In mathematics, an elementary topos (plural toposes or topoi[1]) is a category which has properties making it resemble the category of sets. Elementary toposes can be used as models of intuitionistic higher-order logic.
Introduction
An elementary topos (hereafter just topos) can be pictured as an alternate mathematical universe.[2] It is a category which is sufficiently like the category of sets (the “standard mathematical universe”) that common mathematical constructions can be carried in it, such as subsets, function spaces, etc.
More precisely, a topos supports intuitionistic higher-order logic as an internal language[3]. As such, toposes are widely used as models of constructive mathematics.
Definition
As originally defined by Lawvere and Tierney, an elementary topos is a category such that:[4][5]
- is finitely complete (it has all finite limits). Equivalently, it has a terminal object, binary products, and binary equalizers.
- is finitely cocomplete (it has all finite colimits). Equivalently, it has an initial object, binary coproducts and binary coequalizers.
- has exponential objects (in addition to binary products, which are required for defining exponential objects). Together with having a terminal object and binary products, this means that is Cartesian closed.
- has a subobject classifier.
The requirement that be finitely cocomplete was later observed to be redundant, as it follows from the rest.[6][7][8] (However, the general construction of finite colimits is relatively complicated; in most examples, one can find a more concrete description, which simplifies the translation of statements from the internal language.)
An even more minimalistic definition is possible: an elementary topos is a finitely complete category with power objects.[9] (This essentially replaces the requirements to have exponentials and a subobject classifier with just the special case of exponentials .)
Many other properties follow, such as being regular and moreover exact[10][11].
Logical functors
A logical 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, and exponential objects.[12]
Examples
Having a subobject classifier is a strong requirement, which rules out most categories of algebraic structures such as groups, rings, etc.[13]
Every Grothendieck topos (a category equivalent to the category of sheaves on a site) is an elementary topos. Important special cases include:
- For any small category , the presheaf category .
- As a further special case of the previous example, for any group , the category of -sets, i.e., sets equipped with an action of , with equivariant maps as morphisms.
- For any topological space , the category of sheaves on .
- The classifying topos for a geometric theory.
The category of finite sets is an elementary topos[14] (however, it lacks a natural numbers object), and similarly for finite -sets.
The effective topos is an important example of an elementary topos with natural numbers object. It is not a Grothendieck topos.[15] It can be viewed as a universe of computable mathematics. This example is generalized by realizability toposes.
A source of examples is the fundamental theorem of topos theory, which states that for every elementary topos and every object , the slice category is an elementary topos.
See also
References
- ↑ Johnstone 2014, p. xx.
- ↑ Blechschmidt, Ingo (2022). "Exploring mathematical objects from custom-tailored mathematical universes". In Oliveri, Gianluigi; Ternullo, Claudio; Boscolo, Stefano (eds.). Objects, structures, and logics (FilMat studies in the philosophy of mathematics). Springer Cham. arXiv:2204.00948. doi:10.1007/978-3-030-84706-7.
- ↑ Lambek & Scott 1986, part II.
- ↑ Goldblatt 1984, p. 84.
- ↑ Johnstone 2002a, p. 85.
- ↑ Johnstone 2002a, section A2.2.
- ↑ van Oosten 2024, section 3.2.
- ↑ Mac Lane & Moerdijk 2012, section IV.5.
- ↑ Johnstone 2002a, p. 92.
- ↑ Johnstone 2014, p. 41.
- ↑ van Oosten 2024, p. 45.
- ↑ McLarty 1992, p. 159
- ↑ "subobject classifier". CatDat. Retrieved 2026-07-26.
- ↑ Mac Lane & Moerdijk 2012, p. 27.
- ↑ van Oosten, Jaap (2008). Realizability: an introduction to its categorical side. Elsevier. p. 133. ISBN 9780444515841.
Bibliography
- McLarty, Colin (1992). Elementary Categories, Elementary Toposes. Clarendon Press. ISBN 978-0-19-158949-2.
- MacLane, Saunders; Moerdijk, Ieke (2012) [1994]. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer. ISBN 978-1-4612-0927-0.
- Johnstone, Peter T. (2014) [1977]. Topos Theory. Courier. ISBN 978-0-486-49336-7.
- Johnstone, Peter T. (2002a). Sketches of an Elephant: A Topos Theory Compendium. Vol. 1. Clarendon Press. ISBN 978-0-19-853425-9.
- Johnstone, Peter T. (2002b). Sketches of an Elephant: A Topos Theory Compendium. Vol. 2. Clarendon Press. ISBN 978-0-19-851598-2.
- Lambek, Joachim; Scott, Philip J. (1986). Introduction to higher order categorical logic. Cambridge University Press. ISBN 978-0-521-35653-4. MR 0856915.
- Goldblatt, Robert (1984). Topoi: the categorical analysis of logic. Studies in logic and the foundation of mathematics. Vol. 98. Elsevier. ISBN 0-444-86711-2.
- van Oosten, Jaap (2024). "Topos theory" (PDF). Archived (PDF) from the original on 25 March 2024.
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