A basic universal map from exponent \(m\) to \(n\), is an \(n \times m\)-matrix with coefficients in \(\mathbb Z\).

A universal map from exponent \(m\) to \(n\), is a formal \(\mathbb Z\)-linear combination of basic universal maps from exponent \(m\) to \(n\).

If \(f\) is a basic universal map, then we write \([f]\) for the corresponding universal map.

Let \(f = \sum _g n_g[g]\) be a universal map. We say that \(f\) is bound by a natural number \(N\) if \(\sum _g |n_g| \le N\).

In other words, we are considering the following two categories:

• the category whose objects are natural numbers, and whose morphisms are matrices;

• the category with the same objects, but with Hom-sets replaced by the free abelian groups generated by the sets of matrices. We denote this latter category \(\textnormal{FreeMat}\).

Let \(N\) be a natural number, and \(i {\lt} N\). Then \(\pi '_{N,i}\) denotes the basic universal map from exponent \(N\) to \(1\)

where \(a_j = \delta _{ij}\).

Let \(N\) and \(n\) be natural numbers. Then \(\pi ^N_n\) denotes the universal map from exponent \(N \cdot n\) to \(n\) given by \(\sum _{i {\lt} N} [\pi '_{N,i} \otimes n]\).

(On \(\mathbb {Z}[A^{N \cdot n}] \to \mathbb {Z}[A^n]\) this map is the formal sum of the maps \(\mathbb {Z}[A^{N \cdot n}] \to \mathbb {Z}[A^n]\) induced by the projection maps \(A^{N \cdot n} = (A^n)^N \to A^n\).)

Let \(N\) and \(n\) be natural numbers. Then \(\sigma ^N_n\) denotes the universal map from exponent \(N \cdot n\) to \(n\) given by \([\sum _{i {\lt} N} \pi '_{N,i} \otimes n]\).

(On \(\mathbb {Z}[A^{N \cdot n}] \to \mathbb {Z}[A^n]\) this map is induced by the summation map \(A^{N \cdot n} = (A^n)^N \to A^n\).)

A Breen–Deligne data is a chain complex in \(\textnormal{FreeMat}\).

Concretely, this means that it consists of a sequence of exponents \(n_0, n_1, n_2, \dots \in \mathbb N\), and universal maps \(f_i\) from exponent \(n_{i+1}\) to \(n_i\), such that for all \(i\) we have \(f_i \circ f_{i+1} = 0\).

A morphism of Breen–Deligne data is a morphism of chain complexes.

For every natural numbers \(N\), the endofunctor \(N \otimes \_ \) on \(\textnormal{FreeMat}\) induces an endofunctor of Breen–Deligne data.

Concretely, it maps a pair \((n, f)\) of Breen–Deligne data, to the pair \(N \otimes (n,f)\) consisting of exponents \(N \cdot n_i\) and universal maps \(N \otimes f_i\).

A Breen–Deligne package consists of Breen–Deligne data \(\mathsf{BD}\) together with a homotopy \(h\) between \(\pi ^2_\mathsf{BD}\) and \(\sigma ^2_\mathsf{BD}\).

Let \(\mathsf{BD}\) be a Breen–Deligne package and \(N\) a power of \(2\). Then the homotopy \(h\) induces a homotopy between \(\pi ^N_\mathsf{BD}\) and \(\sigma ^N_\mathsf{BD}\) by iterative composition of the homotopy packaged in \(\mathsf{BD}\).

We will now construct an example of a Breen–Deligne package. In some sense, it is the “easiest” solution to the conditions posed above. The exponents will be \(n_i = 2^i\), and the homotopies \(h_i\) will be the identity. Under these constraints, we recursively construct the universal maps \(f_i\):

We leave it as exercise for the reader, to verify that with these definitions \((n, f, h)\) forms a Breen–Deligne package.

Let \(f\) be a basic universal map from exponent \(m\) to \(n\). Let \(c_1, c_2 \in \mathbb R_{\ge 0}\). We say that \((c_1, c_2)\) is \(f\)-suitable, if for all \(i\)

Let \(f\) be a universal map from exponent \(m\) to \(n\). Let \(c_1, c_2 \in \mathbb R_{\ge 0}\). We say that \((c_1, c_2)\) is \(f\)-suitable, if for all basic universal maps \(g\) that occur in the formal sum \(f\), the pair of nonnegative reals \((c_1, c_2)\) is \(g\)-suitable.

Let \(f\) be a universal map and let \(r, r', c_1, c_2 \in \mathbb R_{\ge 0}\). We say that \((c_1, c_2)\) is very suitable for \((f, r, r')\) if there exist \(N, b \in \mathbb {N}\) and \(c' \in \mathbb {R}_{\ge 0}\) such that:

• \(f\) is bound by \(N\) (see Definition 1.1.3)

• \((c_1, c')\) is \(f\)-suitable

• \(r ^b N ≤ 1\)

• \(c' ≤ (r') ^b c_2\)

Let \(\mathsf{BD}= (n, f)\) be Breen–Deligne data, let \(r, r' \in \mathbb {R}_{\ge 0}\), and let \(\kappa = (\kappa _0, \kappa _1, \dots )\) be a sequence of nonnegative real numbers. We say that \(\kappa \) is \(\mathsf{BD}\)-suitable (resp. very suitable for \((\mathsf{BD}, r, r')\)), if for all \(i\), the pair \((\kappa _{i+1}, \kappa _i)\) is \(f_i\)-suitable (resp. very suitable for \((f_i, r, r')\)).

(Note! The order \((\kappa _{i+1}, \kappa _i)\) is contravariant compared to Definition 1.1.14. This is because of the contravariance of \(\hat V(\_ )\); see Definition 1.5.9.)

Let \(\mathsf{BD}\) be a Breen–Deligne package with data \((n,f)\) and homotopy \(h\). Let \(\kappa , \kappa '\) be sequences of nonnegative real numbers. (In applications \(\kappa \) is a \((n,f)\)-suitable sequence.)

Then \(\kappa '\) is adept to \((\mathsf{BD}, \kappa )\) if for all \(i\) the pair \((\kappa _i / 2, \kappa '_{i+1} \kappa _{i+1})\) is \(h_i\)-suitable. (Recall that \(h_i\) is the homotopy map \(n_i \to n_{i+1}\).)