Mathematical category
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 notion of localization; they are a direct generalization of
point-set topology
.
[1]
The
Grothendieck topoi
find applications in
algebraic geometry
; the more general
elementary topoi
are used in
logic
.
The mathematical field that studies topoi is called
topos theory
.
Grothendieck topos (topos in geometry)
[
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]
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by
Alexander Grothendieck
by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is the
etale topos
of a
scheme
. Another illustration of the capability of Grothendieck topoi to incarnate the “essence” of different mathematical situations is given by their use as bridges for connecting theories which, albeit written in possibly very different languages, share a common mathematical content.
[2]
[3]
Equivalent definitions
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]
A Grothendieck topos is a
category
C
which satisfies any one of the following three properties. (A
theorem
of
Jean Giraud
states that the properties below are all equivalent.)
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
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]
Giraud's axioms for a
category
C
are:
- C
has a small set of
generators
, and admits all small
colimits
. Furthermore,
fiber products
distribute over coproducts. That is, given a set
I
, an
I
-indexed coproduct mapping to
A
, and a morphism
A'
→
A
, the pullback is an
I
-indexed coproduct of the pullbacks:
![{\displaystyle \left(\coprod _{i\in I}B_{i}\right)\times _{A}A'\cong \coprod _{i\in I}(B_{i}\times _{A}A').}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57fb6db9c16c4a07a802a640eacb67f60f4d9507)
- Sums in
C
are disjoint. In other words, the fiber product of
X
and
Y
over their sum is the
initial object
in
C
.
- All
equivalence relations
in
C
are
effective
.
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
![{\displaystyle R\to X\times _{X/R}X\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/252e844f8c87ca8ace8cf5c9ef99feb3226d487e)
is an isomorphism.
Examples
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]
Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often give
rise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory.
Category of sets and G-sets
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]
The
category
of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point since functors on the singleton category with a single object and only the identity morphism are just specific sets in the category of sets.
Similarly, there is a topos
for any
group
which is equivalent to the category of
-sets. We construct this as the category of presheaves on the category with one object, but now the set of morphisms is given by the
group
. Since any functor must give a
-action on the target, this gives the category of
-sets. Similarly, for a
groupoid
the category of presheaves on
gives a collection of sets indexed by the set of objects in
, and the automorphisms of an object in
has an action on the target of the functor.
Topoi from ringed spaces
[
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]
More exotic examples, and the
raison d'etre
of topos theory, come from algebraic geometry. The basic example of a topos comes from the Zariski topos of a
scheme
. For each scheme
there is a site
(of objects given by open subsets and morphisms given by inclusions) whose category of presheaves forms the Zariski topos
. But once distinguished classes of morphisms are considered, there are multiple generalizations of this which leads to non-trivial mathematics. Moreover, topoi give the foundations for studying schemes purely as functors on the category of algebras.
To a scheme and even a
stack
one may associate an
etale
topos, an
fppf
topos, or a
Nisnevich
topos. Another important example of a topos is from the
crystalline site
. In the case of the etale topos, these form the foundational objects of study in
anabelian geometry
, which studies objects in algebraic geometry that are determined entirely by the structure of their
etale fundamental group
.
Pathologies
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]
Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of
pathological
behavior. For instance, there is an example due to
Pierre Deligne
of a nontrivial topos that has no points (see below for the definition of points of a topos).
Geometric morphisms
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]
If
and
are topoi, a
geometric morphism
is a pair of
adjoint functors
(
u
?
,
u
?
) (where
u
?
:
Y
→
X
is left adjoint to
u
?
:
X
→
Y
) such that
u
?
preserves finite limits. Note that
u
?
automatically preserves colimits by virtue of having a right adjoint.
By
Freyd's adjoint functor theorem
, to give a geometric morphism
X
→
Y
is to give a functor
u
?
:
Y
→
X
that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of
locales
.
If
and
are topological spaces and
is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi for the sites
.
Points of topoi
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]
A point of a topos
is defined as a geometric morphism from the topos of sets to
.
If
X
is an ordinary space and
x
is a point of
X
, then the functor that takes a sheaf
F
to its stalk
F
x
has a right adjoint
(the "skyscraper sheaf" functor), so an ordinary point of
X
also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map
x
:
1
→
X
.
For the etale topos
of a space
, a point is a bit more refined of an object. Given a point
of the underlying scheme
a point
of the topos
is then given by a separable field extension
of
such that the associated map
factors through the original point
. Then, the factorization map
![{\displaystyle {\text{Spec}}(k)\to {\text{Spec}}(\kappa (x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6265cb00e55213986be0b74ec1fd7eebaf706e92)
is an
etale morphism
of schemes.
