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See: Category theory, Yoneda Lemma A Grothendieck topos is a reflective subcategory of a presheaf category such that the reflection functor preserves finite limits. Presheaf category is the total information. Topos is a weakened, limited set of information. Pullback is crucial because its adjoints are exists and for all. Pullback (and limits) are the basis for making distinctions. What are the adjoints for the push forward? The logic of geometry is based on local coherence. And the global consequences...? are topology? Morita equivalence. Double negation in the simpler, topos world, almost holds  but when you translate into the richer world  you get a highly nontrivial statement. Example: internal view (of complicated curves) are embedded in an external space (and the latter is removed in the topos). Can define objects and math before logic. Sheaves are not interesting but rather, what we see from our external point of view, and what needs to be removed to see the interesting internal point of view. Combinatorics  create objects "for children"  then count them  the formulas appear in other contexts  gives clues for related toposes. "Sketches of an Elephant: A Topos Theory Compendium 2 Volume Set" Topos:
A topos is a category that has the following two properties:
Grothendieck topos Let C be a category. A theorem of Giraud states that the following are equivalent:
A category with these properties is called a "(Grothendieck) topos". Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf. Giraud's axioms for a category C are:
The last axiom needs the most explanation. If X is an object of C, an "equivalence relation" R on X is a map R→X×X in C such that for any object Y in C, the induced map Hom(Y,R)→Hom(Y,X)×Hom(Y,X) gives an ordinary equivalence relation on the set Hom(Y,X). Since C has colimits we may form the coequalizer of the two maps R→X; call this X/R. The equivalence relation is "effective" if the canonical map {$R → X × _{X / R}X$} is an isomorphism. Sources
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