[citation needed] During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found.
The question of points was close to resolution by 1950; Alexander Grothendieck took a sweeping step (invoking the Yoneda lemma) that disposed of it—naturally at a cost, that every variety or more general scheme should become a functor.
In the light of later work (c. 1970), 'descent' is part of the theory of comonads; here we can see one way in which the Grothendieck school bifurcates in its approach from the 'pure' category theorists, a theme that is important for the understanding of how the topos concept was later treated.
An abelian category is supposed to be closed under certain category-theoretic operations—by using this kind of definition one can focus entirely on structure, saying nothing at all about the nature of the objects involved.
The idea of a Grothendieck topology (also known as a site) has been characterised by John Tate as a bold pun on the two senses of Riemann surface.
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.
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).
On the other hand Brouwer's long efforts on 'species', as he called the intuitionistic theory of reals, are presumably in some way subsumed and deprived of status beyond the historical.
The later work on étale cohomology has tended to suggest that the full, general topos theory isn't required.
What results is essentially an intuitionistic (i.e. constructive logic) theory, its content being clarified by the existence of a free topos.
In that book, the talk is about constructive mathematics; but in fact this can be read as foundational computer science (which is not mentioned).
If one wants to discuss set-theoretic operations, such as the formation of the image (range) of a function, a topos is guaranteed to be able to express this, entirely constructively.
The point of view is written up in Peter Johnstone's Stone Spaces, which has been called by a leader in the field of computer science 'a treatise on extensionality'.
Granted the general view of Saunders Mac Lane about ubiquity of concepts, this gives them a definite status.