This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Topos" – news ·newspapers ·books ·scholar ·JSTOR(July 2015) (Learn how and when to remove this message)

Inmathematics, atopos (US:/ˈtɒpɒs/,UK:/ˈtoʊpoʊs,ˈtoʊpɒs/; pluraltopoi/ˈtɒpɔɪ/ or/ˈtoʊpɔɪ/, ortoposes) is acategory that behaves like the category ofsheaves ofsets on atopological space (or more generally, on asite). Topoi behave much like thecategory of sets and possess a notion of localization.^[1] TheGrothendieck topoi find applications inalgebraic geometry, and more generalelementary topoi are used inlogic.

The mathematical field that studies topoi is calledtopos theory.

Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded byAlexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is theétale topos of ascheme. Another illustration of the capability of Grothendieck topoi to incarnate the “essence” of different mathematical situations is given by their use as "bridges" for connecting theories which, albeit written in possibly very different languages, share a common mathematical content.^[2][3]

A Grothendieck topos is acategory${\displaystyle C}$ which satisfies any one of the following three properties. (Atheorem ofJean Giraud states that the properties below are all equivalent.)

• There is asmall category${\displaystyle D}$ and an inclusion${\displaystyle C\hookrightarrow \mathrm{Presh}(D)}$ that admits a finite-limit-preservingleft adjoint.
• ${\displaystyle C}$ is the category of sheaves on aGrothendieck site.
• ${\displaystyle C}$ satisfies Giraud's axioms, below.

Here${\displaystyle \mathrm{Presh}(D)}$ denotes the category ofcontravariant functors from${\displaystyle D}$ to the category of sets; such a contravariant functor is frequently called apresheaf.

Giraud's axioms for acategory${\displaystyle C}$ are:

• ${\displaystyle C}$ has a small set ofgenerators, and admits all smallcolimits. Furthermore,fiber products distribute over coproducts; that is, given a set${\displaystyle I}$, an${\displaystyle I}$-indexed coproduct mapping to${\displaystyle A}$, and a morphism${\displaystyle A^{\prime }\to A}$, the pullback is an${\displaystyle I}$-indexed coproduct of the pullbacks:$${\displaystyle \left(\coprod _{i\in I}{B}_i\right){\times }_A{A}^{\prime }\cong \coprod _{i\in I}(B_i{\times }_A{A}^{\prime }).}$$
• Sums in${\displaystyle C}$ are disjoint. In other words, the fiber product of${\displaystyle X}$ and${\displaystyle Y}$ over their sum is theinitial object in${\displaystyle C}$.
• Allequivalence relations in${\displaystyle C}$ areeffective.

The last axiom needs the most explanation. IfX is an object ofC, an "equivalence relation"R onX is a mapR →X ×X inCsuch that for any objectY inC, the induced map Hom(Y,R) → Hom(Y,X) × Hom(Y,X) gives an ordinary equivalence relation on the set Hom(Y,X). SinceC has colimits we may form thecoequalizer of the two mapsR →X; call thisX/R. The equivalence relation is "effective" if the canonical map

${\displaystyle R\to X{\times }_{X/R}X}$

is an isomorphism.

Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often giverise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory.

Thecategory of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point since functors on the singleton category with a single object and only the identity morphism are just specific sets in the category of sets.

Similarly, there is a topos${\displaystyle BG}$ for anygroup${\displaystyle G}$ which is equivalent to the category of${\displaystyle G}$-sets. We construct this as the category of presheaves on the category with one object, but now the set of morphisms is given by thegroup${\displaystyle G}$. Since any functor must give a${\displaystyle G}$-action on the target, this gives the category of${\displaystyle G}$-sets. Similarly, for agroupoid${\displaystyle \mathcal{G}}$ the category of presheaves on${\displaystyle \mathcal{G}}$ gives a collection of sets indexed by the set of objects in${\displaystyle \mathcal{G}}$, and the automorphisms of an object in${\displaystyle \mathcal{G}}$ has an action on the target of the functor.