More precisely, those are the
global
points. They are not adequate in themselves for displaying the space-like aspect of a topos, because a non-trivial topos may fail to have any.
Generalized
points are geometric morphisms from a topos
Y
(the
stage of definition
) to
X
. There are enough of these to display the space-like aspect. For example, if
X
is the
classifying topos
S
[
T
] for a geometric theory
T
, then the universal property says that its points are the models of
T
(in any stage of definition
Y
).
Essential geometric morphisms
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]
A geometric morphism (
u
?
,
u
?
) is
essential
if
u
?
has a further left adjoint
u
!
, or equivalently (by the adjoint functor theorem) if
u
?
preserves not only finite but all small limits.
Ringed topoi
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]
A
ringed topos
is a pair (
X
,
R
), where
X
is a topos and
R
is a commutative
ring object
in
X
. Most of the constructions of
ringed spaces
go through for ringed topoi. The category of
R
-module
objects in
X
is an
abelian category
with enough injectives. A more useful abelian category is the subcategory of
quasi-coherent
R
-modules: these are
R
-modules that admit a presentation.
Another important class of ringed topoi, besides ringed spaces, are the etale topoi of
Deligne?Mumford stacks
.
Homotopy theory of topoi
[
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]
Michael Artin
and
Barry Mazur
associated to the site underlying a topos a
pro-simplicial set
(up to
homotopy
).
[4]
(It's better to consider it in Ho(pro-SS); see Edwards) Using this
inverse system
of simplicial sets one may
sometimes
associate to a
homotopy invariant
in classical topology an inverse system of invariants in topos theory. The study of the pro-simplicial set associated to the etale topos of a scheme is called
etale homotopy theory
.
[5]
In good cases (if the scheme is
Noetherian
and
geometrically unibranch
), this pro-simplicial set is
pro-finite
.
Elementary topoi (topoi in logic)
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]
Introduction
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]
Since the early 20th century, the predominant axiomatic foundation of mathematics has been
set theory
, in which all mathematical objects are ultimately represented by sets (including
functions
, which map between sets). More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set-theoretic mathematics. But one could instead choose to work with many alternative topoi. A standard formulation of the
axiom of choice
makes sense in any topos, and there are topoi in which it is invalid.
Constructivists
will be interested to work in a topos without the
law of excluded middle
. If symmetry under a particular
group
G
is of importance, one can use the topos consisting of all
G
-sets
.
It is also possible to encode an
algebraic theory
, such as the theory of groups, as a topos, in the form of a
classifying topos
. The individual models of the theory, i.e. the groups in our example, then correspond to
functors
from the encoding topos to the category of sets that respect the topos structure.
Formal definition
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]
When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise:
A topos is a category that has the following two properties:
- All
limits
taken over finite index categories exist.
- Every object has a power object. This plays the role of the
powerset
in set theory.
Formally, a
power object
of an object
is a pair
with
, which classifies relations, in the following sense.
First note that for every object
, a morphism
("a family of subsets") induces a subobject
. Formally, this is defined by pulling back
along
. The universal property of a power object is that every relation arises in this way, giving a bijective correspondence between relations
and morphisms
.
From finite limits and power objects one can derive that
In some applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived.
Logical functors
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]
A
logical functor
is a functor between topoi that preserves finite limits and power objects. Logical functors preserve the structures that topoi have. In particular, they preserve finite colimits,
subobject classifiers
, and
exponential objects
.
[6]
Explanation
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]
A topos as defined above can be understood as a Cartesian closed category for which the notion of subobject of an object has an
elementary
or first-order definition. This notion, as a natural categorical abstraction of the notions of
subset
of a set,
subgroup
of a group, and more generally
subalgebra
of any
algebraic structure
, predates the notion of topos. It is definable in any category, not just topoi, in
second-order
language, i.e. in terms of classes of morphisms instead of individual morphisms, as follows. Given two monics
m
,
n
from respectively
Y
and
Z
to
X
, we say that
m
≤
n
when there exists a morphism
p
:
Y
→
Z
for which
np
=
m
, inducing a
preorder
on monics to
X
. When
m
≤
n
and
n
≤
m
we say that
m
and
n
are equivalent. The subobjects of
X
are the resulting equivalence classes of the monics to it.
In a topos "subobject" becomes, at least implicitly, a first-order notion, as follows.
As noted above, a topos is a category
C
having all finite limits and hence in particular the empty limit or final object 1. It is then natural to treat morphisms of the form
x
: 1 →
X
as
elements
x
∈
X
. Morphisms
f
:
X
→
Y
thus correspond to functions mapping each element
x
∈
X
to the element
fx
∈
Y
, with application realized by composition.
