# On a Completion of Cohomological Functors Generalising Tate Cohomology I

Max Gheorghiu

(7 June 2024)

###### Abstract

Viewing group cohomology as a so-called cohomological functor, G. Mislin has
generalised Tate cohomology from finite groups to all discrete groups by
defining a completion for cohomological functors in [28]. If
$T^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$ is any cohomological functor,
then we construct its Mislin completion
$\widehat{T}^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$ under the assumption
that the abelian category $\mathcal{C}$ has enough projectives and that in the
abelian category $\mathcal{D}$ all countable direct limits exist and are
exact. This takes Tate cohomology to settings where it has never been
introduced such as in condensed mathematics. Through the latter, one can
define Tate cohomology for any $T1$ topological group. More specifically, we
generalise four constructions of Mislin completions from the literature, prove
that they yield isomorphic cohomological functors and provide explicit
formulae for their connecting homomorphisms. For any morphism $f:M\rightarrow
N$ in $\mathcal{C}$ we develop formulae for
$\widehat{T}^{n}(f):\widehat{T}^{n}(M)\rightarrow\widehat{T}^{n}(N)$ in terms
of each construction.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation and terminology
  2. 2 Outline of constructions
  3. 3 The satellite functor construction and Mislin completions
  4. 4 The resolution and the satellite functor construction
    1. 4.1 Isomorphism of cohomology groups
    2. 4.2 Induced morphisms and connecting homomorphisms
    3. 4.3 Isomorphisms of cohomological functors
    4. 4.4 Countably many distinct constructions of Mislin completions
  5. 5 The naïve and the resolution construction
    1. 5.1 Performing the naïve construction
    2. 5.2 Isomorphism of cohomology groups
    3. 5.3 Connecting homomorphisms of the naïve construction
  6. 6 The hypercohomology and the resolution construction
    1. 6.1 Defining Ext-functors through chain maps
    2. 6.2 Connecting homomorphisms through chain maps
    3. 6.3 Isomorphism of cohomology groups
    4. 6.4 Isomorphism of cohomological functors
  7. 7 Complete cohomology in condensed mathematics
  8. Acknowledgements

##  1 Introduction

Because Tate cohomology and its generalisations are the main motivation of
this work, we first outline Tate cohomology for a finite group $G$ based on
[1, p. 78–79]. For any module $M$ over the group ring $\mathbb{Z}[G]$ and any
integer $n\in\mathbb{Z}$ there is a Tate cohomology group
$\widehat{H}^{n}(G,M)$. Note that $\widehat{H}^{n}(G,M)=H^{n}(G,M)$ for $n\geq
1$ and $\widehat{H}^{n}(G,M)=H_{-n-1}(G,M)$ for $n\leq-2$, whence Tate
cohomology unites both group cohomology and group homology. According to [6,
p. 136], Tate cohomology of $G$ satisfies dimension shifting and possesses cup
products. The former means that for any $M$ there are $\mathbb{Z}[G]$-modules
$M_{\ast}$, $M^{\ast}$ such that
$\widehat{H}^{n-1}(G,M_{\ast})\cong\widehat{H}^{n}(G,M)$ and
$\widehat{H}^{n+1}(G,M^{\ast})\cong\widehat{H}^{n}(G,M)$ for every
$n\in\mathbb{Z}$. Thus, if one desires to establish a property for every
degree $\widehat{H}^{n}(G,-)$ of $\widehat{H}^{\bullet}(G,-)$, then one only
needs to establish it for a carefully chosen degree $n$. If $N$ is a
$\mathbb{Z}[G]$-module and $m\in\mathbb{Z}$ another integer, then a cup
product is a homomorphism of the form

$\smile:\widehat{H}^{n}(G,M)\otimes\widehat{H}^{m}(G,N)\rightarrow\widehat{H}^{m+n}(G,M\otimes
N)$

where the tensor product $M\otimes N$ is a $\mathbb{Z}[G]$-module endowed with
the diagonal action of $G$. Importantly for us, Tate cohomology for $G$
vanishes on projectives according to [1, p. 79], meaning that
$\widehat{H}^{n}(G,P)=0$ for any projective $\mathbb{Z}[G]$-module $P$ and
$n\in\mathbb{Z}$. Then complete cohomology generalises Tate cohomology for
finite groups and was introduced for all discrete groups by G. Mislin, P.
Vogel, D. J. Benson and J. F. Carlson using different constructions all
yielding isomorphic cohomology groups [25, p. 196–197].

On the one hand we generalise these constructions to a previously unknown
extent. On the the other hand, we develop for each construction explicit
formulae for connecting homomorphisms and induced morphisms that have not been
covered in the literature. In particular, since our generalisation takes Tate
cohomology to condensed mathematics, one can construct Tate cohomology for any
topological group in which every point is closed, meaning for any $T1$
topological group. More specifically, condensed mathematics is a novel theory
developed by D. Clausen and P. Scholze in 2018 [35]. In [36], P. Scholze
writes that he wants “to make the strong claim that in the foundations of
mathematics, one should replace topological spaces with condensed sets”.
Independently of whether this claim turns out to be true, it is a very
promising tool for the study of continuous cohomology of topological groups
and the study of representations of topological groups to topological abelian
groups [35, p. 6]. It also serves as a foundation for a very general form of
analytic geometry [9, p. 6]. More fundamentally, it provides a unified
approach for studying topological groups, rings and modules [35, p. 6]. In the
subsequent paper [16], we show that complete cohomology detects whether a
group has finite cohomological dimension, establish a version of dimension
shifting, an Eckmann-Shapiro result as well as cohomology products such as
external products, cup products and Yoneda products.

Let us provide more detail on the nature of our constructions. As in the case
of G. Mislin’s work [28], the notion of a cohomological functor is essential.
More specifically, if $\mathcal{C}$, $\mathcal{D}$ denote two abelian
categories, then a family of additive functors
$(T^{n}:\mathcal{C}\rightarrow\mathcal{D})_{n\in\mathbb{Z}}$ is a
cohomological functor if there are so-called connecting homomorphisms
$(\delta^{n}:T^{n}\rightarrow T^{n+1})_{n\in\mathbb{Z}}$ satisfying two
(natural) axioms [25, p. 201–202]. In particular, if $G$ is a discrete group,
$R$ a discrete ring and $A$ a discrete $R$-module, then setting
$H_{R}^{n}(G,-)=0$ and $\mathrm{Ext}_{R}^{n}(A,-)=0$ for $n<0$ renders group
cohomology and Ext-functors into cohomological functors [25, p. 201], [28, p.
295]. We refer the reader unfamiliar with abelian categories to the account in
[27, p. 249–257], [38, Tag 00ZX] or [42, Section A.4]. G. Mislin generalised
Tate cohomology to all discrete groups by defining complete cohomology
$\widehat{H}_{R}^{\bullet}(G,-)$ as a specific completion of ordinary group
cohomology $H_{R}^{\bullet}(G,-)$ in his paper [28]. Following the convention
from [25, p. 197/202], we call such a completion of a cohomological functor a
Mislin completion. Formally, a Mislin completion of
$T^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$ is a cohomological functor
$\widehat{T}^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$ together with a
morphism $\Phi^{\bullet}:T^{\bullet}\rightarrow\widehat{T}^{\bullet}$ such
that $\widehat{T}^{n}(P)=0$ for any projective $P\in\mathrm{obj}(\mathcal{C})$
and $n\in\mathbb{Z}$ and such that any morphism $T^{\bullet}\rightarrow
V^{\bullet}$ to a cohomological functor
$V^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$ also vanishing on projectives
factors uniquely through $\Phi^{\bullet}$ [28, Definition 2.1]. By its
universal property, any Mislin completion is unique up to isomorphism [25, p.
202].

