Topos

A category that is equivalent to the category of sheaves of sets on some topologized category. Another definition: A topos is a category $\mathcal C$ such that any sheaf for the canonical topology on $\mathcal C$ is representable. For the objects of a topos (which are sheaves of sets) the usual constructions of the category of sets can be defined. For this reason topoi may serve as non-standard models of set theory. Here it is more convenient to use a more general definition: An elementary topos is a category $\mathcal C$ with products and final object, with a contravariant functor $\mathcal P\colon \mathcal C\to\mathcal C$ (here $\mathcal P(X)$ for $X\in\mathcal C$ is understood as the object of subobjects of $X$) and with monomorphisms $\Sigma_X\to X\times\mathcal P(X)$, where $\Sigma$ is the graph of the relation of membership. The object $\mathcal P(1)$ serves as the natural domain of values of propositional logic in the topos $\mathcal C$.

References

[1]	M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Théorie des topos et cohomologie étale des schémas (SGA 4) , Lect. notes in math. , 269; 270; 305 , Springer (1972–1973)
[2]	P. Deligne, "Cohomologie étale (SGA 4 1/2)" , Lect. notes in math. , 569 , Springer (1977)
[3]	P. Cartier, "Catégories, logique et faisceaux: modèles de la théorie des ensembles" , Sem. Bourbaki 1977/78 , Lect. notes in math. , 710 , Springer (1979)
[4]	R.I. Goldblatt, "Topoi: the categorical analysis of logic" , North-Holland (1979)

Comments

A topological category is better known as a site.

The notion of a topos was introduced around 1963 by A. Grothendieck [1] in connection with certain cohomology theories used in algebraic geometry, in particular the étale and crystalline cohomologies of a scheme (cf. Etale topology). Although such cohomology can be described directly in terms of a given site, the topos yields a more invariant description; in contrast to the "classical" case of sheaves on a topological space (cf. Sheaf theory), many different sites can give rise to the same topos of sheaves, and hence to the same cohomology theory. Later, M. Artin and B. Mazur [a1] showed how to define the homotopy groups of a topos, and established "Whitehead theorems" relating the homotopy and cohomology of a given topos (cf. Homotopy type of a topological category).

The notion of elementary topos was introduced in 1969 by F.W. Lawvere and M. Tierney (cf. [a2]). Their idea was to isolate those elementary (i.e. first-order) properties of the category of sets which are also enjoyed by sheaf categories, so enabling the latter to be viewed as models of a "generalized set theory" . The Lawvere–Tierney axioms are both simpler and more general than those introduced by Grothendieck; for this reason, the unadorned term "topos" is nowadays generally taken to mean "elementary topos" , and topoi in the older sense (i.e. categories equivalent to the category of sheaves on a site) are known as Grothendieck topoi. The axioms for an elementary topos can serve as a foundation for mathematics, alternative to the traditional set-theoretic axiom systems [a3], [a4]. Related to this, each elementary topos can be viewed as a model of higher-order intuitionistic type theory (cf. Types, theory of) (see [a5], [a6]), in such a way that many of the traditional models studied in intuitionism, such as Kripke models and Beth models, become special cases. These traditional models do not always fit the narrower framework of Grothendieck topoi. For example, Kleene's realizability interpretation gives rise to the effective topos introduced by J.M.E. Hyland [a7], which is not a Grothendieck topos. The effective topos has recently been of considerable interest in theoretical computer science, since it contains "natural" models for the polymorphic -calculus and related theories [a8], [a9].

Since the introduction of the Lawvere–Tierney axioms, an important aspect of the development of topos theory has been the interaction between its logical side, as indicated in the previous paragraph, and its geometrical side as represented by the earlier researches of Grothendieck and his followers. In this connection, a key role has been played by investigations of the theory of locales (cf. Locale), and by the generalization of topological concepts from the category of locales to that of (Grothendieck) topoi. An important example is the work of A. Joyal and Tierney [a10] on open mappings and descent theory for locales and topoi, leading to a representation of Grothendieck topoi in terms of internal groupoids (cf. Groupoid) in the category of locales [a11], [a12]. Similarly, generalizing the notion of local compactness for locales led to a characterization of the exponentiable objects (i.e. those possessing "function-spaces" ) in the category of Grothendieck topoi [a13]. More recent investigations [a14], [a15], [a16] have shown that the homotopy and cohomology theory of Grothendieck topoi is "no more complicated" than that of locales, in the sense that every topos can be covered by a locale with the same "weak homotopy type" .

