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.
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 $C$ is a category, $c,d$ objects in $C$, the set of morphisms from $c$ to $d$ may be denoted by one of the 5 standard notations $Hom(c,d)= Mor(c,d) = Hom_C(c,d) = Mor_C(c,d) = C(c,d)$. For example, $Top$ 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 $F:C\to D$ is full (resp. faithful, fully faithful) if for each pair $c,c'\in Ob(C)$, functor $F$ sends $Hom_C(c,c')$ into $Hom_D(F(c),F(c'))$ injectively (resp. surjectively, bijectively); and we say that $F$ is a essentially surjective if for every $d\in Ob(D)$ there exists $c\in Ob(D)$ and an invertible morphism $f:F(c)\to d$. 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 $C$, one arrives at the dual or opposite category. It is convenient to look at contravariant functors from category $C$ as functors from the opposite category $C^{op}$; such functors, especially when the codomain is the category $Set$ of sets and functions, or $Grp$ of groups and homomorphisms of groups are often called presheaves. In particular, a functor $F:C^{op}\to Set$ is said to be a presheaf of sets on $C$.
I assume that the reader is familiar with the notion of a natural transformation $\alpha : F\to G$ between two functors $F,G: C\to D$, which is given by its components $\{\alpha_c\}_{c\in C}$, which are morphisms in $D$ such that for each morphism $f: c\to c'$, $G(f)\circ \alpha_c = \alpha_{c'}\circ F(f):F(c)\to G(c')$ (what is usually drawn as a commutative square). Given a pair of small categories $C,D$, the functors from $C$ to $D$ as objects together with natural transformations as morphisms, form a category $Fun(C,D)=D^C$.
An object $c$ in a category $C$ is (universal) initial (resp. terminal) object if for every other object $x$ in $C$ there exist a unique morphism $c\to x$ (resp. $x\to c$). 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 $C$, a diagram in $C$ is a functor $d:D\to C$ whose domain is a small category $D$. Given a diagram $d:D\to C$, a cone with vertex $x$ is a natural transformation $\alpha$ from a constant functor $const_x:D\to C$, $z\mapsto x$ for all $z\in Ob(D)$. A morphism from a cone $\alpha:const_x\to d$ to a cone $\beta:const_y\to d$, is a map $f: x\to y$ such that the transformation $const_f: const_x\to const_y$ whose every component is $f$ and for which $\beta\circ const_f = \alpha$. A terminal object in the category of cones over $d$ is called the limit cone or limit of the diagram $d$; if it exists it is unique up to a unique isomorphism in the category of cones over $d$. 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 $a,b$ and two distinct nontrivial arrows $a\to b$ (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 $c$ be an object in a category $C$; then we define the corresponding representable presheaf of sets $h_c = Hom(-,c):C^{op}\to Set$ by
on objects: $h_c(d) = Hom(d,c)$
on morphisms: for $f:d\to d'$, define $h_c(f)(g) = Hom(f,c)(g) = g\circ f:d\to c$ for every $g:d'\to c$.
A representable presheaf of sets on $C$ is a presheaf of sets $F$ together with natural isomorphism of functors (i.e. invertible natural transformation) $\alpha: F\cong h_c$ for some $c\in Ob(C)$.
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 $F$ to a representable functor $h_c$ ($c$ a fixed object in $C$) are in a natural bijection with the set $F(c)$, i.e. there is a bijection $F(c) \cong Nat(h_c,F)$ which is natural in $C$ (i.e. it is a component of natural transformation at $c$). It is easy to prove this as an exercise. In particular, take $F = h_e$ for some other object $e\in C$; then $Nat(h_e,h_c) \cong Hom(e,c)$; therefore one obtains the weak Yoneda lemma: the functor $h: c\mapsto h_c$ is fully faithful (covariant) functor $C\to Fun(C^{op},Set)$. The category of presheaves of sets $Fun(C^{op},Set)$ is also denoted by $C^\hat{}$. The weak Yoneda lemma, thus, means that every small category $C$ embeds as a full and faithful subcategory of the category of presheaves of sets $C^\hat{}$, this embedding $h:c\mapsto h_c$ is called the Yoneda embedding $C\hookrightarrow C^{\hat{}}$. 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 $Set$ has it, and we compute objectwise for presheaves) over small diagrams; one has more ‘space’ to do various constructions in the bigger category $C^\hat{}=Set^{C^{op}}$. Moreover, one can check directly that the Yoneda embedding $C\hookrightarrow Set^{C^{op}}$ preserves limits, but not necessarily the colimits. There are refinements of Yoneda embedding $C\hookrightarrow Ind C\hookrightarrow C^{\hat{}}_{sm}\hookrightarrow C^{\hat{}}$, where $Ind C$ is the full subcategory whose objects are the ind-objects in $C$, i.e. those presheaves $C^{op}\to Set$ which are colimits of small filtered diagrams of representables and $C^{\hat{}}$ the full subcategory whose objects are the small presheaves, i.e. those presheaves which are colimits of small diagrams of representables.