We generalise two constructions of Mislin completions due to G. Mislin [28]
under the assumption that the domain category $\mathcal{C}$ has enough
projective objects and that in the codomain category $\mathcal{D}$ all
countable direct limits exist and are exact (Lemma 3.2 and Theorem 4.5).
Direct limits are a specific kind of colimits [34, p. 14] and are vital to
these constructions. In doing so, we demonstrate that there are countably many
distinct constructions of Mislin completions (Lemma 4.10). If $\mathbf{Ab}$
denotes the category of abelian groups, then we generalise P. Vogel’s [17] as
well as D. J. Benson and J. F. Carlson’s construction [5] mentioned above so
that they give rise to the Mislin completion of (unenriched) Ext-functors
$\mathrm{Ext}_{\mathcal{C}}^{\bullet}(A,-):\mathcal{C}\rightarrow\mathbf{Ab}$
that we call completed (unenriched) Ext-functors (Theorem 5.10 and Corollary
6.15). Especially in the context of condensed mathematics, we also consider
Hom-functors of the form
$\mathrm{Hom}_{\mathcal{C}}:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathcal{D}$
called enriched Hom-functors where the category $\mathcal{D}$ does not need to
be $\mathbf{Ab}$ and the corresponding enriched Ext-functors
$\mathrm{Ext}_{\mathcal{C}}^{\bullet}(A,-):\mathcal{C}\rightarrow\mathcal{D}$
together with their completed enriched Ext-functors
$\widehat{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(A,-)$. For any morphism
$f:M\rightarrow N$ in $\mathcal{C}$ we develop formulae for the induced
morphism $\widehat{T}^{n}(f):\widehat{T}^{n}(M)\rightarrow\widehat{T}^{n}(N)$
of the Mislin completion in terms of each construction (Proposition 3.4,
Proposition 4.2, Definition 5.4, Definition 6.2). Using each construction, we
provide explicit formulae for the connecting homomorphisms of
$\widehat{T}^{\bullet}$. (Lemma 3.3, Definition 4.3, Lemma 5.9, Definition
6.5).

Before presenting some examples, we note that that complete cohomology
generalises Tate-Farrell cohomology, which we prove in [16]. More
specifically, F. T. Farrell generalised Tate cohomology of finite groups to
discrete groups having a finite index subgroup of finite cohomological
dimension in his paper [14] by using so-called complete resolutions. A
complete resolution of an object $A$ in $\mathcal{C}$ is a particular acyclic
chain complex $(\overline{A}_{n})_{n\in\mathbb{Z}}$ of projective objects in
$\mathcal{C}$ that agrees with a projective resolution of $M$ in sufficiently
high degree [10, Definition 1.1]. Following [14, p. 158], Tate-Farrell Ext-
functors are defined as the cohomology of the cochain complex
$\mathrm{Hom}_{\mathcal{C}}(\overline{A}_{\bullet},B)$ where we the Hom-
functor is unenriched. Using F. T. Farrell’s work, P. Symonds deduces in
[40, p. 34] that Tate-Farrell cohomology exists for every profinite group
having an open subgroup of finite cohomological dimension. In [39], he
computes the Tate-Farrell cohomology of the Morava stabiliser group $S_{p-1}$
with coefficients in the moduli space $E_{p-1}$ for odd primes $p$.

However, there are groups for which their complete cohomology cannot be
calculated by a complete resolution. For instance, a free abelian group of
countable rank $\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$, $GL_{n}(K)$ for $K$ a
subfield of the algebraic closure of $\mathbb{Q}$, the Thompson group $F$ and
the free product
$\mathop{\scalebox{2.0}{\raisebox{-0.43057pt}{$\ast$}}}_{n\in\mathbb{N}}\mathbb{Z}^{n}$
do not admit complete resolutions [10, Example 5.3], [11, p. 119–120]. By [28,
p. 297–298], the cohomology of the first three examples is isomorphic to their
complete cohomology while all cohomology groups of the last group are distinct
from its complete cohomology groups [23, p. 432]. The zeroeth complete
cohomology group of this last example is calculated in [11, Corollary 2.4 and
Theorem A]. In [12], F. Dembegioti calculates the zeroeth complete cohomology
group of a class of discrete polycyclic groups, but also explains why one
cannot determine a general formula calculating the zeroeth complete cohomology
groups for all polycyclic groups. Lastly, if $p$ is a prime number, then
pro-$p$ groups are a class of profinite groups for which there is a rich
theory of their group cohomology. See for instance [37, Chapter I] or [41].
One example of a pro-$p$ group that is also a pro-$p$ ring are the $p$-adic
integers $\mathbb{Z}_{p}$ which are studied in [43, Section 1.5]. These
examples give rise to the following questions.

######  Question 1.1.

As in [34, Example 3.3.8(c)], let $\prod_{\mathbb{N}}\mathbb{Z}_{p}$ denote
the free abelian pro-$p$ group over $\mathbb{N}$. Is its cohomology
$H_{\mathbb{Z}_{p}}^{\bullet}(\prod_{\mathbb{N}}\mathbb{Z}_{p},-)$ isomorphic
to its complete cohomology
$\widehat{H}_{\mathbb{Z}_{p}}^{\bullet}(\prod_{\mathbb{N}}\mathbb{Z}_{p},-)$
as a cohomological functor?

######  Question 1.2.

In accordance with [33, p. 137–140], denote by
$G=\bigsqcup_{n\in\mathbb{N}\cup\\{\infty\\}}G_{n}$ the following free pro-$p$
product over the one-point compactification of the integers
$\mathbb{N}\cup\\{\infty\\}$. Set $G_{n}=\mathbb{Z}_{p}^{n}$ for
$n\in\mathbb{N}$ and $G_{\infty}=\\{1\\}$. Can one compute the zeroeth
complete cohomology $\widehat{H}_{\mathbb{Z}_{p}}^{0}(G,A)$ for an
$\mathbb{Z}_{p}[[G]]$-module $A$?

