adjoint pair of functors between presheaves of sets on XX and topological spaces (Y,τ Y)(Y,\tau_Y) over (X,τ)(X,\tau). Thus the sheaves can be considered as local homeomorphisms over (X,τ)(X,\tau), also called etale spaces.

Grothendieck considers a more general category CC instead of Ouv X,τOuv_{X,\tau} and wants to utilize more general families of maps {U αY} α\{U_\alpha\to Y\}_\alpha as covers than just the families of inclusions whose union is YY (the latter conditions even do not make sense literally for a general category). He axiomatized such covering families; the resulting notion is called a (Grothendieck) pretopology on CC. The axioms in a category admiting pullbacks:

  • (isomorphism) any singleton {VU}\{V\to U\} consisting of an isomorphism is covering;
  • (stability) a pullback of a covering family is covering;
  • (transitivity) if {U iU} iI\{U_i\to U\}_{i\in I} is covering family and for each jj, the family {U ijU j} iI j\{ U_{i j}\to U_j\}_{i\in I_j} is covering then the family of all compositions {U ijU jU} jJ,iI j\{U_{i j}\to U_j\to U\}_{j\in J, i\in I_j} is covering family.

A slightly different version of this notion is called Grothendieck topology and is described in terms of sieves; every pretopology generates a Grothendieck topology; the category equipped with a Grothendieck pretopology is called a Grothendieck site. The epipresheaf and monopresheaf conditions easily generalize to from Ouv X,τOuv_{X,\tau} to general (Grothendieck) sites. Again the sheaves on a general site (C,τ)(C,\tau) form a subcategory of the category of presheaves on CC which has a left adjoint “sheafification” functor, which is not only preserving colimits but is itself left exact (preserves finite limits). We say that the Grothendieck site (C,τ)(C,\tau) is subcanonical if every representable presheaf of sets on CC is a sheaf. Then the Yoneda embedding followed by sheafification embeds CC into the category of sheaves Sh X,τSh_{X,\tau} on (X,τ)(X,\tau). The composition is not only left exact but it also preserves the covers (this property is close to preserving certain chosen colimits). The covers may be viewed as distinguished cones over the discrete diagram; one can consider more general cones over non-discrete diagrams, and such may form so-called Q-category, which is a way generalizing Grothendieck topologies. Generalized sheaves can then be defined in this context.

The reader is likely familiar with the notion of a manifold: a manifold is glued from local pieces which look like the local model which is the real nn-dimensional space R n\mathbf{R}^n. One also knows what the morphisms of real spaces are and uses them to define the maps of manifolds passing to pairs of local charts. Grothendieck similarly starts with a subcanonical site (Aff,τ)(Aff,\tau) and defines the corresponding notion of a space as a sheaf of sets on (Aff,τ)(Aff,\tau) which is locally affine, i.e. locally representable. Namely one has the induced Grothendieck topology on Sh X,τSh_{X,\tau} such that the covers of representables are the same as before; a sheaf FSh X,τF\in Sh_{X,\tau} is locally representable if there is a cover GFG\to F in induced pretopology such that GG is representable.

Some applications require even larger embeddings than the classical Yoneda, but in similar spirit, for instance into (a 2-category of) functors into the (2-)category of groupoids. For motivation, consider the problem of quotients. If one creates naively a quotient of a space by an action of an appropriate version of a group object, than the quotient is often not well behaved or even it does not exist in the category one starts with. Quotient is a special case of a colimit construction. Colimits commute with other colimits, hence but not necessarily with limits.

In homotopy theory and homological algebra, one has a way to correct functors which are left exact (preserve finite limits) or right exact (preserve finite colimits), but not exact (i.e. both). This is the theory of derived functors which seek best approximation (so called Kan extension) to the exact functor. The right derived functor is in classical mathematics described via a sequence of corrections called classical right derived functors; various cohomologies come as examples. Left derived functors are homology like and appear in geometry when we want to correct left exact constructions like intersections. Corrected quotients in geometry are described as geometric stacks. As Yoneda embedding sends object to presheaves of sets; one has higher Yoneda embeddings into higher categorical version of presheaves into (the higher category of) GpdGpd higher groupoids instead of sets. A higher categorical version of sheaf conditions then singles out stacks. One can further introduce the representability (affinity) leading to affine or geometric stacks. Many natural objects (for example moduli spaces) are represented by geometric stacks. This point of view taken systematically in geometry amounts to the derived geometry.