While in a category (synonym:1-category) there are objects and morphisms, one can consider $n$-categories where one has objects, morphisms, 2-morphisms which are morphisms among morphisms and so on till $n$-morphisms, together with structure of various compositions. $k$-morphisms are also called $k$-cells, and objects $0$-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 $(n,1)$-categories, which are $n$-categories such that all higher $k$-morphisms for $k\gt 1$ are invertible. It may look suprising, but natural, that various approaches to higher categories lead to equivalent theories. For $(\infty,1)$-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. $(\infty,1)$-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 $\Delta$ whose objects are nonempty finite totally ordered sets $0\lt 1\lt\ldots \lt n$ and morphisms are the monotone maps. The $\infinity$-groupoids correspond those simplicial sets which satisfy so called Kan conditions; the $(\infinity,1)$-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.
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 $X$ be equipped with topology $\tau\subset \mathcal{P}(X)$. The topology is the set of objects of a category $Ouv_X = Ouv_{X,\tau}$ where the morphisms $U\to V$ are the inclusions of $\tau$-open sets; in other words the morphism set $Hom(U,V)$ is either empty or a singleton. A presheaf of sets $F: Ouv_{X,\tau}^{op}\to Set$ is said to be a presheaf of sets on the topological space $(X,\tau)$. Given the inclusion $i_{U V}: U\subset V$ the map $F(i_{U V}):F(V)\to F(U)$ is called a restriction map $r_{U V}$ from $V$ to $U$; for each $s\in F(U)$ one also denotes the restriction $r_{U V}(s) = s|_U\in F(U)$. A presheaf $F$ on $(X,\tau)$ is a monopresheaf or separated presheaf if for every family $\{U_\alpha\}_{\alpha\in A}$ of open sets two elements $s,t\in F(\cup_\alpha U_\alpha)$ are different whenever $s|_{U_\alpha} = t|_{U_\alpha}$ for all $\alpha \in A$. The presheaf $F$ is an epipresheaf if for every family $\{U_\alpha\}_{\alpha\in A}$ of open sets and for every family $s_\alpha \in F(U_\alpha)$ such that $s_\alpha|_{U_\alpha\cap U_\beta} = s_\beta|_{U_\alpha\cap U_\beta}$ for all pairs $(\alpha,\beta)$, there is at least one $s\in F(\cup_\alpha U_\alpha)$ such that $s_\alpha = s|_{U_\alpha}$. 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, $\{U_\alpha\}_\alpha$ is a cover of $\cup_\alpha U_\alpha$ in the classical sense.
There is an equivalence between the sheaves of sets on $(X,\tau)$ and surjective local homeomorphisms $(Y,\tau_Y)\to (X,\tau)$ which comes as a restriction of an adjoint pair of functors between presheaves of sets on $X$ and topological spaces $(Y,\tau_Y)$ over $(X,\tau)$. Thus the sheaves can be considered as local homeomorphisms over $(X,\tau)$, also called etale spaces.
