In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a classifying topos for some sort of structure is a
topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...
''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''.
Examples
*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
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
.
*The classifying topos for
linear order
In mathematics, a total order 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 ( ref ...
s with distinct largest and smallest elements of a topos is the topos of
simplicial set
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs.
Every simplicial set gives rise to a "n ...
s.
*If ''G'' is a
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...
, the classifying topos for ''G''-
torsor
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 no ...
s over a topos is the topos ''BG'' of ''G''-sets.
*The
classifying space
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 ...
of
topological group
In mathematics, topological groups are 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 structures ...
s in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
.
References
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External links
*
Topos theory
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