More exotic examples, and theraison d'être of topos theory, come from algebraic geometry. The basic example of a topos comes from the Zariski topos of ascheme. For each scheme${\displaystyle X}$ there is a site${\displaystyle \text{Open}(X)}$ (of objects given by open subsets and morphisms given by inclusions) whose category of presheaves forms the Zariski topos${\displaystyle (X{)}_{Zar}}$. But once distinguished classes of morphisms are considered, there are multiple generalizations of this which leads to non-trivial mathematics. Moreover, topoi give the foundations for studying schemes purely as functors on the category of algebras.

To a scheme and even astack one may associate anétale topos, anfppf topos, or aNisnevich topos. Another important example of a topos is from thecrystalline site. In the case of the étale topos, these form the foundational objects of study inanabelian geometry, which studies objects in algebraic geometry that are determined entirely by the structure of theirétale fundamental group.

Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances ofpathological behavior. For instance, there is an example due toPierre Deligne of a nontrivial topos that has no points (see below for the definition of points of a topos).

If${\displaystyle X}$ and${\displaystyle Y}$ are topoi, ageometric morphism${\displaystyle u:X\to Y}$ is a pair ofadjoint functors (u^∗,u_∗) (whereu^∗ :Y →X is left adjoint tou_∗ :X →Y) such thatu^∗ preserves finite limits. Note thatu^∗ automatically preserves colimits by virtue of having a right adjoint.

ByFreyd's adjoint functor theorem, to give a geometric morphismX →Y is to give a functoru^∗: Y →X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps oflocales.

If${\displaystyle X}$ and${\displaystyle Y}$ are topological spaces and${\displaystyle u}$ is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi for the sites${\displaystyle \text{Open}(X),\text{Open}(Y)}$.

A point of a topos${\displaystyle X}$ is defined as a geometric morphism from the topos of sets to${\displaystyle X}$.

IfX is an ordinary space andx is a point ofX, then the functor that takes a sheafF to its stalkF_x has a right adjoint(the "skyscraper sheaf" functor), so an ordinary point ofX also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous mapx: 1 →X.

For the etale topos${\displaystyle (X{)}_{et}}$ of a space${\displaystyle X}$, a point is a bit more refined of an object. Given a point${\displaystyle x:\text{Spec}(\kappa (x))\to X}$ of the underlying scheme${\displaystyle X}$ a point${\displaystyle x^{\prime }}$ of the topos${\displaystyle (X{)}_{et}}$ is then given by a separable field extension${\displaystyle k}$ of${\displaystyle \kappa (x)}$ such that the associated map${\displaystyle x^{\prime }:\text{Spec}(k)\to X}$ factors through the original point${\displaystyle x}$. Then, the factorization map$${\displaystyle \text{Spec}(k)\to \text{Spec}(\kappa (x))}$$ is anetale morphism of schemes.

More precisely, those are theglobal points. They are not adequate in themselves for displaying the space-like aspect of a topos, because a non-trivial topos may fail to have any.Generalized points are geometric morphisms from a toposY (thestage of definition) toX. There are enough of these to display the space-like aspect. For example, ifX is theclassifying toposS[T] for a geometric theoryT, then the universal property says that its points are the models ofT (in any stage of definitionY).

A geometric morphism (u^∗,u_∗) isessential ifu^∗ has a further left adjointu_!, or equivalently (by the adjoint functor theorem) ifu^∗ preserves not only finite but all small limits.

Aringed topos is a pair (X,R), whereX is a topos andR is a commutativering object inX. Most of the constructions ofringed spaces go through for ringed topoi. The category ofR-module objects inX is anabelian category with enough injectives. A more useful abelian category is the subcategory ofquasi-coherentR-modules: these areR-modules that admit a presentation.

Another important class of ringed topoi, besides ringed spaces, are the étale topoi ofDeligne–Mumford stacks.

Michael Artin andBarry Mazur associated to the site underlying a topos apro-simplicial set (up tohomotopy).^[4] (It's better to consider it in Ho(pro-SS); see Edwards) Using thisinverse system of simplicial sets one maysometimes associate to ahomotopy invariant in classical topology an inverse system of invariants in topos theory. The study of the pro-simplicial set associated to the étale topos of a scheme is calledétale homotopy theory.^[5] In good cases (if the scheme isNoetherian andgeometrically unibranch), this pro-simplicial set ispro-finite.