Let us elaborate on the implementation of complete cohomology into condensed
mathematics. If $\mathbf{Pro}$ denotes the category of profinite spaces, then,
informally, a condensed set can be thought as a contravariant functor
$X:\mathbf{Pro}\rightarrow\mathbf{Set}$ satisfying certain compatibility
conditions [35, p. 7]. Given any $T1$ topological space $Y$, one can turn it
into a condensed set

$\underline{Y}:\mathbf{Pro}^{\mathrm{op}}\rightarrow\mathbf{Set},S\mapsto\\{f:S\rightarrow
Y\text{ continuous}\\}$

called the condensate of $T$ [35, p. 15/16 and Proposition 1.7]. Condensed
groups (resp. rings, modules etc.) are defined analogously [35, p. 7] and
condensates of $T1$ topological groups are condensed groups (resp. rings,
modules etc) [35, p. 8]. Because we define condensed sets and modules by using
sheaves, they possess excellent category theoretic properties such as the
existence of all small limits and colimits [35, p. 11/15] as well as enough
projectives [9, p. 16]. Moreover, all products, direct sums and direct limits
of condensed modules are exact [35, Theorem 2.2]. J. Anschütz and A.-C. Le
Bras have defined condensed group cohomology over any condensed group ring
$\mathcal{R}[\mathcal{G}]$ in [2, p. 5] since condensed abelian groups form an
abelian category contrary to topological abelian groups [35, p. 6/11]. We note
that for condensed $\mathcal{R}$-modules over a condensed ring $\mathcal{R}$
there is not only an unenriched Hom-functor
$\mathrm{Hom}_{\mathrm{Cond}(\mathcal{R})}(-,-)$ mapping to abelian groups,
but also an internal Hom-functor
$\underline{\mathrm{Hom}}_{\mathrm{Cond}(\mathcal{R})}(-,-)$ mapping to
condensed abelian groups [35, p. 13].

######  Theorem 1.3.

(= Theorem 7.1) Let $\mathcal{R}$ be a condensed ring and $A$ a condensed
$\mathcal{R}$-module. Then there are completed condensed unenriched Ext-
functors
$\widehat{\mathrm{Ext}}_{\mathrm{Cond}(Mod(\mathcal{R}))}^{\bullet}(A,-)$ and
completed condensed internal Ext-functors
$\widehat{\underline{\mathrm{Ext}}}_{\mathrm{Cond}(Mod(\mathcal{R}))}^{\bullet}(A,-)$.
If $\mathcal{G}$ is a condensed group, then complete condensed unenriched
group cohomology $\widehat{H}_{\mathcal{R}}^{\bullet}(\mathcal{G},-)$ and
complete condensed internal group cohomology
$\widehat{\underline{H}}_{\mathcal{R}}^{\bullet}(\mathcal{G},-)$ can be
defined.

######  Corollary 1.4.

One can define group cohomology and thus complete cohomology for any $T1$
topological group $G$ by taking its condensate $\underline{G}$.

Specifically for profinite groups, it is more natural to consider so-called
solid modules. In the same manner as a completed group ring of a profinite
group over a profinite ring can be seen as a completion of an ordinary group
ring [43, Proposition 7.1.2], one can introduce a notion of completion for
condensed modules over a condensed ring $\mathcal{R}$. We call such completed
condensed modules solid [35, p. 32/44] and $\mathcal{R}$ analytic if it admits
a choice of solid modules [35, Definition 7.4].

######  Lemma 1.5.

(= Lemma 7.3) Let $\mathcal{A}$ be an analytic ring and $M$ a solid
$\mathcal{A}$-module. Then there are completed solid unenriched Ext-functors
$\widehat{\mathrm{Ext}}_{\mathrm{Solid}(\mathcal{A})}^{\bullet}(M,-)$.

For a profinite group $G$ and for analytic rings $\mathcal{R}$ of a specific
form, J. Anschütz and A.-C. Le Bras define in [2, p. 2/5] solid group
cohomology $H_{\mathrm{Solid}(\mathcal{R})}^{\bullet}(\underline{G},-)$ of the
condensate of $G$ taking coefficients in solid
$\mathcal{R}[\underline{G}]$-modules. They mention some profinite rings $R$
whose condensates $\underline{R}$ are analytic rings of this form [2, p. 2].
As any condensate $\underline{M}$ of a profinite or discrete $R$-module $M$ is
a solid $\underline{R}$-module for such rings [2, p. 4], one has
$H_{\mathrm{Solid}(\underline{R})}^{\bullet}(\underline{G},\underline{M})\cong
H_{R}^{\bullet}(G,M)$ [2, Lemma 2.1]. We prove that there exists complete
solid cohomology
$\widehat{H}_{\mathrm{Solid}(\mathcal{R})}^{\bullet}(\underline{G},-)$ mapping
to abelian groups and complete solid internal group cohomology
$\widehat{\underline{H}}_{\mathrm{Solid}(\mathcal{R})}^{\bullet}(\underline{G},-)$
mapping to solid $\underline{\mathbb{Z}}$-modules whenever the analytic ring
$\mathcal{R}$ is of the required form (Lemma 7.5). In particular, the
following question arises.

######  Question 1.6.

When is the complete cohomology of a profinite group
$\widehat{H}_{R}^{\bullet}(G,-)$ isomorphic to the complete solid cohomology
of its condensate
$\widehat{H}_{\mathrm{Solid}(\underline{R})}^{\bullet}(\underline{G},-)$ as a
cohomological functor?

We refer the reader to M. Land’s notes [26] and P. Scholze’s notes [35] for
more detail on condensed mathematics. For results on condensed and solid group
cohomology the reader is referred to J. Anschütz and A.-C. Le Bras’ preprint
[2].

Let us detail how our work compares to the literature. Recently, S. Guo and L.
Liang have generalised in [19] the above constructions by G. Mislin [28], by
D. J. Benson and J. F. Carlson [5] and by P. Vogel [17] for unenriched
relative Ext-functors over any abelian category $\mathcal{C}$ with a special
precovering subcategory $\mathcal{W}$ and only proved that the resulting
completed relative Ext-functors
$\widehat{\mathrm{Ext}}_{\mathcal{WC}}^{n}(A,-):\mathcal{C}\rightarrow\mathbf{Ab}$
are naturally isomorphic for every $n\in\mathbb{Z}$. Although they prove that
their completed Ext-functors $\widehat{\mathrm{Ext}}_{\mathcal{WC}}^{n}(A,-)$
form a cohomological functor, we do not know whether they also form a Mislin
completion of the Ext-functors $\mathrm{Ext}_{\mathcal{WC}}^{\bullet}(A,-)$
(Question 6.19). On the other hand, A. Beligiannis and I. Reiten establish
specific Mislin completions in their book on torsion theories [4]. In the
elegant account from their Section IX.2, they consider an abelian category
$\mathcal{C}$ containing a particular pair of subcategories
$(\mathcal{X},\mathcal{Y})$, define a completed unenriched relative Ext-
functor $\widehat{\mathrm{Ext}}_{(\mathcal{X},\mathcal{Y})}^{\bullet}(-,-)$
using a version of D. J. Benson and J. F. Carlson’s construction from [5]
and prove that
$\mathrm{Ext}_{\mathcal{X}\cap\mathcal{Y}}^{\bullet}(A,-)\rightarrow\widehat{\mathrm{Ext}}_{(\mathcal{X},\mathcal{Y})}^{\bullet}(A,-)$
forms a Mislin completion for any $A\in\mathrm{obj}(\mathcal{C})$. Lastly, J.
Hu et al. use in [22] a version of P. Vogel’s construction from [17] to
establish completed unenriched Ext-functors
$\widehat{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(-,-)$ for an extriangulated
category $\mathcal{C}$ where extriangulated categories are a generalisation of
exact categories and triangulated categories. Among their applications, they
provide a criterion for the validity of the Wakamatsu Tilting Conjecture. As a
common application of completed unenriched Ext-functors, all above mentioned
authors consider various notions of homological dimension for which they
characterise when a homological dimension of an object in $\mathcal{C}$ is
finite.