Grothendieck considers a more general category $C$ instead of $Ouv_{X,\tau}$ and wants to utilize more general families of maps $\{U_\alpha\to Y\}_\alpha$ as covers than just the families of inclusions whose union is $Y$ (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 $C$. The axioms in a category admiting pullbacks:
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 $Ouv_{X,\tau}$ to general (Grothendieck) sites. Again the sheaves on a general site $(C,\tau)$ form a subcategory of the category of presheaves on $C$ 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 $(C,\tau)$ is subcanonical if every representable presheaf of sets on $C$ is a sheaf. Then the Yoneda embedding followed by sheafification embeds $C$ into the category of sheaves $Sh_{X,\tau}$ on $(X,\tau)$. 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 $n$-dimensional space $\mathbf{R}^n$. 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 $(Aff,\tau)$ and defines the corresponding notion of a space as a sheaf of sets on $(Aff,\tau)$ which is locally affine, i.e. locally representable. Namely one has the induced Grothendieck topology on $Sh_{X,\tau}$ such that the covers of representables are the same as before; a sheaf $F\in Sh_{X,\tau}$ is locally representable if there is a cover $G\to F$ in induced pretopology such that $G$ 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) $Gpd$ 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.
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 $C^\ast$-algebra $A$ can be used to relate $A$ to the algebra to a topological space, the Gel’fand spectrum of $A$. More precisely, the category of commutative unital $C^\ast$-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.
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.
While the spectrum of a commutative ring $R$ is obtained just from studying the ideal in a ring, that is $R$-submodules in $R$, 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) $R$-modules, thus not necessarily submodules of $R$. A similar thing is with the reconstruction of commutative schemes: the whole abelian category of quasicoherent sheaves? of $\mathcal{O}$-modules, not only the quasicoherent $\mathcal{O}$-submodules of $\mathcal{O}$, is needed for the reconstruction.
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 1.1) A category $C$ 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 $C$ is the full subcategory $\mathfrak{Spec}^1(C)\hookrightarrow C$ whose objects are all $x$ in $Ob C$ such that the undercategory? $x\backslash C$ is local.
(RosenbergSpectraNSp 1.3) If $x$ is an object in $C$, its support in $C$ is the full subcategory $\mathfrak{Supp}_C(x)\subset C$ whose objects are all objects $y$ in $C$ such that $C(x,y)=\emptyset$.
(RosenbergSpectraNSp 1.4) $\mathfrak{Spec}^0(C)\subset C$ is the full subcategory of $C$ generated by those objects $x$ in $C$ whose support $\mathfrak{Supp}_C(x)$ has a final object, $\hat{x}$ (in particular the support is nonempty).
(RosenbergSpectraNSp 1.4.4) A choice of a final object $\hat{x}$ for every object $x$ in $\mathfrak{Spec}^0(C)$ extends to a functor $\theta_c : \mathfrak{Spec}^0(C)\to C$.
(RosenbergSpectraNSp 1.4.6) Now specialize to the case where $C$ is a preorder? category having finite coproducts (equivalently supremum for every pair of objects). Then the functor $\theta_c : \mathfrak{Spec}^0(C)\to C$ factors through the embedding $\mathfrak{Spec}^1(C)\to C$. Consequently, it corestricts to a functor $\mathfrak{Spec}^0(C)\to \mathfrak{Spec}^1(C)$ which may also be denoted as $\theta_C$.
(RosenbergSpectraNSp 1.6) Given a functor between small categories $F: C\to D$ the relative spectra $\mathfrak{Spec}^0(C,F), \mathfrak{Spec}^1(C,F)$ are defined as isocomma objects (“2-categorical pullbacks”)
It appears that, for a fixed target category $D$, the constructions $(C,F)\mapsto \mathfrak{Spec}^0(C,F)$ and $(C,F)\mapsto \mathfrak{Spec}^1(C,F)$ extend to pseudofunctors $Cat/D\to Cat$. If $D$ is a preorder with finite coproducts then every functor $F: C\to D$ induces a canonical functor
which are the components of a natural transformation of pseudofunctors $Cat/D\to Cat$
Starting with an Abelian category $A$ we proceed in three steps. In the first step we construct some preorder $C_A$ from $A$, or of a functor $F_A: C_A\to D_A$ with a preorder category $D_A$ and then apply the spectral cookbook. The objects of $C_A$ may be some special subcategories of $A$ (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.
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.
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.
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Last revised on October 9, 2016 at 14:15:33. See the history of this page for a list of all contributions to it.