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Nerve (category theory)
Encyclopedia
In category theory
, the nerve N(C) of a small category C is a simplicial set
constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space
, called the classifying space of the category C. These closely related objects can provide information about some familiar and useful categories using algebraic topology
, most often homotopy theory.
s. If X is an object of C, its moduli space should somehow encode all objects isomorphic to X and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the answers to such questions provide interesting information about the original category C, or about related categories.
The notion of nerve is a direct generalization of the classical notion of classifying space
of a discrete group; see below for details.
The diagram suggests our course of action: add a 2-simplex for this commutative triangle. Every 2-simplex of N(C) comes from a pair of composable morphisms in this way. Note that the addition of these 2-simplices does not erase or otherwise disregard morphisms obtained by composition, it merely remembers that that is how they arise.
In general, N(C)k consists of the k-tuples of composable morphisms
of C. To complete the definition of N(C) as a simplicial set, we must also specify the face and degeneracy maps. These are also provided to us by the structure of C as a category. The face maps
are given by composition of morphisms at the ith object (or dropping an object on the end, when i is 0 or k). This means that di sends the k-tuple
to the (k-1)-tuple
.
That is, the map di composes the morphisms Ai-1 → Ai and Ai → Ai+1 into the morphism Ai-1 → Ai+1, yielding a (k−1)-tuple for every k-tuple.
Similarly, the degeneracy maps
are given by inserting an identity morphism at the object Ai.
Recall that simplicial sets may also be regarded as functors Δop → Set, where Δ is the category of totally ordered finite sets and order-preserving morphisms. Every partially ordered set P yields a (small) category i(P) with objects the elements of P and with a unique morphism from p to q whenever p ≤ q in P. We thus obtain a functor i from the category Δ to the category of small categories. We can now describe the nerve of the category C as the functor Δop → Set
.
This description of the nerve makes functoriality quite transparent; for example, a functor between small categories C and D induces a map of simplicial sets N(C) → N(D). Moreover a natural transformation between two such functors induces a homotopy between the induced maps. This observation can be regarded as the beginning of one of the principles of higher category theory
. It follows that adjoint functors induce homotopy equivalences. In particular, if C has an initial
or final object, its nerve is contractible.
structure on infinite-dimensional real projective space
. The latter is the most popular model for the classifying space of the group with two elements. See (Segal 1968) for further details and the relationship of the above to Milnor's join construction of BG.
of K, and thus its realization is homeomorphic to X, because X is the realization of K by hypothesis and barycentric subdivision does not change the homeomorphism type of the realization.
is obtained from the above definitions by replacing the cover with the category obtained by regarding the cover as a partially ordered set with relation that of set inclusion. Note that the realization of this nerve is not generally homeomorphic to X (or even homotopy equivalent).
such that the morphisms U → X and Y → V are isomorphisms in D. Morphisms in C(X,Y) are diagrams of the following shape:
Here, the indicated maps are to be isomorphisms or identities. The nerve of C(X,Y) is the moduli space
of maps X → Y. In the appropriate model category
setting, this moduli space is weak homotopy equivalent to the simplicial set of morphisms of D from X to Y.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, the nerve N(C) of a small category C is a simplicial set
Simplicial set
In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, called the classifying space of the category C. These closely related objects can provide information about some familiar and useful categories using algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, most often homotopy theory.
Motivation
The nerve of a category is often used to construct topological versions of moduli spaceModuli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s. If X is an object of C, its moduli space should somehow encode all objects isomorphic to X and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the answers to such questions provide interesting information about the original category C, or about related categories.
The notion of nerve is a direct generalization of the classical notion of classifying space
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 by a free action of G...
of a discrete group; see below for details.
