Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a classifying topos for some sort of structure is a topos
''T'' such that there is a natural equivalence between geometric morphisms from a cocomplete topos ''E'' to ''T'' and the category of models for the structure in ''E''.
*The classifying topos for objects of a topos is the topos of presheaves
over the opposite of the category of finite sets.
*The classifying topos for rings of a topos is the topos of presheaves over the opposite of the category of finitely presented rings.
*The classifying topos for local rings of a topos is the topos of sheaves over the opposite of the category of finitely presented rings with the Zariski topology
*The classifying topos for
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...
s with distinct largest and smallest elements of a topos is the topos of simplicial set
*If ''G'' is a discrete group
, the classifying topos for ''G''-
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
s over a topos is the topos ''BG'' of ''G''-sets.
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...