Introduction 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 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 X is a pair (PX,\ni_X) with {\ni_X}\subseteq PX\times X, which classifies relations, in the following sense. First note that for every object I, a morphism r\colon I\to PX ("a family of subsets") induces a subobject \{(i,x)~|~x\in r(i)\}\subseteq I\times X. Formally, this is defined by pulling back \ni_X along r\times X:I\times X\to PX\times X. The universal property of a power object is that every relation arises in this way, giving a bijective correspondence between relations R\subseteq I \times X and morphisms r\colon I\to PX. From finite limits and power objects one can derive that • All
colimits taken over finite index categories exist. • The category has a
subobject classifier. • The category is
Cartesian closed. 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 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.
Explanation 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. 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 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 SetC (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
SetC, 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
Cop embeds in
SetC as a full subcategory. In the graph example the embedding represents
Cop as the subcategory of
SetC 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 SetC
, which we can identify with Grph, is not made concrete by either V'
or E'
alone, the functor U
: Grph → Set2 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 PX were the power object of the pointed set X, and 1 denotes the pointed singleton, then there is only one point-preserving function r\colon 1\to PX, but the relations in 1\times X are as numerous as the pointed subsets of X. The
category of abelian groups is also not a topos, for a similar reason: every group homomorphism must map 0 to 0. ==See also==