Construction
Let C be a small category. It is easy to define the sets N(C)k for small k, which leads to the general definition. In particular, there is a 0-simplex of N(C) for each object of C. There is a 1-simplex for each morphism f: x → y in C. Now suppose that f: x → y and g: y → z are morphisms in C. Then we also have their composition gf: x → z.In general, N(C)k consists of the k-tuples of composable morphisms
of C. To complete the definition of N(C) as a simplicial set, we must also specify the face and degeneracy maps. These are also provided to us by the structure of C as a category. The face maps
are given by composition of morphisms at the ith object (or dropping an object on the end, when i is 0 or k). This means that di sends the k-tuple
to the (k-1)-tuple
.That is, the map di composes the morphisms Ai-1 → Ai and Ai → Ai+1 into the morphism Ai-1 → Ai+1, yielding a (k−1)-tuple for every k-tuple.
Similarly, the degeneracy maps
are given by inserting an identity morphism at the object Ai.
Recall that simplicial sets may also be regarded as functors Δop → Set, where Δ is the category of totally ordered finite sets and order-preserving morphisms. Every partially ordered set P yields a (small) category i(P) with objects the elements of P and with a unique morphism from p to q whenever p ≤ q in P. We thus obtain a functor i from the category Δ to the category of small categories. We can now describe the nerve of the category C as the functor Δop → Set
.This description of the nerve makes functoriality quite transparent; for example, a functor between small categories C and D induces a map of simplicial sets N(C) → N(D). Moreover a natural transformation between two such functors induces a homotopy between the induced maps. This observation can be regarded as the beginning of one of the principles of higher category theory
Higher category theory
Higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.- Strict higher categories :...
. It follows that adjoint functors induce homotopy equivalences. In particular, if C has an initial
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
or final object, its nerve is contractible.
Examples
The primordial example is the classifying space of a discrete group G. We regard G as a category with one object whose endomorphisms are the elements of G. Then the k-simplices of N(G) are just k-tuples of elements of G. The face maps act by multiplication, and the degeneracy maps act by insertion of the identity element. If G is the group with two elements, then there is exactly one nondegenerate k-simplex for each nonnegative integer k, corresponding to the unique k-tuple of elements of G containing no identities. After passing to the geometric realization, it is not hard to see that this k-tuple can be identified with the unique k-cell in the usual CWCW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
structure on infinite-dimensional real projective space
Real projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
. The latter is the most popular model for the classifying space of the group with two elements. See (Segal 1968) for further details and the relationship of the above to Milnor's join construction of BG.
Most spaces are classifying spaces
It is well known that every "reasonable" topological space is homeomorphic to the classifying space of a small category. Here, "reasonable" means that the space in question is the geometric realization of a simplicial set. This is obviously a necessary condition; it is perhaps surprising that it is also sufficient. Indeed, let X be the geometric realization of a simplicial set K. The set of simplices in K is partially ordered, by the relation x ≤ y if and only if x is a face of y. Of course, we may consider this partially ordered set as a category. The nerve of this category is the barycentric subdivisionBarycentric subdivision
In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.The name...
of K, and thus its realization is homeomorphic to X, because X is the realization of K by hypothesis and barycentric subdivision does not change the homeomorphism type of the realization.
The nerve of an open covering
If X is a topological space with open cover Ui, the nerve of the coverNerve of an open covering
In mathematics, the nerve of an open covering is a construction in topology, of an abstract simplicial complex from an open covering of a topological space X.The notion of nerve was introduced by Pavel Alexandrov....
is obtained from the above definitions by replacing the cover with the category obtained by regarding the cover as a partially ordered set with relation that of set inclusion. Note that the realization of this nerve is not generally homeomorphic to X (or even homotopy equivalent).
A moduli example
One can use the nerve construction to recover mapping spaces, and even get "higher-homotopical" information about maps. Let D be a category, and let X and Y be objects of D. One is often interested in computing the set of morphisms X → Y. We can use a nerve construction to recover this set. Let C = C(X,Y) be the category whose objects are diagramssuch that the morphisms U → X and Y → V are isomorphisms in D. Morphisms in C(X,Y) are diagrams of the following shape:
Here, the indicated maps are to be isomorphisms or identities. The nerve of C(X,Y) is the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of maps X → Y. In the appropriate model category
Model category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
setting, this moduli space is weak homotopy equivalent to the simplicial set of morphisms of D from X to Y.