Extract
Traditionally, mathematics is constructed using set theory, and all objects studied in mathematics are ultimately sets and functions. It is said that category theory could provide better foundations for mathematics. Analyzing exactly which properties of the category of sets and functions are necessary to express the math, you come to the definition of moles, and can thus formulate the mathematics of any moles.
Of course, the category of sets as moles, trivially. In a more interesting topos, the axiom of choice may not be valid, or the law of excluded middle (every statement is true or false) may fail. It is thus of some interest to collect those theorems which are valid in all moles, the moles solamentente not set. One can also work in private moles to focus only on certain items.
For example, constructivists may be interested in the moles of all sets and functions "buildable" in some sense. If relevant under a certain symmetry group G, one can use the moles that consists of all G-spaces. Another important example of moles (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 as the theory of all groups in moles. The individual models of the theory, ie groups in our example, then correspond to functors of moles of coding the category of sets which respect the structure of moles.
The origins of topos theory is the algebraic geometry. Alexander Grothendieck generalized the concept of beam. The result is the category of sheaves with respect to a Grothendieck topology - also called Grothendieck topos. FW Lawvere chose the logical content of this structure, and its axioms led to the current notion. Note that the notion of Lawvere, initially called elementary topos is wider than that of Grothendieck, and is today called simply "moles." Formal definition
A topos is a category that has the following two properties: All limits on a finite set of indices exist. Each object has a power object.
From here we can derive the following facts, some, like the subobject classifier, very important for understanding the concept of topos:
1. All colimits over a finite set of indices exist.
2. The category has a subobject classifier.
3. Any two objects have exponential object.
4. The category is Cartesian closed. Additional examples
There
an important class of examples of moles which was not presented in the introduction: if C is a small category, then the functor category Setco (consisting of all covariant functors from C to sets, with natural transformations as morphisms) are moles. For example, the category of all directed graphs are moles. A graph consists of two sets, a set of arrows and a set of vertices, and two functions between those sets, assigning to each arrow its initial and final vertices. The category of graphs is thus equivalent to the category Setco functor, where C with two objects linked by two morphisms. The categories of finite sets of finite G-spaces and finite directed graphs are also moles. References
• John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html (http://math.ucr.edu/home/baez/topos . html). An introduction sets.
• Robert Goldblatt: Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics vol. 98.), North-Holland, New York, 1984. A good start.
or My book "Topoi: the Categorial Analysis of Logic (North-Holland 1979, Revised Edition 1984) is now officially out of circulation, and the copyright is now only mine. (Http://www.mcs.vuw.ac.nz/ ~ rob /)
The book (http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id = 3)
• Ieke Saunders Mac Lane and Moerdijk: Sheaves in Geometry and Logic: a First Introduction to Topos Theory, Springer, New York, 1992. More complete and more difficult to read.
• Michael Barr and Charles Wells: Toposes, Triples and Theories, Springer, 1985. Accessible online at: http://www.cwru.edu/artsci/math/wells/pub/ttt.html (http://www.cwru.edu/artsci/math/wells/pub/ttt.html). More concise than the previous one: Sheaves in Geometry and Logic.
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