Examples Nerve (category theory)

1 examples

1.1 spaces classifying spaces
1.2 nerve of open covering
1.3 moduli example

examples

the primordial example classifying space of discrete group g. regard g category 1 object endomorphisms elements of g. k-simplices of n(g) k-tuples of elements of g. face maps act multiplication, , degeneracy maps act insertion of identity element. if g group 2 elements, there 1 nondegenerate k-simplex each nonnegative integer k, corresponding unique k-tuple of elements of g containing no identities. after passing geometric realization, k-tuple can identified unique k-cell in usual cw structure on infinite-dimensional real projective space. latter popular model classifying space of group 2 elements. see (segal 1968) further details , relationship of above milnor s join construction of bg.

most spaces classifying spaces

every reasonable topological space homeomorphic classifying space of small category. here, reasonable means space in question geometric realization of simplicial set. necessary condition; sufficient. indeed, let x geometric realization of simplicial set k. set of simplices in k partially ordered, relation x ≤ y if , if x face of y. may consider partially ordered set category. nerve of category barycentric subdivision of k, , realization homeomorphic x, because x realization of k hypothesis , barycentric subdivision not change homeomorphism type of realization.

the nerve of open covering

if x topological space open cover ui, nerve of cover obtained above definitions replacing cover category obtained regarding cover partially ordered set relation of set inclusion. note realization of nerve not homeomorphic x (or homotopy equivalent).

a moduli example

one can use nerve construction recover mapping spaces, , higher-homotopical information maps. let d category, , let x , y objects of d. 1 interested in computing set of morphisms x → y. can use nerve construction recover set. let c = c(x,y) category objects diagrams

x
⟵
u
⟶
v
⟵
y


{\displaystyle x\longleftarrow u\longrightarrow v\longleftarrow y}

such morphisms u → x , y → v isomorphisms in d. morphisms in c(x, y) diagrams of following shape:

here, indicated maps isomorphisms or identities. nerve of c(x, y) moduli space of maps x → y. in appropriate model category setting, moduli space weak homotopy equivalent simplicial set of morphisms of d x to y.