Simplicial objects, simplicial commutative rings and homotopy groups

Let $\mathrm{\Delta }$ be the simplex category: its objects are the finite nonempty ordered sets $[n]=\{0<1<\cdots <n\}$ for $n\ge 0$, and its morphisms are the order-preserving maps. A simplicial object in a category $\mathcal{C}$ is a functor ${\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}\to \mathcal{C}$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection, Covariant functor, identity functor, composite functor, and contravariant functor). A simplicial set is a simplicial object in sets; a simplicial commutative ring is a simplicial object in commutative rings (Commutative ring), so that each $A_n$ is a commutative ring and the structure maps are ring homomorphisms (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$); a simplicial module over a simplicial ring $A_{\bullet }$ is a simplicial object $M_{\bullet }$ in abelian groups together with a compatible $A_{\bullet }$-action, and the resulting category of simplicial $A_{\bullet }$-modules is abelian.

Writing $d_i\text{\hspace{0.17em}⁣}:M_n\to M_{n-1}$ for the image of the injection $[n-1]\to [n]$ omitting $i$ (a face map) and $s_i\text{\hspace{0.17em}⁣}:M_n\to M_{n+1}$ for the image of the surjection $[n+1]\to [n]$ repeating $i$ (a degeneracy map), these maps satisfy the usual simplicial identities $d_i{d}_j=d_{j-1}{d}_i(i<j),s_i{s}_j=s_{j+1}{s}_i(i\le j),d_i{s}_j=\{\begin{array}{ll}{s}_{j-1}{d}_i,& i<j,\\ \mathrm{i}\mathrm{d},& i=j\text{ or }i=j+1,\\ s_j{d}_{i-1},& i>j+1.\end{array}$ Conversely, such a sequence of face and degeneracy maps determines the functor.

The Moore complex of a simplicial abelian group or module $M_{\bullet }$ is the chain complex concentrated in nonnegative degrees with $M_n$ in degree $n$ and differential ${\mathrm{\partial }}_n={\sum }_{i=0}^n(-1{)}^i{d}_i\text{\hspace{0.17em}⁣}:M_n\to M_{n-1}(n\ge 1),$ with ${\mathrm{\partial }}_0=0$; the $d_i$ are the face maps (Chain complex in an abelian category); the simplicial identities give ${\mathrm{\partial }}_{n-1}{\mathrm{\partial }}_n=0$. Its homology is written ${\pi }_n(M_{\bullet })=H_n(M_{\bullet })(n\ge 0)$ (Homology object of a chain complex). By the Dold-Kan normalization theorem this agrees with the classical homotopy-group definition $H_n(N{M}_{\bullet })$ via the normalized subcomplex, so the convention is canonical; no model-category machinery is introduced here.

For a simplicial commutative ring $A_{\bullet }$, each ${\pi }_n(A_{\bullet })$ is defined as above, and ${\pi }_0(A_{\bullet })=A_0\mathrm{/}(d_0-d_1)(A_1)$ is a commutative ring. The image is an ideal: for $a\in A_0$, multiplying a representative $b\in A_1$ by $s_{0}a$ gives $a(d_{0}b-d_{1}b)$. Every ${\pi }_n(A_{\bullet })$ is a ${\pi }_0(A_{\bullet })$-module. Indeed multiplication by the totally degenerate simplex of a vertex $a\in A_0$ is a chain map on the Moore complex: its faces are the corresponding totally degenerate simplex of $a$ in the preceding degree. If two vertices are the endpoints of $b\in A_1$, multiplication by the simplicial path defined by $b$ gives a homotopy between these chain maps (the alternating prism sum inserts the degeneracies of $b$). Consequently $(d_0-d_1)(b)$ acts as zero on homology, so the action factors through the displayed quotient. Additivity, associativity and the unit descend from levelwise ring multiplication.

A morphism of simplicial commutative rings is a natural transformation $A_{\bullet }\to B_{\bullet }$ (Natural transformation and its components). Such a morphism is a weak equivalence when it induces isomorphisms ${\pi }_n(A_{\bullet })\to {\pi }_n(B_{\bullet })$ for all $n\ge 0$. The category of derived rings is the localization of the category of simplicial commutative rings at the weak equivalences; constructions on derived rings are used below only through statements that are independent of the chosen replacement up to canonical isomorphism.