One might then think to define a subobject of
X
as an equivalence class of monics
m
:
X′
→
X
having the same
image
{
mx
|
x
∈
X′
}. The catch is that two or more morphisms may correspond to the same function, that is, we cannot assume that
C
is concrete in the sense that the functor
C
(1,-):
C
→
Set
is faithful. For example the category
Grph
of
graphs
and their associated
homomorphisms
is a topos whose final object 1 is the graph with one vertex and one edge (a self-loop), but is not concrete because the elements 1 →
G
of a graph
G
correspond only to the self-loops and not the other edges, nor the vertices without self-loops. Whereas the second-order definition makes
G
and the subgraph of all self-loops of
G
(with their vertices) distinct subobjects of
G
(unless every edge is, and every vertex has, a self-loop), this image-based one does not. This can be addressed for the graph example and related examples via the
Yoneda Lemma
as described in the
Further examples
section below, but this then ceases to be first-order. Topoi provide a more abstract, general, and first-order solution.
Figure 1.
m
as a pullback of the generic subobject
t
along
f
.
As noted above, a topos
C
has a subobject classifier Ω, namely an object of
C
with an element
t
∈ Ω, the
generic subobject
of
C
, having the property that every
monic
m
:
X′
→
X
arises as a pullback of the generic subobject along a unique morphism
f
:
X
→ Ω, as per Figure 1. Now the pullback of a monic is a monic, and all elements including
t
are monics since there is only one morphism to 1 from any given object, whence the pullback of
t
along
f
:
X
→ Ω is a monic. The monics to
X
are therefore in bijection with the pullbacks of
t
along morphisms from
X
to Ω. The latter morphisms partition the monics into equivalence classes each determined by a morphism
f
:
X
→ Ω, the characteristic morphism of that class, which we take to be the subobject of
X
characterized or named by
f
.
All this applies to any topos, whether or not concrete. In the concrete case, namely
C
(1,-) faithful, for example the category of sets, the situation reduces to the familiar behavior of functions. Here the monics
m
:
X′
→
X
are exactly the injections (one-one functions) from
X′
to
X
, and those with a given image {
mx
|
x
∈
X′
} constitute the subobject of
X
corresponding to the morphism
f
:
X
→ Ω for which
f
−1
(
t
) is that image. The monics of a subobject will in general have many domains, all of which however will be in bijection with each other.
To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monics to
X
as had previously been defined explicitly by the second-order notion of subobject for any category. The notion of equivalence relation on a class of morphisms is itself intrinsically second-order, which the definition of topos neatly sidesteps by explicitly defining only the notion of subobject
classifier
Ω, leaving the notion of subobject of
X
as an implicit consequence characterized (and hence namable) by its associated morphism
f
:
X
→ Ω.
Further examples and non-examples
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]
Every Grothendieck topos is an elementary topos, but the converse is not true (since every Grothendieck topos is cocomplete, which is not required from an elementary topos).
The categories of finite sets, of finite
G
-sets (
actions of a group
G
on a finite set), and of finite graphs are elementary topoi that are not Grothendieck topoi.
If
C
is a small category, then the
functor category
Set
C
(consisting of all covariant functors from
C
to sets, with
natural transformations
as morphisms) is a topos. For instance, the category
Grph
of graphs of the kind permitting multiple directed edges between two vertices is a topos. Such a graph consists of two sets, an edge set and a vertex set, and two functions
s,t
between those sets, assigning to every edge
e
its source
s
(
e
) and target
t
(
e
).
Grph
is thus
equivalent
to the functor category
Set
C
, where
C
is the category with two objects
E
and
V
and two morphisms
s,t
:
E
→
V
giving respectively the source and target of each edge.
The
Yoneda lemma
asserts that
C
op
embeds in
Set
C
as a full subcategory. In the graph example the embedding represents
C
op
as the subcategory of
Set
C
whose two objects are
V'
as the one-vertex no-edge graph and
E'
as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms from
V'
to
E'
(both as natural transformations). The natural transformations from
V'
to an arbitrary graph (functor)
G
constitute the vertices of
G
while those from
E'
to
G
constitute its edges. Although
Set
C
, which we can identify with
Grph
, is not made concrete by either
V'
or
E'
alone, the functor
U
:
Grph
→
Set
2
sending object
G
to the pair of sets (
Grph
(
V'
,
G
),
Grph
(
E'
,
G
)) and morphism
h
:
G
→
H
to the pair of functions (
Grph
(
V'
,
h
),
Grph
(
E'
,
h
)) is faithful. That is, a morphism of graphs can be understood as a
pair
of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of
generalized
elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of topoi by allowing an object to have multiple underlying sets, that is, to be multisorted.
