Editing History of topos theory (section)

==From pure category theory to categorical logic==
{{See|Categorical logic}}

The current definition of [[topos]] goes back to [[William Lawvere]] and [[Myles Tierney]]. While the timing follows closely on from that described above, as a matter of history, the attitude is different, and the definition is more inclusive. That is, there are examples of '''toposes''' that are not a '''Grothendieck topos'''. What is more, these may be of interest for a number of [[mathematical logic|logical]] disciplines.

Lawvere and Tierney's definition picks out the central role in topos theory of the '''[[sub-object classifier]]'''. In the usual category of sets, this is the two-element set of Boolean [[truth-value]]s, '''true''' and '''false'''. It is almost tautologous to say that the subsets of a given set ''X'' are ''the same as'' (just as good as) the functions on ''X'' to any such given two-element set: fix the 'first' element and make a subset ''Y'' correspond to the function sending ''Y'' there and its complement in ''X'' to the other element.

Now sub-object classifiers can be found in [[sheaf (mathematics)|sheaf]] theory. Still tautologously, though certainly more abstractly, for a [[topological space]] ''X'' there is a direct description of a sheaf on ''X'' that plays the role with respect to all sheaves of sets on ''X''. Its set of sections over an open set ''U'' of ''X'' is just the set of open subsets of ''U''. The [[sheaf (mathematics)|space associated with a sheaf]], for it, is more difficult to describe.

Lawvere and Tierney therefore formulated '''axioms for a topos''' that assumed a sub-object classifier, and some limit conditions (to make a [[cartesian-closed category]], at least). For a while this notion of topos was called 'elementary topos'.

Once the idea of a connection with logic was formulated, there were several developments 'testing' the new theory:

* models of [[set theory]] corresponding to proofs of the independence of the [[axiom of choice]] and [[continuum hypothesis]] by [[Paul Cohen]]'s method of [[forcing (mathematics)|forcing]].
* recognition of the connection with [[Kripke semantics]], the [[intuitionistic logic|intuitionistic]] [[existential quantifier]] and [[intuitionistic type theory]].
* combining these, discussion of the [[intuitionistic theory of real numbers]], by sheaf models.