Topos

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For discussion of topoi in literary theory, see literary topos. For discussion of topoi in rhetorical invention, see Inventio.

In the mathematical field of category theory, a topos is a type of category that behaves like the category of sheaves of sets on a topological space.

Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on that space. This idea was expounded by Alexander Grothendieck by introducing the notion of a topos (plural: topoi or toposes — this is a contentious topic). The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest success of this yoga to date has been the introduction of the étale topos of a scheme.

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Introduction

A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (even functions 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 alternate topoi. A topos exists in which the axiom of choice is invalid. Constructivists will be interested to work in a topos without the law of excluded middle. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets. Another important example of a topos (and historically the first) is the category of all sheaves of sets on a given topological space.

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

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Formal definition

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, if not illuminating:

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

From this one can derive that

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

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Further examples

There is one major class of examples of topoi that wasn't listed in the introduction: if C is a small category, then the functor category SetC (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category of all directed graphs is a topos. A graph consists of two sets, an arrow set and a vertex set, and two functions between those sets, assigning to every arrow its start and end vertex. The category of graphs is thus equivalent to the functor category SetC, where C is the category with two objects joined by two morphisms.

The categories of finite sets, of finite G-sets and of finite directed graphs are also topoi.

Example from logic should go here
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History

Main article: Background and genesis of topos theory

The historical origin of topos theory is algebraic geometry. Alexander Grothendieck generalized the concept of a sheaf. The result is the category of sheaves with respect to a Grothendieck topology - also called a Grothendieck topos. F. W. Lawvere realized the logical content of this structure, and his axioms led to the current notion. Note that Lawvere's notion, initially called elementary topos, is more general than Grothendieck's, and is the one that's nowadays simply called "topos".

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References

The following textbooks provide easy paced first introductions (including basics of category theory). They should be suitable for students of various—even non-mathematical—disciplines:

  • F. William Lawvere and Stephen H. Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
  • F. William Lawvere and Robert Rosebrugh: Sets for Mathematics, Cambridge University Press, Cambridge, 2003. Discusses the foundations of mathematics from a categorical perspective. A book "for students who are beginning the study of advanced mathematical subjects".

The original work of Grothendieck

  • Grothendieck and Verdier: Théorie des topos et cohomologie étale des schémas (known as SGA4)". New York/Berlin: Springer, ??. (Lecture notes in mathematics, 269–270)

Interesting research books that are provide introductions to topos theory (or to a specific aspect of it), but which do not primarily cater to students. The given order roughly (!) reflects the difficulty of the level of exposition:

  • Colin McLarty: Elementary Categories, Elementary Toposes, Clarendon Press, Oxford, 1992. Includes a nice introduction of the basic notions of category theory, topos theory, and topos logic. Assumes very few prerequisites.
  • Robert Goldblatt: Topoi, the Categorial Analysis of Logic. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.
This book is now out of print and the copyright has reverted to the author. It can be accessed freely on Robert Goldblatt's homepage: Topoi, the Categorical Analysis of Logic.

Works which serve as a reference for experts in the field rather than as a treatment suitable for first introduction:

  • Francis Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves, Volume 52 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1994. 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.
  • Peter T. Johnstone: Topos Theory, L. M. S. Monographs no. 10, Academic Press, 1977. For a long time the standard compendium on topos theory. However, it has also been described as "far too hard to read, and not for the faint-hearted", as quoted by Johnstone himself.
  • Peter T. Johnstone: Sketches of an Elephant: A Topos Theory Compendium, Oxford Science Publications, Oxford, 2002. Johnstone’s overwhelming compendium. As of early 2006, two of the scheduled three volumes were available.

Books that target special applications of topos theory:

  • Maria Cristina Pedicchio and Walter Tholen (editors): Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Volume 97 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004. Includes many interesting special applications.de:Topos (Mathematik)

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