Apart from providing different applications in [16], our contribution mainly
consists in establishing a Mislin completion whenever we are given any
cohomological functor between two particular abelian categories. For instance,
we can define completed internal condensed Ext-functors which cannot be
established through the previous frameworks (Remark 7.2). Moreover, in none of
the above sources, connecting homomorphisms or induced morphisms of Mislin
completions are constructed explicitly.

For an overview of various other generalisations of Tate cohomology and their
applications, the reader is referred to M. Paganin’s survey paper [31]. To
mention two developments that are not contained in it, a version of completed
Ext-functors for complexes of modules over a ring is defined in [3] and a
generalisation of Tate cohomology in the language of spectra can be found in
[24]. B. E. A. Nucinkis observed first in [29] that all construction can be
performed dually by assuming that there are enough injectives instead of
enough projectives. More specifically, if $\mathcal{C}$ is an abelian category
with enough injectives and $\mathcal{D}$ is an abelian category in which all
countable direct limits exist and are exact, then one can define for a
cohomological functor $T^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$ a
“Nucinkis completion” $T^{\bullet}\rightarrow\widetilde{T}^{\bullet}$ such
that $\widetilde{T}^{\bullet}$ vanishes on injectives and is universal with
respect to this property. For Ext-functors, this completion takes the form
$\mathrm{Ext}_{\mathcal{C}}^{\bullet}(-,A)\rightarrow\widetilde{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(-,A)$.
S. Guo and L. Liang perform all their constructions of completed unenriched
relative Ext-functors in a dual manner in [19]. Together with X. Yang they
provide in [20, Main Theorem] a criterion for when
$\widehat{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(-,-)$ is isomorphic to
$\widetilde{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(-,-)$ as well as A.
Beligiannis and I. Reiten in [4, Theorem IX.4.4] as well as S. Hu et al. in
[21, Theorem 4.4].

After outlining four constructions of Mislin completions in Section 2, we
prove in the subsequent four sections that they result in isomorphic
cohomological functors and provide explicit formulae of their connecting
homomorphisms and induced morphisms. More specifically, in Section 3 we
demonstrate that one construction in [28] by G. Mislin gives rise to Mislin
completions. In Section 4 we prove that another construction of a
cohomological functor from [28] is isomorphic to the former. Thereby we deduce
that there are countably many distinct constructions of Mislin completions.
Then in Section 5 we show that P. Vogel’s construction in [17] gives rise to
completed unenriched Ext-functors that are isomorphic to the ones from the
latter construction by G. Mislin. In Section 6 we prove that the completed
unenriched Ext-functors resulting from D. J. Benson and J. F. Carlson’s
construction in [5] are also isomorphic to the ones from the latter
construction by G. Mislin. Lastly, in Section 7, we outline how one can
introduce completed Ext-functors and complete cohomology in condensed
mathematics.

###  1.1 Notation and terminology

We adopt the convention that the natural numbers $\mathbb{N}$ include $1$, but
do not include $0$. We write $\mathbb{N}_{0}$ for $\mathbb{N}\cup\\{0\\}$.
Moreover, we adopt B. Poonen’s convention from [32] that a ring is an abelian
group together with a totally associative product, meaning a binary
associative relation admitting an identity element. In particular, ring
homomorphisms are understood to map identity elements to identity elements. We
use the symbol $\varinjlim$ only to denote direct limits. If $\mathcal{D}$ is
a category, then $\varinjlim_{\mathcal{D}}$ denotes a direct limit in this
category and if $I$ is a directed set, then $\varinjlim_{i\in I}$ denotes a
direct limit indexed over $I$. We write
$\varinjlim_{k\in\mathbb{N}}(M_{k},\mu_{k})$ if $\mu_{k}:M_{k}\rightarrow
M_{k+1}$ are the morphisms giving rise to the direct limit. Lastly, we use the
same numbering to label diagrams and equations.

##  2 Outline of constructions

To showcase the relevant constructions of Mislin completions, let us
axiomatically define cohomological functors. For two abelian categories
$\mathcal{C}$, $\mathcal{D}$ we call a family of additive functors
$(T^{n}:\mathcal{C}\rightarrow\mathcal{D})_{n\in\mathbb{Z}}$ a cohomological
functor if it satisfies the following two axioms [25, p. 201–202].

######  Axiom 2.1.

For every $n\in\mathbb{Z}$ and short exact sequence $0\rightarrow A\rightarrow
B\rightarrow C\rightarrow 0$ in $\mathcal{C}$, there is natural connecting
homomorphism $\delta^{n}:T^{n}(C)\rightarrow T^{n+1}(A)$.

Being natural means in this context that for every commuting diagram in
$\mathcal{C}$

${0}$${A}$${B}$${C}$${0}$${0}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$${0}$$\scriptstyle{f}$$\scriptstyle{g}$

with exact rows there is a commuting diagram

${T^{n}(A)}$${T^{n+1}(A)}$${T^{n}(A^{\prime})}$${T^{n+1}(C^{\prime})}$$\scriptstyle{\delta^{n}}$$\scriptstyle{T^{n}(g)}$$\scriptstyle{T^{n+1}(f)}$$\scriptstyle{\delta^{n}}$

in $\mathcal{D}$.

