## 1.1 Breen–Deligne data

The goal of this subsection is to a give a precise statement of a variant of the Breen–Deligne resolution. This variant is not actually a resolution, but it is sufficient for our purposes, and is much easier to state and prove.

We first recall the original statement of the Breen–Deligne resolution.

For an abelian group \(A\), there is a resolution, functorial in \(A\), of the form

What does a homomorphism \(f \colon \mathbb Z[A^m] \to \mathbb Z[A^n]\) that is functorial in \(A\) look like? We should perhaps say more precisely what we mean by this. The idea is that \(m\) and \(n\) are fixed, and for each abelian group \(A\) we have a group homomorphism \(f_A\colon \mathbb Z[A^m] \to \mathbb Z[A^n]\) such that if \(\phi \colon A\to B\) is a group homomorphism inducing \(\phi _i \colon \mathbb {Z}[A^i]\to \mathbb {Z}[B^i]\) for each natural number \(i\) then the obvious square commutes: \(\phi _n \circ f_A = f_B \circ \phi _m\).

The map \(f_A\) is specified by what it does to the generators \((a_1, a_2, a_3, \dots , a_m)\in A^m\). It can send such an element to an arbitrary element of \(\mathbb Z[A^n]\), but one can check that universality implies that \(f_A\) will be a \(\mathbb Z\)-linear combination of “basic universal maps”, where a “basic universal map” is one that sends \((a_1, a_2, \dots , a_m)\) to \((t_1, \dots , t_n)\), where \(t_i\) is a \(\mathbb Z\)-linear combination \(c_{i,1} \cdot a_1 + \dots + c_{i,m} \cdot a_m\). So a “basic universal map” is specified by the \(n \times m\)-matrix \(c\).

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 \(\sum _g |n_g| \le N\).

We point out that basic universal maps can be composed by matrix multiplication, and this formally induces a composition of universal maps. As mentioned above, one can also check (this has been formalised in Lean) that this construction gives a bijection between universal maps from exponent \(m\) to \(n\) and functorial collections \(f_A \colon \mathbb {Z}[A^m]\to \mathbb {Z}[A^n]\).

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 \(\textnormal{FreeMat}\).

Both categories naturally come with a monoidal structure: for the first it is given by the Kronecker product of matrices (a.k.a. tensor product of linear maps) which induces a monoidal structure on \(\textnormal{FreeMat}\). As usual, we will denote this monoidal structure \(\_ \otimes \_ \). For example, if \(f\) is a basic universal map, then \(2 \otimes f\) denotes the block matrix

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 \(N \otimes f_i\).

Let \(\mathsf{BD}\) be Breen–Deligne data. The universal maps \(\sigma ^N\) and \(\pi ^N\) defined above, induce morphisms \(\sigma ^N_\mathsf{BD},\pi ^N_\mathsf{BD}\colon N \otimes \mathsf{BD}\to \mathsf{BD}\).

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.

We now make definitions that will make precise some conditions between constants that will be needed when we construct Breen–Deligne complexes of normed abelian groups.

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 \(i\)

To orient the reader: later on we will be considering maps on normed abelian groups induced from universal maps, and this inequality will guarantee that if \(\| m\| \leq c_1\) then \(\| f(m)\| \leq c_2\).

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 \(n_i \to n_{i+1}\).)

Let \(\mathsf{BD}\) be a Breen–Deligne package, \(N\) a power of \(2\), and let \(\kappa , \kappa '\) be sequences of nonnegative real numbers. Assume that \(\kappa '\) is adept to \((\mathsf{BD}, \kappa )\). Let \(h^N\) be the homotopy between \(\pi ^N_\mathsf{BD}\) and \(\sigma ^N_\mathsf{BD}\) defined in Def 1.1.11.

For all \(i\), the pair \((\kappa _i / N, \kappa '_{i+1} \kappa _{i+1})\) is \(h^N_i\)-suitable.

Omitted. (But done in Lean.)

Let \(\mathsf{BD}\) be a Breen–Deligne package, and let \(r, r'\) be nonnegative reals, such that \(r {\lt} 1\) and \(r' {\gt} 0\).

There exists a sequence \(\kappa \) of positive real numbers such that \(\kappa \) is very suitable for \((\mathsf{BD}, r, r')\).

The sequence can be constructed recursively, which we leave as exercise for the reader. (It has been done in Lean.)

Let \(\mathsf{BD}\) be a Breen–Deligne package, and let \(r, r'\) be nonnegative reals, such that \(0 {\lt} r {\lt} 1\) and \(0 {\lt} r' \le 1\). Let \(\kappa \) be any sequence of positive reals.

There exists a sequence \(\kappa '\) of nonnegative real numbers that is adept to \((\mathsf{BD}, \kappa )\).

The sequence can be constructed recursively, which we leave as exercise for the reader. (It has been done in Lean.)