Nerve Functor
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In
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a discipline within mathematics, the nerve ''N''(''C'') of a
small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
''C'' is a
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 ...
constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, 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 is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, most often
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 ...
.


Motivation

The nerve of a category is often used to construct topological versions of
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
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 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 a discrete group; see below for details.


Construction

Let ''C'' be a small category. 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''. 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. The addition of these 2-simplices does not erase or otherwise disregard morphisms obtained by composition, it merely remembers that this is how they arise. In general, ''N''(''C'')''k'' consists of the ''k''-tuples of composable morphisms :A_0 \to A_1 \to A_2 \to \cdots \to A_ \to A_k 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 :d_i \colon N(C)_k\to N(C)_ are given by composition of morphisms at the ''i''th object (or removing the ''i''th object from the sequence, when ''i'' is 0 or ''k''). This means that ''d''''i'' sends the ''k''-tuple :A_0 \to \cdots \to A_ \to A_i \to A_ \to \cdots \to A_k to the (''k'' − 1)-tuple :A_0 \to \cdots \to A_ \to A_ \to \cdots \to A_k. That is, the map ''d''''i'' composes the morphisms ''A''''i''−1 → ''A''''i'' and ''A''''i'' → ''A''''i''+1 into the morphism ''A''''i''−1 → ''A''''i''+1, yielding a (''k'' − 1)-tuple for every ''k''-tuple. Similarly, the degeneracy maps :s_i:N(C)_k\to N(C)_ are given by inserting an identity morphism at the object ''A''''i''. Simplicial sets may also be regarded as
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s 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 :N(C)(\_) = \mathrm(i(\_),C). \, This description of the nerve makes functoriality 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 In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
. It follows that
adjoint functor In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
s induce
homotopy equivalence In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
s. In particular, if ''C'' has an
initial In a written or published work, an initial is a letter at the beginning of a word, a chapter (books), chapter, or a paragraph that is larger than the rest of the text. The word is ultimately derived from the Latin ''initiālis'', which means '' ...
or
final object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
, 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, this ''k''-tuple can be identified with the unique ''k''-cell in the usual CW structure on infinite-dimensional
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properti ...
. 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

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 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''. We may consider this partially ordered set as a category with the relations as morphisms. The nerve of this category is the
barycentric subdivision In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is a canonical method to refining them. Therefore, the barycentric subdivision is an important tool ...
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 ''U''''i'', the nerve of the cover is obtained from the above definitions by replacing the cover with the category obtained by regarding the cover as a partially ordered set with set inclusions as relations (and hence morphisms). Note that the realization of this nerve is not generally homeomorphic to ''X'' (or even homotopy equivalent): homotopy equivalence will usually hold only for a
good cover In mathematics, an open cover of a topological space X is a family of open subsets such that X is the union of all of the open sets. A good cover is an open cover in which all sets and all non-empty intersections of finitely-many sets are contracti ...
by contractible sets having contractible intersections.


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 diagrams :X \longleftarrow U \longrightarrow V \longleftarrow Y 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 In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of maps ''X'' → ''Y''. In the appropriate
model category A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...
setting, this moduli space is weak homotopy equivalent to the simplicial set of morphisms of ''D'' from ''X'' to ''Y''.


Nerve theorem

The next theorem is due to Grothendieck. See also:
Segal space In mathematics, a Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered as a simplicial discrete space, satisfies the Segal cond ...
,
Homotopy category of an ∞-category In mathematics, especially category theory, the homotopy category of an ∞-category ''C'' is the category where the objects are those in ''C'' but the hom-set from ''x'' to ''y'' is the quotient of the set of morphisms from ''x'' to ''y'' in ''C'' ...


Homotopy coherent nerve

Like a nerve functor, but more generally, the homotopy coherent nerve functor associates to a
simplicially enriched category In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies t ...
a simpllicial set; i.e.,Lurie, https://kerodon.net/tag/00KM :N^ : \textbf \to \textbf. For example, the
Duskin nerve In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation between functors. ...
of a strict 2-category is a special case of this. See also https://ncatlab.org/nlab/show/homotopy+coherent+nerve for more details.


References

* Blanc, D., W. G. Dwyer, and P.G. Goerss. "The realization space of a \Pi-algebra: a moduli problem in algebraic topology." Topology 43 (2004), no. 4, 857–892. * Goerss, P. G., and M. J. Hopkins.
Moduli spaces of commutative ring spectra
" ''Structured ring spectra'', 151–200, London Math. Soc. Lecture Note Ser., 315, Cambridge Univ. Press, Cambridge, 2004. * Segal, Graeme. "Classifying spaces and spectral sequences." Inst. Hautes Études Sci. Publ. Math. No. 34 (1968) 105–112. * {{nlab, id=nerve, title=Nerve Category theory Simplicial sets