######  Axiom 2.2.

For every short exact sequence $0\rightarrow
A\xrightarrow{\iota}B\xrightarrow{\pi}C\rightarrow 0$ in $\mathcal{C}$ there
is a long exact sequence

$\dots\xrightarrow{T^{n-1}(\pi)}T^{n-1}(C)\xrightarrow{\delta^{n-1}}T^{n}(A)\xrightarrow{T^{n}(\iota)}T^{n}(B)\xrightarrow{T^{n}(\pi)}T^{n}(C)\xrightarrow{\delta^{n}}T^{n+1}(A)\xrightarrow{T^{n+1}(\iota)}{}\dots$

Assume that $(T^{\bullet},\delta^{\bullet})$ and
$(U^{\bullet},\varepsilon^{\bullet})$ are cohomological functors from
$\mathcal{C}$ to $\mathcal{D}$. Then a family of natural transformations
$(\nu^{n}:T^{n}\rightarrow U^{n})_{n\in\mathbb{Z}}$ is a morphism of
cohomological functors if it satisfies the axiom [25, p. 202]

######  Axiom 2.3.

For every $n\in\mathbb{Z}$ and short exact sequence $0\rightarrow A\rightarrow
B\rightarrow C\rightarrow 0$ in $\mathcal{C}$, the following square commutes:

${T^{n}(C)}$${T^{n+1}(A)}$${U^{n}(C)}$${U^{n+1}(A)}$$\scriptstyle{\delta^{n}}$$\scriptstyle{\nu^{n}}$$\scriptstyle{\nu^{n+1}}$$\scriptstyle{\varepsilon^{n}}$

Let us generalise G. Mislin’s Definition 2.1 from [28].

######  Definition 2.4 (Mislin completion).

Let $(T^{\bullet},\delta^{\bullet})$ be a cohomological functor from
$\mathcal{C}$ to $\mathcal{D}$. Then its Mislin completion is a cohomological
functor $(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$ from
$\mathcal{C}$ to $\mathcal{D}$ together with a morphism
$\nu^{\bullet}:T^{\bullet}\rightarrow\widehat{T}^{\bullet}$ satisfying the
following two conditions:
1. For every $n\in\mathbb{Z}$ and every projective $P\in\mathrm{obj}(\mathcal{C})$ we have $\widehat{T}^{n}(P)=0$.
2. If $(U^{\bullet},\varepsilon^{\bullet})$ is any cohomological functor vanishing on projectives, then each morphism $T^{\bullet}\rightarrow U^{\bullet}$ factors uniquely as $T^{\bullet}\xrightarrow{\nu^{\bullet}}\widehat{T}^{\bullet}\rightarrow U^{\bullet}$.

Hence, if $(U^{\bullet},\varepsilon^{\bullet})$ is another Mislin completion
of $(T^{\bullet},\delta^{\bullet})$, then there is an isomorphism
$\mu^{\bullet}:\widehat{T}^{\bullet}\rightarrow U^{\bullet}$, meaning that
$\mu^{n}(M):\widehat{T}^{n}(M)\rightarrow U^{n}(M)$ is an isomorphism for any
$n\in\mathbb{Z}$ and $M\in\mathrm{obj}(\mathcal{C})$ [25, p. 202]. This allows
us to state G. Mislin’s definition from [28, p. 297] in greater generality.

######  Definition 2.5 (Axiomatic, Mislin).

For any $A\in\mathrm{obj}(\mathcal{C})$ we extend the (enriched or unenriched)
Ext-functors to cohomological functor by setting
$\mathrm{Ext}_{\mathcal{C}}^{n}(A,-)=0$ for $n<0$ and define completed Ext-
functors as the Mislin completion
$(\widehat{\mathrm{Ext}}_{\mathcal{C}}^{\bullet}(A,-),\widehat{\delta}^{\bullet})$.
Analogously, if $G$ is a group object in $\mathcal{C}$ and $R$ a ring object,
then we extend group cohomology to a cohomological functor
$(H_{R}^{\bullet}(G,-),\delta^{\bullet})$ by imposing $H_{R}^{n}(G,-)=0$ for
$n<0$ and define complete cohomology as the Mislin completion
$(\widehat{H}_{R}^{\bullet}(G,-),\widehat{\delta}^{\bullet})$.

To ensure that complete cohomology exists, we introduce left satellite
functors to present a construction due to G. Mislin. As we assume that
$\mathcal{C}$ has enough projective objects, there is for any
$M\in\mathrm{obj}(\mathcal{C})$ a short exact sequence $0\rightarrow
K\rightarrow P\rightarrow M\rightarrow 0$ in $\mathcal{C}$ with $P$
projective. For a cohomological functor $(T^{\bullet},\delta^{\bullet})$, we
define the zeroeth left satellite functor of $T^{n}$ as $S^{0}T^{n}:=T^{n}$,
the first left satellite functor as

$S^{-1}T^{n}(M):=\mathrm{Ker}\big{(}T^{n}(K)\rightarrow T^{n}(P)\big{)}$

and the $k^{\text{th}}$ left satellite functor as
$S^{-k}T^{n}:=S^{-1}(S^{-k+1}T^{n})$ for $k\geq 2$ [7, p. 36]. It is shown in
[7, Section III.1] that left satellite functors do not depend on the choice of
short exact sequence. Since they have been defined as kernels, it follows from
Axiom 2.2 that $\delta^{n}:T^{n}\rightarrow T^{n+1}$ induces a morphism
$\underline{\delta}^{n}:T^{n}(M)\rightarrow S^{-1}T^{n+1}(M)$ and therefore
$S^{-k}\underline{\delta}^{n+k}:S^{-k}T^{n+k}(M)\rightarrow
S^{-k-1}T^{n+k+1}(M)$ for any $k\in\mathbb{N}$ [25, p. 207–208]. Here, our
assumption hits in that all countable direct limits exist in the codomain
category $\mathcal{D}$ of $T^{\bullet}$.

######  Definition 2.6.

A partially ordered set $(I,\leq)$ is a directed set if for every $i,j\in I$
there is $k\in I$ such that $i,j\leq k$ [34, p. 1]. According to [34, p. 14],
a diagram $\\{D_{i}\\}_{i\in I}$ in $\mathcal{D}$ indexed over a directed set
is called a direct system in $\mathcal{D}$. More formally, $I$ can be turned
into a category whose objects are its elements and there is a unique morphism
$i\rightarrow j$ whenever $i\leq j\in I$. Then a direct system is a covariant
functor $I\rightarrow\mathcal{D},i\mapsto D_{i}$. A direct limit
$\varinjlim_{i\in I}D_{i}$ in $\mathcal{D}$ is a colimit of a direct system
$\\{D_{i}\\}_{i\in I}$. Direct limits in $\mathcal{D}$ are called exact if for
every direct system of short exact sequence $\\{0\rightarrow A_{i}\rightarrow
B_{i}\rightarrow C_{i}\rightarrow 0\\}_{i\in I}$ also

$0\rightarrow\varinjlim_{i\in I}A_{i}\rightarrow\varinjlim_{i\in
I}B_{i}\rightarrow\varinjlim_{i\in I}C_{i}\rightarrow 0$

is a short exact sequence [38, Tag 079A].

Hence, we can extend G. Mislin’s construction from [28, p. 293].

