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There are two related kinds of categories which are called topoi (or alternatively toposes). First, there is the Grothendieck topos, which was developed by Grothendieck as part of his general reconstruction of algebraic geometry. Second, there is the elementary topos, which was introduced by Lawvere as a setting for work in categorical logic. We give a brief overview of each kind of topos.
A Grothendieck topos is a category naturally equivalent to the category of sheaves on some site.
An elementary topos is a category
 which:
The first assumption guarantees the existence of finite limits and colimits as well as power objects. This allows
 to model basic constructions of set theory such as products, disjoint unions, intersections, and powersets. It also guarantees that
 has a terminal object  , which corresponds to a singleton set in
 . We can model elements of an object  by morphisms  .
The second assumption means that
 has a notion of “truth”. In particular,
 must have a truth object  and a morphism
 such that if
 is any monomorphism of
 , then there is a unique associated characteristic morphism
 such that the diagram
is a pullback square. Speaking loosely, this says that a subobject of  arises as a collection of elements of  satisfying a particular predicate  . The converse of this assumption
corresponds to the comprehension axiom of set theory and follows from Cartesian closedness.
An elementary topos is a Boolean topos if its truth object has exactly two elements, “true”
 and “false”
 . It has choice (admits the axiom of choice) if every epimorphism is split. It is a fact that every elementary topos with choice is Boolean. Note that not every elementary topos has choice. So elementary topoi can be used to model intuitionistic logic.
The category of sets is the canonical example of a Boolean topos.
Remarks.
- A category
 is a topos iff it is finitely complete and has power objects.
- If
 and
 are topoi, so is
 .
- If
 is a topos and  is an object of
 , then the comma category
 is a topos.
- Every Grothendieck topos is also an elementary topos.
- 1
- M. Barr and C. Wells. Toposes, Triples and Theories. Montreal: McGill University, 2000.
- 2
- J. Lambek and P. J. Scott. Introduction to higher order categorical logic. Cambridge University Press, 1986.
- 3
- S. Mac Lane. Categories for the Working Mathematician, 2nd ed. Springer-Verlag, 1997
- 4
- S. Mac Lane and I. Moerdijk. Sheaves and Geometry in Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.
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Cross-references: comma category, finitely complete, iff, canonical, category of sets, intuitionistic logic, Boolean, epimorphism, axiom of choice, comprehension axiom, converse, predicate, collection, subobject, pullback square, diagram, characteristic morphism, monomorphism, truth object, morphisms, object, singleton, terminal object, powersets, intersections, disjoint unions, products, set theory, power objects, colimits, limits, finite, Cartesian closed category, site, sheaves, naturally equivalent, logic, algebraic geometry, categories
There are 30 references to this entry.
This is version 12 of topos, born on 2007-01-19, modified 2008-08-16.
Object id is 8796, canonical name is Topos.
Accessed 4376 times total.
Classification:
| AMS MSC: | 14F20 (Algebraic geometry :: homology theory :: Étale and other Grothendieck topologies and cohomologies) | | | 03G30 (Mathematical logic and foundations :: Algebraic logic :: Categorical logic, topoi) | | | 18B25 (Category theory; homological algebra :: Special categories :: Topoi) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A\ar[d]_{m}\ar[r] & 1\ar[d]^{\top} \ B\ar[r]^{\chi} & \Omega } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/8796/l2h/img15.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A\ar[d]_{m}\ar[r] & 1\ar[d]^{\top} \ B\ar[r]^{\chi} & \Omega } } \end{xy}$")