Since the early 20th century, the predominant axiomaticfoundation of mathematics has beenset theory, in which all mathematical objects are ultimately represented by sets (includingfunctions, which map between sets). More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set-theoretic mathematics. But one could instead choose to work with many alternative topoi. A standard formulation of theaxiom of choice makes sense in any topos, and there are topoi in which it is invalid.Constructivists will be interested to work in a topos without thelaw of excluded middle. If symmetry under a particulargroupG is of importance, one can use the topos consisting of allG-sets.

It is also possible to encode analgebraic theory, such as the theory of groups, as a topos, in the form of aclassifying topos. The individual models of the theory, i.e. the groups in our example, then correspond tofunctors from the encoding topos to the category of sets that respect the topos structure.

When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise:

A topos is a category that has the following two properties:

• Alllimits taken over finite index categories exist.
• Every object has a power object. This plays the role of thepowerset in set theory.

Formally, apower object of an object${\displaystyle X}$ is a pair${\displaystyle (PX,{\ni }_X)}$ with${\displaystyle {\ni }_X\subseteq PX\times X}$, which classifies relations, in the following sense. First note that for every object${\displaystyle I}$, a morphism${\displaystyle r:I\to PX}$ ("a family of subsets") induces a subobject${\displaystyle \{(i,x)|x\in r(i)\}\subseteq I\times X}$. Formally, this is defined by pulling back${\displaystyle {\ni }_X}$ along${\displaystyle r\times X:I\times X\to PX\times X}$. The universal property of a power object is that every relation arises in this way, giving a bijective correspondence between relations${\displaystyle R\subseteq I\times X}$ and morphisms${\displaystyle r:I\to PX}$.

From finite limits and power objects one can derive that

• Allcolimits taken over finite index categories exist.
• The category has asubobject classifier.
• The category isCartesian closed.

In some applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived.

Alogical functor is a functor between topoi that preserves finite limits and power objects. Logical functors preserve the structures that topoi have. In particular, they preserve finite colimits,subobject classifiers, andexponential objects.^[6]

A topos as defined above can be understood as a Cartesian closed category for which the notion of subobject of an object has anelementary or first-order definition. This notion, as a natural categorical abstraction of the notions ofsubset of a set,subgroup of a group, and more generallysubalgebra of anyalgebraic structure, predates the notion of topos. It is definable in any category, not just topoi, insecond-order language, i.e. in terms of classes of morphisms instead of individual morphisms, as follows. Given two monicsm,n from respectivelyY andZ toX, we say thatm ≤n when there exists a morphismp:Y →Z for whichnp =m, inducing apreorder on monics toX. Whenm ≤n andn ≤m we say thatm andn are equivalent. The subobjects ofX are the resulting equivalence classes of the monics to it.

In a topos "subobject" becomes, at least implicitly, a first-order notion, as follows.

As noted above, a topos is a categoryC having all finite limits and hence in particular the empty limit or final object 1. It is then natural to treat morphisms of the formx: 1 →X aselementsx ∈X. Morphismsf:X →Y thus correspond to functions mapping each elementx ∈X to the elementfx ∈Y, with application realized by composition.

One might then think to define a subobject ofX as an equivalence class of monicsm:X′ →X having the sameimage {mx |x ∈X′ }. The catch is that two or more morphisms may correspond to the same function, that is, we cannot assume thatC is concrete in the sense that the functorC(1,-):C →Set is faithful. For example the categoryGrph ofgraphs and their associatedhomomorphisms is a topos whose final object 1 is the graph with one vertex and one edge (a self-loop), but is not concrete because the elements 1 →G of a graphG correspond only to the self-loops and not the other edges, nor the vertices without self-loops. Whereas the second-order definition makesG and the subgraph of all self-loops ofG (with their vertices) distinct subobjects ofG (unless every edge is, and every vertex has, a self-loop), this image-based one does not. This can be addressed for the graph example and related examples via theYoneda Lemma as described in theFurther examples section below, but this then ceases to be first-order. Topoi provide a more abstract, general, and first-order solution.