######  Definition 2.7 (Via satellite functors, Mislin).

The Mislin completion of a cohomological functor
$(T^{\bullet},\delta^{\bullet})$ can be defined as
$\widehat{T}^{n}(M):=\varinjlim_{k\in\mathbb{N}_{0}}(S^{-k}T^{n+k}(M),S^{-k}\underline{\delta}^{n+k})$
for any $M\in\mathrm{obj}(\mathcal{C})$ and $n\in\mathbb{Z}$. Accordingly,
$\widehat{H}_{R}^{n}(G,M):=\varinjlim_{k\in\mathbb{N}_{0}}(S^{-k}H_{R}^{n+k}(G,M),S^{-k}\underline{\delta}^{n+k})$
is a definition of complete cohomology.

In order that the above forms a cohomological functor, one needs to impose
that all direct limits in $\mathcal{D}$ are exact. The reader might be aware
that one can define cohomology of discrete groups and cohomology of profinite
groups taking discrete coefficients by using that the respective category
$\mathcal{C}$ has enough injectives [6, p. 61], [37, p. 9]. One could assume
instead that $\mathcal{C}$ has enough injectives and in the category
$\mathcal{D}$ all countable inverse limits exist and are exact. One could then
perform a construction via right satellite functors dual to the above. This
completion of group cohomology would have a universal property as a Mislin
completion, except that it would vanish on injective objects instead of
projective ones. However, cohomology of discrete groups as well as of
profinite groups taking discrete coefficients already vanishes on injectives
[6, p. 61], [37, p. 9]. Thus, such a completion would not yield Tate
cohomology whence we assume that $\mathcal{C}$ has enough projectives instead
of enough injectives.

######  Notation 2.8.

For the rest of the paper, $\mathcal{C}$ always denotes an abelian category
with enough projectives and $\mathcal{D}$ an abelian category in which all
countable direct limits exist and are exact.

Let us go over to what we term the resolution construction that occurs in [8,
Lemma B.3] and can be retrieved from page 299 in G. Mislin’s paper [28]. If
$(M_{n})_{n\in\mathbb{N}_{0}}$ is a projective resolution of
$M\in\mathrm{obj}(\mathcal{C})$, let us define $\widetilde{M}_{0}:=M$ and
$\widetilde{M}_{k}:=\mathrm{Ker}(M_{k-1}\rightarrow\widetilde{M}_{k-1})$ for
$k\in\mathbb{N}$. This is called the $k^{\text{th}}$ syzygy of $M_{\bullet}$
in the Gorenstein context [31, p. 89]. The choice of our notation is meant to
reflect that our syzygies do not necessarily arise from a specific choice of
projective resolution as in [25] and [28]. Since for every
$k\in\mathbb{N}_{0}$ we have the short exact sequence
$0\rightarrow\widetilde{M}_{k+1}\rightarrow
M_{k}\rightarrow\widetilde{M}_{k}\rightarrow 0$, there is a connecting
homomorphism $\delta^{n+k}:T^{n+k}(\widetilde{M}_{k})\rightarrow
T^{n+k+1}(\widetilde{M}_{k+1})$ for every $n\in\mathbb{Z}$. Then the following
definition makes it more apparent why Mislin completions vanish on projective
objects.

######  Definition 2.9 (Resolutions, Mislin).

The Mislin completion of a cohomological functor
$(T^{\bullet},\delta^{\bullet})$ can be defined as
$\widehat{T}^{n}(M):=\varinjlim_{k\in\mathbb{N}_{0}}(T^{n+k}(\widetilde{M}_{k}),\delta^{n+k})$
for any $n\in\mathbb{Z}$ and $M\in\mathrm{obj}(\mathcal{C})$. Accordingly,
$\widehat{H}_{R}^{n}(G,M):=\varinjlim_{k\in\mathbb{N}_{0}}(H_{R}^{n+k}(G,\widetilde{M}_{k}),\delta^{n+k})$
is a definition of complete cohomology.

The next two constructions only give rise to completed unenriched Ext-
functors. Let $A_{\bullet}$, $B_{\bullet}$ are projective resolutions of
$A,B\in\mathrm{obj}(\mathcal{C})$ and let
$\widetilde{f}_{n+k}:\widetilde{A}_{n+k}\rightarrow\widetilde{B}_{k}$ be a
morphism for $n\in\mathbb{Z}$ and $k\in\mathbb{N}_{0}$ such that $n+k\geq 0$.
Then we can write the commuting diagram

${0}$${\widetilde{A}_{n+k+1}}$${A_{n+k}}$${\widetilde{A}_{n+k}}$${0}$${0}$${\widetilde{B}_{k+1}}$${B_{k+1}}$${\widetilde{B}_{k}}$${0}$$\scriptstyle{\widetilde{f}_{k+1}}$$\scriptstyle{f_{k+1}}$$\scriptstyle{\widetilde{f}_{k}}$

whose terms arise as follows. Since the bottom row is exact and the term
$A_{n+k}$ projective, there is a lift $f_{k}$ of $\widetilde{f}_{k}$ making
the right-hand square commute. Because $\widetilde{B}_{k+1}\rightarrow
B_{k+1}$ is a kernel, there is a morphism $\widetilde{f}_{k+1}$ making the
left-hand side commute. If $\mathrm{Hom}_{\mathcal{C}}(-,-)$ denotes the
(unenriched) Hom-functor in $\mathcal{C}$, then
$\mathrm{Hom}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})$ is an
abelian group by virtue of $\mathcal{C}$ being an abelian category. We define
$\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})$ to be the
subgroup of
$\mathrm{Hom}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})$ consisting
of all morphisms factoring through a projective object and write the quotient
as
$[\widetilde{A}_{n+k},\widetilde{B}_{k}]_{\mathcal{C}}:=\mathrm{Hom}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})/\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})$
[25, p. 203]. As in the case of modules over a ring covered by [25, p. 204],
we prove that

$\displaystyle
t_{\widetilde{A}_{n+k},\widetilde{B}_{k}}:[\widetilde{A}_{n+k},\widetilde{B}_{k}]_{\mathcal{C}}$
$\displaystyle\rightarrow[\widetilde{A}_{n+k+1},\widetilde{B}_{k+1}]_{\mathcal{C}},$
$\displaystyle\widetilde{f}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})$
$\displaystyle\mapsto\widetilde{f}_{k+1}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{n+k+1},\widetilde{B}_{k+1})$

is a well defined homomorphism. Through this we generalise the following
definition from D. J. Benson and J. F. Carlson’s paper [5, p. 109].

######  Definition 2.10 (Naïve construction, Benson & Carlson).

We can define the $n^{\text{th}}$ completed unenriched Ext-functor as
$\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B):=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
0}([\widetilde{A}_{n+k},\widetilde{B}_{k}]_{\mathcal{C}},t_{\widetilde{A}_{n+k},\widetilde{B}_{k}})$
for $n\in\mathbb{Z}$. In particular, if $R_{\bullet}$ is a projective
$R[G]$-resolution of $R\in\mathrm{obj}(\mathcal{C}_{R,G})$, we can define
complete unenriched cohomology as
$\widehat{H}_{R}^{n}(G,B):=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
0}([\widetilde{R}_{n+k},\widetilde{B}_{k}]_{\mathcal{C}},t_{\widetilde{R}_{n+k},\widetilde{B}_{k}})$.

