Algebraic Topology: Lecture 11

We introduced the category of simplicial sets. This is the category of diagrams of sets indexed by the opposite of following category &Delta. The objects of &Delta are the finite ordered sets [n] = {0,1,...,n}, where n is a non-negative integer, and the morphisms of &Delta are the increasing maps u : [n] &rarr [m]. A map is increasing if i =< j implies f(i) =< f(j). More generally, a simplicial object in a category C is a functor

    X[-] : &Delta^op &rarr C,

and a cosimplicial object in a category C is a functor

    X[-] : &Delta &rarr C.

Let &Delta[n] be the topological standard n-simplex defined as the convex hull of the standard basis in Rⁿ⁺¹. A map u : [m] &rarr [n] induces a map from the standard basis in R^m+1 to the standard basis in Rⁿ⁺¹, and we define u_* : &Delta[m] &rarr &Delta[n] to be the induced affine map. With these definition, we get a cosimplicial space &Delta[-] : &Delta &rarr T. Now, if Y is a topological space, we define the singular set of Y to be the simplicial set Sin Y[-] : &Delta^op &rarr Sets defined by

    Sin Y[-] = Hom_T(&Delta[-],Y).

Conversely, we define the geometric realization of a simplicial set X[-] to be the topological space |[k] |-> X[k]| defined to be the quotient space of the disjoint union of the product spaces X[n] x &Delta[n] (here X[n] is given by discrete topology) by the equivalence relation generated by the relation that for every map u : [n] &rarr [m] in &Delta, every x∈X[m], and every z∈&Delta[n] identifies

    (u^*x,z)∈X[n] x &Delta[n]

and

    (x,u_*z)∈X[m] x &Delta[m].

We say that x∈X[n] is degenerate if x is in the image of u^* : X[m] &rarr X[n], for some m < n, and that x∈X[n] is non-degenerate if x is not degenerate. One can show that, if x∈X[n] is non-degenerate, then the map

    &iota_x : &Delta[n] &rarr |[k] |-> X[k]|

that maps z to the class of (x,z)∈X[n] x &Delta[n] is an inclusion on the interior &Delta[n]\&part&Delta[n]. So a non-degenerate x∈X[n] gives a "cell" in |[k] |-> X[k]|. Conversely, for every point &xi∈|[k] |-> X[k]|, there exists a unique non-degenerate x∈X[n] and a unique z∈&Delta[n]\&part&Delta[n] such that &iota_x(z)=&xi .

The simplicial standard n-simplex is the simplicial set &Delta[n][-] defined by

    &Delta[n][m] = Hom_&Delta([m],[n]).

The boundary &part&Delta[n][-] of &Delta[n][-] is the simplicial subset that consists of the maps u : [m] &rarr [n] that are not surjective. And for 0 =< k =< n, the kth horn &Lambda_k[n][-] is the simplicial subset of &Delta[n][-] that consists of all maps u : [m] &rarr [n] such that the image of u does not fully contain the subset {0,1,...,k-1,k+1,...,n} of [n].

Theorem The category of simplicial sets has a cofibrantly generated model structure, where the weak equivalences are the maps f of simplicial sets such that the induced map |f| of geometric realizations is a weak equivalence of topological spaces, and where

    I = { &part&Delta[n][-] &rarr &Delta[n] | n non-negative integer}

    J = { &Lambda_k[n][-] &rarr &Delta[n] | n non-negative integer, 0 =< k =< n}

are sets of generating cofibrations and generating trivial cofibrations.