Spectra in geometry and duality between spaces and algebras

Newton in a prism experiment decomposed the sun light into colors and called the pattern spectrum, after the Latin word for ghost. In physics, the spectral lines correspond to frequences of light, or equivalently to quanta of energy. Energies of a quantum system correspond to the eigenvalues stationary solutions of wave equations, generalizing the eigenvalues of finite matrices. For a set of commuting operators, one can consider their joint eigenvalues, this generalizes to the spectrum of an operator algebra. Gel’fand and Naimark have shown that the appropriate generalization of the spectral theory of an operator to a C *C^\ast-algebra AA can be used to relate AA to the algebra to a topological space, the Gel’fand spectrum of AA. More precisely, the category of commutative unital C *C^\ast-algebras is antiequivalent to the category of compact Hausdorff topological spaces. This example has some of the more general features of the theory, namely the points of the space recovered are the (continuous) characters into the ground field which are then identified with the evaluation functionals in points of the original space. In commutative algebra there is similar duality between commutative unital rings and affine schemes, extending the duality between affine algebraic varieties over a field and commutative Noetherian rings with no nilpotent elements. For algebraic varieties the points come from maximal ideals which are again evaluation functionals; for the scheme case one includes prime ideals which are a bigger class and which has functorial behaviour, unlike the case of varieties. This role of evaluation functionals in geometry is similar to the most important triviality in mathematics, the Yoneda lemma, where the objects of the original category are obtained as the analogues of the evaluation functionals – the representable presheaves.

Now, in noncommutative algebraic geometry, it appears that the the analogues of prime ideals, which are in the noncommutative case the one sided completely prime ideals, are insufficient to supply a rich geometry for most noncommutative rings. Thus one soon realizes that the lattice of left ideals should be replaced by the full category of left modules. Inside one again stumbles into objects which appear in some sense the most primitive; which can not be approached from below in certain finiteness-involving technical sense, very much alike the limit cardinals do. This leads to the whole machinery of spectral theories. Good spectra of (a suitable class of) abelian categories were devised and may be equipped with a topology and structure stack of categories. If the abelian category if the category of quasicoherent sheaves on an algebraic scheme, then one can go a step further and take a construction of a center of a fiber of such a stack to simplify it into a set and as a result one will get a locally ringed space, which under mild conditions, for a suitable variant of a spectrum, reconstructs the scheme. But in noncommutative case, one really can not pass with allowing only for a sheaf, one must work at the stack level.

Spectra of categories

Typically spectra of categories involve utilization of some preordering and looking for “almost final” objects in that preordering and declaring them points of spectra. For example, one can look at some class of topologizing subcategories and equip that class with a preordering. A deeper inspection shows that an additional functor may be involved.

Historical motivation for spectra of categories

While the spectrum of a commutative ring RR is obtained just from studying the ideal in a ring, that is RR-submodules in RR, the structure of various sets of ideals in a noncommutative ring usually is too small and otherwise inadequate for geometric purposes. One needs to consider not only ideals, but the whole category of (say left) RR-modules, thus not necessarily submodules of RR. A similar thing is with the reconstruction of commutative schemes: the whole abelian category of quasicoherent sheaves? of 𝒪\mathcal{O}-modules, not only the quasicoherent 𝒪\mathcal{O}-submodules of 𝒪\mathcal{O}, is needed for the reconstruction.

The development of spectral machinery for categories

Pierre Gabriel introduced his spectrum of indecomposable injectives to reconstruct Noetherian separated schemes from their categories of qausicoherent sheaves; now it is often called the Gabriel spectrum. Later many other spectra of abelian categories were invented, including the 1980-s A. L. Rosenberg‘s spectrum used for the reconstruction of quasicompact quasiseparated schemes. Around 2000, Rosenberg noticed that the zoo of many spectra has their common feature and that all the spectra can be produced in analogous way. This pattern for producing spectra is introduced as a spectral cookbook in

  • (RosenbergSpectraNSp) A. L. Rosenberg, Spectra of noncommutative spaces, MPIM2003-110 ps dvi (2003)

The cookbook

Local categories and local spectrum

(RosenbergSpectraNSp 1.1) A category CC is local if the full subcategory generated by all objects which are not initial, has itself an initial object. In particular, every local category has initial objects.

(RosenbergSpectraNSp 1.2) The local spectrum of an arbitrary small category CC is the full subcategory 𝔖𝔭𝔢𝔠 1(C)C\mathfrak{Spec}^1(C)\hookrightarrow C whose objects are all xx in ObCOb C such that the undercategory? x\Cx\backslash C is local.

Support of an object

(RosenbergSpectraNSp 1.3) If xx is an object in CC, its support in CC is the full subcategory 𝔖𝔲𝔭𝔭 C(x)C\mathfrak{Supp}_C(x)\subset C whose objects are all objects yy in CC such that C(x,y)=C(x,y)=\emptyset.

𝔖𝔭𝔢𝔠 0\mathfrak{Spec}^0

(RosenbergSpectraNSp 1.4) 𝔖𝔭𝔢𝔠 0(C)C\mathfrak{Spec}^0(C)\subset C is the full subcategory of CC generated by those objects xx in CC whose support 𝔖𝔲𝔭𝔭 C(x)\mathfrak{Supp}_C(x) has a final object, x^\hat{x} (in particular the support is nonempty).

(RosenbergSpectraNSp 1.4.4) A choice of a final object x^\hat{x} for every object xx in 𝔖𝔭𝔢𝔠 0(C)\mathfrak{Spec}^0(C) extends to a functor θ c:𝔖𝔭𝔢𝔠 0(C)C\theta_c : \mathfrak{Spec}^0(C)\to C.