Lastly, we present what we call the hypercohomology construction of complete
cohomology. We define the chain complex $(A_{n}^{\prime})_{n\in\mathbb{Z}}$ by
$A_{n}^{\prime}=A_{n}$ for $n\geq 0$ and $A_{n}^{\prime}=0$ for $n<0$ and
similarly $(B_{n}^{\prime})_{n\in\mathbb{Z}}$ [25, p. 209]. Define the
hypercohomology complex
$(\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n},d^{n})_{n\in\mathbb{Z}}$
by having $n$-cochains
$\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n}=\prod_{k\in\mathbb{Z}}\mathrm{Hom}_{\mathcal{C}}(A_{k+n}^{\prime},B_{k}^{\prime})$.
To ease notation in the following, we view abelian groups as
$\mathbb{Z}$-modules. If we denote by $a_{n}:A_{n}^{\prime}\rightarrow
A_{n-1}^{\prime}$ and $b_{n}:B_{n}^{\prime}\rightarrow B_{n-1}^{\prime}$ the
differentials induced from the respective projective resolution, we define for
$n\in\mathbb{Z}$ the differential

$\displaystyle
d^{n}:\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n}$
$\displaystyle\rightarrow\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n+1}$
(2.1) $\displaystyle(\varphi_{n+k})_{k\in\mathbb{Z}}$
$\displaystyle\mapsto(b_{k+1}\circ\varphi_{n+k+1}-(-1)^{n}\varphi_{n+k}\circ
a_{n+k+1})_{k\in\mathbb{Z}}$

Let us define the bounded complex
$\mathrm{Bdd}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n\in\mathbb{Z}}$
as the subcomplex of
$\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n\in\mathbb{Z}}$
given by
$\mathrm{Bdd}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n}=\bigoplus_{k\in\mathbb{Z}}\mathrm{Hom}_{\mathcal{C}}(A_{k+n}^{\prime},B_{k}^{\prime})$.
Now we define the Vogel complex
$\mathrm{Vog}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n\in\mathbb{Z}}$
as the quotient complex
$\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n}/\mathrm{Bdd}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{n}$
[25, p. 209]. By this, we generalise Definition 1.2 from F. Goichot’s paper
[17] where he attributes it to P. Vogel on page 39.

######  Definition 2.11 (Hypercohomology, Vogel).

For $n\in\mathbb{Z}$ we can define the $n^{\text{th}}$ completed unenriched
Ext-functor as
$\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B):=H^{n}((\mathrm{Vog}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})_{k},d^{k})_{k\in\mathbb{Z}})$.
We can thus define complete unenriched cohomology as
$\widehat{H}_{R}^{n}(G,M):=H^{n}((\mathrm{Vog}_{R}(R_{\bullet}^{\prime},B_{\bullet}^{\prime})_{k},d^{k})_{k\in\mathbb{Z}})$.

Let us remark why it is unlikely that the previous two constructions could
yield Mislin completions of more general enriched Ext-functors. For the naïve
construction, one aims to find a morphism
$\mathrm{Hom}_{\mathcal{C}}(\widetilde{A}_{n+k},\widetilde{B}_{k})\rightarrow\mathrm{Hom}_{\mathcal{C}}(\widetilde{A}_{n+k+1},\widetilde{B}_{k+1})$.
If one tries to lift along the morphisms induced by
$A_{n+k}\rightarrow\widetilde{A}_{n+k}$ and
$B_{n+k}\rightarrow\widetilde{B}_{n+k}$, one requires that
$\mathrm{Hom}_{\mathcal{C}}(A_{n+k},-)$ preserves epimorphisms. For unenriched
Hom-functors, this role is exactly played by projective objects [42, Lemma
2.2.3]. For the hypercohomology construction we require that the coproduct
$\mathrm{Bdd}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})$ maps
into the product
$\mathrm{Hyp}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})$.
Moreover, as we shall see in Subsection 6.1, the cohomology groups of the
Vogel complex
$\mathrm{Vog}_{\mathcal{C}}(A_{\bullet}^{\prime},B_{\bullet}^{\prime})$
correspond exactly to so-called almost chain maps modulo almost chain
homotopy. However, this implies that the objects
$\mathrm{Hom}_{\mathcal{C}}(A_{p},B_{q})\in\mathrm{obj}(\mathcal{D})$ describe
morphisms of the form $A_{p}\rightarrow B_{q}$ in $\mathcal{C}$.

##  3 The satellite functor construction and Mislin completions

Satellite functors are treated in the third chapter of H. Cartan and S.
Eilenberg’s book on homological algebra [7]. By using morphisms of
cohomological functors, G. Mislin constructs Mislin completions through
satellite functors in the proof of Theorem 2.2 from [28] while P. H.
Kropholler provides more explicit descriptions of the terms of such a Mislin
completion on pages 206–208 of [25]. In this section we generalise G. Mislin
and P. H. Kropholler’s work that is performed only using modules over a
ring. As we also aim to determine explicit formulae for the induced morphisms
and for the connecting homomorphisms of the resulting Mislin completions, we
provide full details on the satellite functor construction. In turn, these
details are needed later to manufacture the induced morphisms and connecting
homomorphisms of all other constructions of Mislin completions.

For the reader’s convenience, we summarise relevant results on left satellite
functors from the third chapter of [7]. They may be working explicitly in a
category of modules over a ring, but A. Grothendieck has generalised their
satellite functors in his Tôhoku paper [18, p. 140–143] and thus, their
arguments pertain to our abelian categories. In the first section, they show
that left satellite functors are well defined and additive functors. More
specifically, let $f:C^{\prime}\rightarrow C$ be a morphism in $\mathcal{C}$.
As we assume that the abelian category $\mathcal{C}$ has enough projectives,
one can take short exact sequences

$0\rightarrow M^{\prime}\rightarrow P^{\prime}\rightarrow
C^{\prime}\rightarrow 0\quad\text{and}\quad 0\rightarrow M\rightarrow
P\rightarrow C\rightarrow 0$

with $P^{\prime}$ and $P$ projective. Using this and that every monomorphism
is normal in an abelian category, we deduce the existence of morphisms
$\overline{f}^{\ast}:P^{\prime}\rightarrow P$ and
$\overline{f}^{\prime}:M^{\prime}\rightarrow M$ rendering the diagram