The category of
pointed sets
with point-preserving functions is
not
a topos, since it doesn't have power objects: if
were the power object of the pointed set
, and
denotes the pointed singleton, then there is only one point-preserving function
, but the relations in
are as numerous as the pointed subsets of
. The
category of abelian groups
is also not a topos, for a similar reason: every group homomorphism must map 0 to 0.
See also
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]
Notes
[
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]
- ^
Illusie 2004
- ^
Caramello, Olivia
(2016).
Grothendieck toposes as unifying `bridges' in Mathematics
(PDF)
(HDR). Paris Diderot University (Paris 7).
- ^
Caramello, Olivia (2017).
Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic `bridges
. Oxford University Press.
doi
:
10.1093/oso/9780198758914.001.0001
.
ISBN
9780198758914
.
- ^
Artin, Michael
;
Mazur, Barry
(1969).
Etale homotopy
. Lecture Notes in Mathematics. Vol. 100.
Springer-Verlag
.
doi
:
10.1007/BFb0080957
.
ISBN
978-3-540-36142-8
.
- ^
Friedlander, Eric M.
(1982),
Etale homotopy of simplicial schemes
, Annals of Mathematics Studies, vol. 104,
Princeton University Press
,
ISBN
978-0-691-08317-9
- ^
McLarty 1992
, p.
159
References
[
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]
- Some gentle papers
The following texts are easy-paced introductions to toposes and the basics of category theory. They should be suitable for those knowing little mathematical logic and set theory, even non-mathematicians.
Grothendieck foundational work on topoi:
The following monographs include an introduction to some or all of topos theory, but do not cater primarily to beginning students. Listed in (perceived) order of increasing difficulty.
- McLarty, Colin
(1992).
Elementary Categories, Elementary Toposes
. Clarendon Press.
ISBN
978-0-19-158949-2
.
A nice introduction to the basics of category theory, topos theory, and topos logic. Assumes very few prerequisites.
- Goldblatt, Robert
(2013) [1984].
Topoi: The Categorial Analysis of Logic
. Courier Corporation.
ISBN
978-0-486-31796-0
.
A good start. Available
online
at
Robert Goldblatt's homepage.
- Bell, John L.
(2001).
"The Development of Categorical Logic"
. In Gabbay, D.M.; Guenthner, Franz (eds.).
Handbook of Philosophical Logic
. Vol. 12 (2nd ed.). Springer. pp. 279?.
ISBN
978-1-4020-3091-8
.
Version available
online
at
John Bell's homepage.
- MacLane, Saunders
;
Moerdijk, Ieke
(2012) [1994].
Sheaves in Geometry and Logic: A First Introduction to Topos Theory
. Springer.
ISBN
978-1-4612-0927-0
.
More complete, and more difficult to read.
- Barr, Michael
;
Wells, Charles
(2013) [1985].
Toposes, Triples and Theories
. Springer.
ISBN
978-1-4899-0023-4
.
(Online version). More concise than
Sheaves in Geometry and Logic
, but hard on beginners.
- Reference works for experts, less suitable for first introduction
- Edwards, D.A.; Hastings, H.M. (1976).
?ech and Steenrod homotopy theories with applications to geometric topology
. Lecture Notes in Maths. Vol. 542. Springer-Verlag.
doi
:
10.1007/BFb0081083
.
ISBN
978-3-540-38103-7
.
- Borceux, Francis (1994).
Handbook of Categorical Algebra: Volume 3, Sheaf Theory
. Encyclopedia of Mathematics and its Applications. Vol. 52. Cambridge University Press.
ISBN
978-0-521-44180-3
.
The third part of "Borceux' remarkable magnum opus", as Johnstone has labelled it. Still suitable as an introduction, though beginners may find it hard to recognize the most relevant results among the huge amount of material given.
- Johnstone, Peter T.
(2014) [1977].
Topos Theory
. Courier.
ISBN
978-0-486-49336-7
.
For a long time the standard compendium on topos theory. However, even Johnstone describes this work as "far too hard to read, and not for the faint-hearted."
- Johnstone, Peter T. (2002).
Sketches of an Elephant: A Topos Theory Compendium
. Vol. 2. Clarendon Press.
ISBN
978-0-19-851598-2
.
As of early 2010, two of the scheduled three volumes of this overwhelming compendium were available.
- Caramello, Olivia (2017).
Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic `bridges
. Oxford University Press.
doi
:
10.1093/oso/9780198758914.001.0001
.
ISBN
9780198758914
.
- Books that target special applications of topos theory