Holmstrom Pointed simplicial set

See also Simplicial set.

A pointed simplicial set is a simplicial object in the category of pointed sets. For these, we can define wedge and smash product; smash is distributive over wedge.

The free/forget adjoint pair between Ab and Set factors through $Set_*$ because abelian groups are pointed at zero. The relevant functor from $Set_*$ to $Ab$ is $X \mapsto \tilde{\mathbb{Z}} = \mathbb{Z}[X] / \mathbb{Z}[*]$. This functor send wedges to direct sums, and smash products to tensor products, and a “subspace sequence” to an exact sequence. It also takes the n-sphere to the n-th integral Eilenberg-MacLane space.

We have mapping spaces, defined by

$Map_*(X,Y)_q = \{ \textrm{pointed simplicial maps} \ X \wedge \Delta[q]_+ \to Y \}$

Just as for simplicial sets, $Hom_*$ is the 0 simplices of $Map_*$, and we have a “composition” and an exponential adjunction with the smash product.

The loop space of a pointed simplicial set is defined by taking geometric realization, applying the topological loop space functor, and then applying $sing$.

The n-th (reduced, integral) cohomology group of a simplicial set $X$ is defined as $\pi_0 Map_*(X, \tilde{\mathbb{Z}}[S^n])$

Quote: “A simplicial abelian group is fibrant, so need not apply $sing \circ | \cdot |$”

nLab page on Pointed simplicial set