For more information on topos theory, including aspects of the subject not covered here, see [a4], [a6], [a17]–[a20].

References

[a1]	M. Artin, B. Mazur, "Etale homotopy" , Lect. notes in math. , 100 , Springer (1969) MR0245577 Zbl 0182.26001
[a2]	F.W. Lawvere, "Quantifiers and sheaves" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 329–334 MR0430021 Zbl 0261.18010
[a3]	F.W. Lawvere, "An elementary theory of the category of sets" Proc. Nat. Acad. Sci. USA , 52 (1964) pp. 1506–1511 MR0396262 MR0172807 Zbl 0141.00603
[a4]	J.L. Bell, "Toposes and local set theories" , Oxford Univ. Press (1988) MR0972257 Zbl 0649.18004
[a5]	M.P. Fourman, "The logic of topoi" J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1977) pp. 1053–1090
[a6]	J. Lambek, P.J. Scott, "Introduction to higher-order categorical logic" , Cambridge Univ. Press (1986) MR0856915 Zbl 0596.03002
[a7]	J.M.E. Hyland, "The effective topos" A.S. Troelstra (ed.) D. van Dalen (ed.) , The L.E.J. Brouwer Centenary Symposium , North-Holland (1982) pp. 165–216 MR0717245 Zbl 0522.03055
[a8]	J.M.E. Hyland, "A small complete category" Ann. Pure Appl. Logic , 40 (1988) pp. 135–165 MR0972520 Zbl 0659.18007
[a9]	A. Carboni, P.J. Freyd, A. Scedrov, "A categorical approach to realizability and polymorphic types" M. Main (ed.) et al. (ed.) , Proc. 3-rd A.C.M. Workshop on Mathematical Foundations of Programming Language Semantics , Lect. notes in comp. science , 298 , Springer (1988) pp. 23–42 MR0948482 Zbl 0651.18004
[a10]	A. Joyal, M. Tierney, "An extension of the Galois theory of Grothendieck" Memoirs Amer. Math. Soc. , 309 (1984) MR0756176 Zbl 0541.18002
[a11]	P.T. Johnstone, "How general is a generalized space?" I.H. James (ed.) E.H. Kronheimer (ed.) , Aspects of Topology: in Memory of Hugh Dowker , Lect. notes London Math. Soc. , 93 , Cambridge Univ. Press (1985) pp. 77–111 MR0787824 Zbl 0556.18003
[a12]	I. Moerdijk, "Toposes and groupoids" F. Borceux (ed.) , Categorical Algebra and its Applications , Lect. notes in math. , 1348 , Springer (1988) pp. 280–298 MR0975977 Zbl 0659.18008
[a13]	P.T. Johnstone, A. Joyal, "Continuous categories and exponentiable toposes" J. Pure Appl. algebra , 25 (1982) pp. 255–296 MR0666021 Zbl 0487.18003
[a14]	A. Joyal, G.C. Wraith, "Eilenberg–Mac Lane toposes and cohomology" J.W. Gray (ed.) , Mathematical Applications of Category Theory , Amer. Math. Soc. (1984) pp. 117–131 MR749771
[a15]	A. Joyal, I. Moerdijk, "Toposes as homotopy groupoids" Adv. Math. , 80 (1990) pp. 22–38 MR1041882 Zbl 0783.18004
[a16]	A. Joyal, I. Moerdijk, "Toposes are cohomologically equivalent to spaces" Amer. J. Math. , 112 (1990) pp. 87–95 MR1037604 Zbl 0713.18004
[a17]	P.J. Freyd, "Aspects of topoi" Bull. Austral. Math. Soc. , 7 (1972) pp. 1–76; 467–480 MR0419550 MR0396714 Zbl 0252.18002 Zbl 0252.18001
[a18]	P.T. Johnstone, "Topos theory" , Acad. Press (1977) MR0470019 Zbl 0368.18001
[a19]	M. Barr, C. Wells, "Toposes, triples and theories" , Springer (1985) MR0771116 Zbl 0567.18001
[a20]	S. MacLane, I. Moerdijk, "A first introduction to topos theory" , Springer (1991) Zbl 0822.18001