**topos** was introduced by
**Alexandre Grothendieck** in the early
1960s, in order to provide a mathematical underpinning for the 'exotic' cohomology theories needed in
algebraic geometry. Every topological space X gives rise to a topos (the category of sheaves of sets on X)
and every topos in Grothendieck’s sense can be considered as a '**generalized space**'.

At the end of the same decade, William **Lawvere** and Myles
**Tierney**
realized that the concept of Grothendieck topos also yielded an abstract
notion of **mathematical universe** within which one could
carry out most familiar set-theoretic constructions, but which also,
thanks to the inherent 'flexibility' of the notion of topos, could be
profitably exploited to construct ‘new mathematical worlds’ having
particular properties.

A few years later, the theory of
**classifying toposes**
added a further fundamental viewpoint to the above-mentioned ones: a
topos can be seen not only as a generalized space or as a mathematical
universe, but also as a suitable kind of **first-order theory**
(considered **up to a general notion of equivalence of theories**).

In fact, every first-order mathematical theory (of a general specified
form - technically speaking, a **geometric theory**) admits
a **canonical topos-theoretic model **lying in the
classifying topos of the theory, satisfying the universal property that
every topos-theoretic model of the theory can be obtained as a pullback
of this canonical model along a unique morphism of toposes. Such
canonical models of mathematical theories do *not* exist in the
classical set-theoretic setting, but any classical (i.e., set-based)
**model** of the theory can be seen as a **point**
of its classifying topos.

**elementary
toposes, **a more general (elementarily axiomatizable) class of
categories in which one can consider models of arbitrary finitary
first-order theories, but in which one cannot do geometry in a
Grothendieckian sense due to the fact that elementary toposes do not in
general admit sites of definition. Even though the study of these more
general concepts has shed important light on the original concept of
topos (in fact, it was seen that several concepts in Grothendieck's
topos theory admitted an 'elementary' formulation), the concept of
elementary topos has not been fruitful as a source of 'concrete'
applications in different fields of Mathematics, essentially because of
the lack of a non-trivial representation theory 'from below' analogous
to that provided by **sites** (which in the context of Grothendieck toposes play the
role of 'carriers of concrete mathematical information').