As noted above, a toposC has a subobject classifier Ω, namely an object ofC with an elementt ∈ Ω, thegeneric subobject ofC, having the property that everymonicm:X′ →X arises as a pullback of the generic subobject along a unique morphismf:X → Ω, as per Figure 1. Now the pullback of a monic is a monic, and all elements includingt are monics since there is only one morphism to 1 from any given object, whence the pullback oft alongf:X → Ω is a monic. The monics toX are therefore in bijection with the pullbacks oft along morphisms fromX to Ω. The latter morphisms partition the monics into equivalence classes each determined by a morphismf:X → Ω, the characteristic morphism of that class, which we take to be the subobject ofX characterized or named byf.

All this applies to any topos, whether or not concrete. In the concrete case, namelyC(1,-) faithful, for example the category of sets, the situation reduces to the familiar behavior of functions. Here the monicsm:X′ →X are exactly the injections (one-one functions) fromX′ toX, and those with a given image {mx |x ∈X′ } constitute the subobject ofX corresponding to the morphismf:X → Ω for whichf⁻¹(t) is that image. The monics of a subobject will in general have many domains, all of which however will be in bijection with each other.

To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monics toX as had previously been defined explicitly by the second-order notion of subobject for any category. The notion of equivalence relation on a class of morphisms is itself intrinsically second-order, which the definition of topos neatly sidesteps by explicitly defining only the notion of subobjectclassifier Ω, leaving the notion of subobject ofX as an implicit consequence characterized (and hence namable) by its associated morphismf:X → Ω.

Every Grothendieck topos is an elementary topos, but the converse is not true (since every Grothendieck topos is cocomplete, which is not required from an elementary topos).

The categories of finite sets, of finiteG-sets (actions of a groupG on a finite set), and of finite graphs are elementary topoi that are not Grothendieck topoi.

IfC is a small category, then thefunctor categorySet^C (consisting of all covariant functors fromC to sets, withnatural transformations as morphisms) is a topos. For instance, the categoryGrph of graphs of the kind permitting multiple directed edges between two vertices is a topos. Such a graph consists of two sets, an edge set and a vertex set, and two functionss,t between those sets, assigning to every edgee its sources(e) and targett(e).Grph is thusequivalent to the functor categorySet^C, whereC is the category with two objectsE andV and two morphismss,t:E →V giving respectively the source and target of each edge.

TheYoneda lemma asserts thatC^op embeds inSet^C as a full subcategory. In the graph example the embedding representsC^op as the subcategory ofSet^C whose two objects areV' as the one-vertex no-edge graph andE' as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms fromV' toE' (both as natural transformations). The natural transformations fromV' to an arbitrary graph (functor)G constitute the vertices ofG while those fromE' toG constitute its edges. AlthoughSet^C, which we can identify withGrph, is not made concrete by eitherV' orE' alone, the functorU:Grph →Set² sending objectG to the pair of sets (Grph(V',G),Grph(E',G)) and morphismh:G →H to the pair of functions (Grph(V',h),Grph(E',h)) is faithful. That is, a morphism of graphs can be understood as apair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts ofgeneralized elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of topoi by allowing an object to have multiple underlying sets, that is, to be multisorted.

The category ofpointed sets with point-preserving functions isnot a topos, since it doesn't have power objects: if${\displaystyle PX}$ were the power object of the pointed set${\displaystyle X}$, and${\displaystyle 1}$ denotes the pointed singleton, then there is only one point-preserving function${\displaystyle r:1\to PX}$, but the relations in${\displaystyle 1\times X}$ are as numerous as the pointed subsets of${\displaystyle X}$. Thecategory of abelian groups is also not a topos, for a similar reason: every group homomorphism must map 0 to 0.

• History of topos theory
• Homotopy hypothesis
• Intuitionistic type theory
• ∞-topos
• Quasitopos
• Geometric logic
• Generalized space

Some gentle papers

• Edwards, D.A.; Hastings, H.M. (Summer 1980)."Čech Theory: its Past, Present, and Future"(PDF).Rocky Mountain Journal of Mathematics.10 (3):429–468.doi:10.1216/RMJ-1980-10-3-429.JSTOR 44236540.
• Baez, John."Topos theory in a nutshell". A gentle introduction.
• Steven Vickers: "Toposes pour les nuls" and "Toposes pour les vraiment nuls." Elementary and even more elementary introductions to toposes as generalized spaces.
• Illusie, Luc (2004)."What is...A Topos?"(PDF).Notices of the AMS.51 (9):160–1.

