Topos

The category that works like a category of sheaves of sets in mathematics on a topological space is referred to as topos. The plural form of the topos is toposes or topoi. The major focus is to study a space by focusing sheaves on that particular space. The notion of topos was first introduced by Alexander Grothendieck. This notion has been utilized more in mathematics where one can find an effect topological intuition and where the honest topological space will be lacking.

The introduction of the etale topos is considered to be the greatest success that has been achieved by this programmatic idea.

The following equivalent formulations were stated by the theorem of Giraud.

• There is a small category D and an inclusion C  Presh(D) that admits a finite-limit-preserving left adjoint.
• On a Grothendieck site, C is the category of sheaves
• C satisfies Giraud's axioms, below.

For a category C, the Giraud’s axioms are that,

C admits all small colimits since it has a small set of generators. In addition to this, the colimit also commutes with the fiber products. The initial object in C is the fiber product of X and Y. In simple words, the sums in the C are just disjoint. 

Most explanation is needed in the last axiom. If X is an object of C, an equivalence relation R on X is a map R→X×X in C such that all the maps Hom(Y,R)→Hom(Y,X)×Hom(Y,X) are the relation set that are of equivalence. Since there are colimits in C, we may form the coequalizer of the two maps R→X; and this can be called X/R. If the colonal map is

then the equivalence relation will be effective.

Geometrics morphisms:

If the topos are X and Y, then the geometric morphism u: X→Y is just a pair of the adjoint functors where  u^* would be made to preserve finite limits. The pair of the adjoint functors here are u^*,u_* By having a right adjoint the u^* preserves the finite limits in it.

According to the Freyd’s adjoint functor theorem, in order to give a functor u^*: Y → X that preserves all small colimits and finite limits, there is a need to give a geometric morphism X → Y.

Ringed Topoi:

The ringed topoi on the other hand is the pair (X,R) where the R is a commutative ring object that is present in X and X is a topos. Usually most constructions that are present in the ringed spaces will go for the ringed topoi.

Questions:

• What is Topos?
• What is ringed topoi?