When CC is a preorder

(RosenbergSpectraNSp 1.4.6) Now specialize to the case where CC is a preorder? category having finite coproducts (equivalently supremum for every pair of objects). Then the functor θ c:𝔖𝔭𝔢𝔠 0(C)C\theta_c : \mathfrak{Spec}^0(C)\to C factors through the embedding 𝔖𝔭𝔢𝔠 1(C)C\mathfrak{Spec}^1(C)\to C. Consequently, it corestricts to a functor 𝔖𝔭𝔢𝔠 0(C)𝔖𝔭𝔢𝔠 1(C)\mathfrak{Spec}^0(C)\to \mathfrak{Spec}^1(C) which may also be denoted as θ C\theta_C.

Relative spectra

(RosenbergSpectraNSp 1.6) Given a functor between small categories F:CDF: C\to D the relative spectra 𝔖𝔭𝔢𝔠 0(C,F),𝔖𝔭𝔢𝔠 1(C,F)\mathfrak{Spec}^0(C,F), \mathfrak{Spec}^1(C,F) are defined as isocomma objects (“2-categorical pullbacks”)

𝔖𝔭𝔢𝔠 0(C,F) θ F C π F 0 F 𝔖𝔭𝔢𝔠 0(D) θ D D𝔖𝔭𝔢𝔠 1(C,F) θ F 1 C π F 1 F 𝔖𝔭𝔢𝔠 1(D) θ D 1 D\array{ \mathfrak{Spec}^0(C,F)&\stackrel{\theta_F}\longrightarrow&C\\ {\pi_F^0}_\mathrlap{}\downarrow&&\downarrow{}_\mathrlap{F}\\ \mathfrak{Spec}^0(D)&\underset{\theta_D}\longrightarrow&D } \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \array{ \mathfrak{Spec}^1(C,F)&\stackrel{\theta^1_F}\longrightarrow&C\\ {\pi_F^1}_\mathrlap{}\downarrow&&\downarrow{}_\mathrlap{F}\\ \mathfrak{Spec}^1(D)&\underset{\theta^1_D}\longrightarrow&D }

It appears that, for a fixed target category DD, the constructions (C,F)𝔖𝔭𝔢𝔠 0(C,F)(C,F)\mapsto \mathfrak{Spec}^0(C,F) and (C,F)𝔖𝔭𝔢𝔠 1(C,F)(C,F)\mapsto \mathfrak{Spec}^1(C,F) extend to pseudofunctors Cat/DCatCat/D\to Cat. If DD is a preorder with finite coproducts then every functor F:CDF: C\to D induces a canonical functor

θ C,F:𝔖𝔭𝔢𝔠 0(C,F)𝔖𝔭𝔢𝔠 1(C,F) \theta_{C,F} : \mathfrak{Spec}^0(C,F)\longrightarrow \mathfrak{Spec}^1(C,F)

which are the components of a natural transformation of pseudofunctors Cat/DCatCat/D\to Cat

θ:𝔖𝔭𝔢𝔠 0𝔖𝔭𝔢𝔠 1 \theta : \mathfrak{Spec}^0\longrightarrow \mathfrak{Spec}^1

Applications to spectra of Abelian categories

Idea

Starting with an Abelian category AA we proceed in three steps. In the first step we construct some preorder C AC_A from AA, or of a functor F A:C AD AF_A: C_A\to D_A with a preorder category D AD_A and then apply the spectral cookbook. The objects of C AC_A may be some special subcategories of AA (e.g. some class of topologizing subcategories), or multiplicative systems of morphisms, as used in the localization theory and so on. In a third step, one often passes to the equivalence classes of objects in the obtained spectra.

Additionally, one may add construction of some sort of topology or additional “structure stack” to obtained spectrum using possibly supplemental knowledge about the input data.

Remark on monoidal case

Some reconstruction theorems (like the theorems of Balmer and of Garkusha) consider abelian symmetric monoidal categories instead; their spectra are very analogous but the subcategories used to construct the intermediate preorder category are monoidal subcategories as well and various constructions respect the monoidal structure. The pattern is still the same.

Spectra versus reconstruction

Spectral cookbook gives just the points of spectra. If one wants to reconstruct, say a scheme from the category of quasicoherent sheaves, one needs also the topology and structure sheaf. Not every spectrum is big enough to reconstruct the space, nor every spectrum has a natural topology, but some do. For the noncommutative spaces, one constructs not the structure sheaf, but a structure stack. In the case of Rosenberg’s spectrum of abelian category, its fibers are local abelian categories. There is a center construction which can be applied to these fibers to obtain commutative algebras. In the case of reconstruction of schemes, the center of a local category will give a local ring; thus one obtains a presheaf of commutative local rings, which one sheafifies to get the reconstruced scheme.

Modern homotopy theory and higher categories

Last revised on October 9, 2016 at 14:15:33. See the history of this page for a list of all contributions to it.