The following texts are easy-paced introductions to toposes and the basics of category theory. They should be suitable for those knowing little mathematical logic and set theory, even non-mathematicians.

• Lawvere, F. William; Schanuel, Stephen H. (1997).Conceptual Mathematics: A First Introduction to Categories. Cambridge University Press.ISBN 978-0-521-47817-5. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
• Lawvere, F. William; Rosebrugh, Robert (2003).Sets for Mathematics. Cambridge University Press.ISBN 978-0-521-01060-3. Introduces the foundations of mathematics from a categorical perspective.

Grothendieck foundational work on topoi:

• Grothendieck, A.;Verdier, J.L. (1972).Théorie des Topos et Cohomologie Etale des Schémas. Lecture notes in mathematics. Vol. 269. Springer.doi:10.1007/BFb0081551.ISBN 978-3-540-37549-4.Tome 2270doi:10.1007/BFb0061319ISBN 978-3-540-37987-4

The following monographs include an introduction to some or all of topos theory, but do not cater primarily to beginning students. Listed in (perceived) order of increasing difficulty.

• McLarty, Colin (1992).Elementary Categories, Elementary Toposes. Clarendon Press.ISBN 978-0-19-158949-2. A nice introduction to the basics of category theory, topos theory, and topos logic. Assumes very few prerequisites.
• Goldblatt, Robert (2013) [1984].Topoi: The Categorial Analysis of Logic. Courier Corporation.ISBN 978-0-486-31796-0. A good start. Availableonline atRobert Goldblatt's homepage.
• Bell, John L. (2001)."The Development of Categorical Logic". In Gabbay, D.M.; Guenthner, Franz (eds.).Handbook of Philosophical Logic. Vol. 12 (2nd ed.). Springer. pp. 279–.ISBN 978-1-4020-3091-8. Version availableonline atJohn Bell's homepage.
• MacLane, Saunders;Moerdijk, Ieke (2012) [1994].Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer.ISBN 978-1-4612-0927-0. More complete, and more difficult to read.
• Barr, Michael;Wells, Charles (2013) [1985].Toposes, Triples and Theories. Springer.ISBN 978-1-4899-0023-4. (Online version). More concise thanSheaves in Geometry and Logic, but hard on beginners.

Reference works for experts, less suitable for first introduction

• Edwards, D.A.; Hastings, H.M. (1976).Čech and Steenrod homotopy theories with applications to geometric topology. Lecture Notes in Maths. Vol. 542. Springer-Verlag.doi:10.1007/BFb0081083.ISBN 978-3-540-38103-7.
• Borceux, Francis (1994).Handbook of Categorical Algebra: Volume 3, Sheaf Theory. Encyclopedia of Mathematics and its Applications. Vol. 52. Cambridge University Press.ISBN 978-0-521-44180-3. The third part of "Borceux' remarkable magnum opus", as Johnstone has labelled it. Still suitable as an introduction, though beginners may find it hard to recognize the most relevant results among the huge amount of material given.
• Johnstone, Peter T. (2014) [1977].Topos Theory. Courier.ISBN 978-0-486-49336-7. For a long time the standard compendium on topos theory. However, even Johnstone describes this work as "far too hard to read, and not for the faint-hearted."
• Johnstone, Peter T. (2002).Sketches of an Elephant: A Topos Theory Compendium. Vol. 2. Clarendon Press.ISBN 978-0-19-851598-2. As of early 2010, two of the scheduled three volumes of this overwhelming compendium were available.
• Caramello, Olivia (2017).Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges. Vol. 1. Oxford University Press.doi:10.1093/oso/9780198758914.001.0001.ISBN 9780198758914.

Books that target special applications of topos theory

• Pedicchio, Maria Cristina; Tholen, Walter; Rota, G.C., eds. (2004).Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Encyclopedia of Mathematics and its Applications. Vol. 97. Cambridge University Press.ISBN 978-0-521-83414-8. Includes many interesting special applications.

• Foundations of mathematics
• Topos theory
• Sheaf theory

• Articles with short description
• Short description is different from Wikidata
• Articles needing additional references from July 2015
• All articles needing additional references