${0}$${M^{\prime}}$${P^{\prime}}$${C^{\prime}}$${0}$${0}$${M}$${P}$${C}$${0}$$\scriptstyle{\overline{f}^{\ast}}$$\scriptstyle{\overline{f}}$$\scriptstyle{f}$
(3.1)

commutative. To ease future notation, let
$T^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$ be a cohomological functor
with connecting homomorphisms $\delta^{\bullet}$. The morphism
$T^{n}(\overline{f}^{\ast}):T^{n}(M^{\prime})\rightarrow T^{n}(M)$ induces the
corresponding morphism between the kernels
$S^{-1}T^{n}(f):S^{-1}T^{n}(C^{\prime})\rightarrow S^{-1}T^{n}(C)$.
Pictorially,

${S^{-1}T^{n}(C^{\prime})}$${T^{n}(M^{\prime})}$${S^{-1}T^{n}(C)}$${T^{n}(M)}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{S^{-1}T^{n}(f)}$$\scriptstyle{T^{n}(\overline{f}^{\ast})}$$\scriptstyle{\varepsilon^{-1}}$

where the morphisms $\varepsilon^{-1}$ are the canonical monomorphisms from
the kernels. Being independent of the choices of short exact sequences, the
morphism $S^{-1}T^{n}(f)$ is well defined. The morphisms
$S^{-k}T^{n}(f):S^{-k}T^{n}(C^{\prime})\rightarrow S^{-k}T^{n}(C)$ for higher
satellite functors are defined in the same manner. In the second section, they
construct for any exact sequence $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ in $\mathcal{C}$ and any $k\in\mathbb{N}_{0}$ a connecting
homomorphism $\varepsilon^{-k-1}:S^{-k-1}T^{n}(C)\rightarrow S^{-k}T^{n}(A)$
in the following manner. If $B$ is projective, then $\varepsilon^{-k-1}$ is
the monomorphism from the kernel $S^{-k-1}T^{n}(C)$. Otherwise let
$0\rightarrow M\rightarrow P\rightarrow C\rightarrow 0$ be a short exact
sequence with $P$ projective. As we have seen above, there are morphisms
$h:P\rightarrow B$ and $h^{\ast}:M\rightarrow A$ extending the identity
morphism $\mathrm{id}_{C}:C\rightarrow C$ making the diagram

${0}$${M}$${P}$${C}$${0}$${0}$${A}$${B}$${C}$${0}$$\scriptstyle{h^{\ast}}$$\scriptstyle{h}$$\scriptstyle{\mathrm{id}_{C}}$
(3.2)

commute. Then $\varepsilon^{-k-1}:S^{-k-1}T^{n}(C)\rightarrow
S^{-k}T^{n}(M)\xrightarrow{S^{-k}T^{n}(h^{\ast})}S^{-k}T^{n}(A)$ is a
connecting homomorphism where the first morphism is the monomorphism of the
kernel $S^{-n-1}T^{n}(C)$. Moreover they show that the every connecting
homomorphism is a natural transformation. Extending into the third section,
they use these to construct the long exact sequence

$\dots\rightarrow
S^{-k-1}T^{n}(C)\xrightarrow{\varepsilon^{-k-1}}S^{-k}T^{n}(A)\rightarrow
S^{-k}T^{n}(B)\rightarrow
S^{-k}T^{n}(C)\xrightarrow{\varepsilon^{-k}}S^{-k+1}T^{n}(A)\rightarrow{}\dots$

where the unlabelled morphisms are induced from the underlying short exact
sequence. In the fifth section, they provide the following crucial result
corresponding to Proposition 5.2 on page 46:

######  Lemma 3.1.

Let $T^{\bullet}$ and $U^{\bullet}$ be cohomological functors and
$\Phi^{0}:T^{0}\rightarrow U^{0}$ be a natural transformation. If for every
short exact sequence $0\rightarrow K\rightarrow P\rightarrow M\rightarrow 0$
with $P$ projective and every $n\in\mathbb{N}$ the sequence $0\rightarrow
U^{-n-1}(M)\rightarrow U^{-n}(K)$ involving the connecting homomorphism is
exact, then $\Phi^{0}$ extends uniquely to a (partial) morphism of
cohomological functors $(\Phi^{n}:T^{n}\rightarrow U^{n})_{n\leq 0}$ only
defined for $n\leq 0$. In particular, if also $T^{\bullet}$ has the property
that the sequence $0\rightarrow T^{-n-1}(M)\rightarrow T^{-n}(K)$ is exact for
$n\in\mathbb{N}$ and $\Phi^{0}$ is an equivalence, then the extension
$\Phi^{\bullet}$ is an isomorphism.

G. Mislin uses this lemma to perform his satellite functor construction on
pages 295–296 in [28] which we generalise in the following. For any
$m\in\mathbb{N}_{0}$ we can define the cohomological functor

$T^{k}\langle m\rangle(M):=\begin{cases}S^{k-m}T^{m}(M)&\text{if }k<m\\\
T^{k}(M)&\text{if }k\geq m\end{cases}$ (3.3)

where the connecting homomorphisms are given by $\delta^{k}\langle
m\rangle=\delta^{k}:T^{k}\rightarrow T^{k+1}$ for $k\geq m$ and
$\delta^{k}\langle m\rangle=\varepsilon^{k-m}:S^{k-m}T^{m}\rightarrow
S^{k-m+1}T^{m}$ for $k<m$. Consider the partial morphism
$(\Phi_{m}^{k}:=\mathrm{id}:T^{k}\rightarrow T^{k}\langle m\rangle)_{m\leq k}$
of cohomological functors defined only in degrees $m\leq k$. According to the
previous lemma we can extend $\Phi_{m}^{\bullet}$ uniquely to a morphism of
cohomological functors defined in all degrees. Using the same arguments we
have for $m\leq n$ a unique morphism $\Phi_{m,n}^{\bullet}:T^{\bullet}\langle
m\rangle\rightarrow T^{\bullet}\langle n\rangle$ that is equal to the identity
morphism in any degree $n\leq k\in\mathbb{Z}$. Using the uniqueness of all
these morphisms, we conclude for $m\leq n\leq o\in\mathbb{N}_{0}$ that
$\Phi_{n}^{\bullet}=\Phi_{m,n}^{\bullet}\circ\Phi_{m}^{\bullet}$ and
$\Phi_{m,o}^{\bullet}=\Phi_{n,o}^{\bullet}\circ\Phi_{m,n}^{\bullet}$. This
yields a morphism to a direct system
$T^{\bullet}\rightarrow(T^{\bullet}\langle
m\rangle,\Phi_{m,n}^{\bullet})_{m\leq n\in\mathbb{N}_{0}}$. This is where
countable direct limits $\varinjlim_{\mathcal{D}}$ in the codomain category
$\mathcal{D}$ come into play. For every $n\in\mathbb{Z}$ we define the
$n^{\text{th}}$ term of the satellite functor construction as

$\widehat{T}^{n}(M):=\varinjlim_{k\in\mathbb{N}_{0}}T^{n}\langle
k\rangle(M)\,.$

Unravelling the definitions and noting that $\\{k\in\mathbb{N}_{0}\mid k\geq
n\\}$ is cofinal in $\mathbb{N}_{0}$, we obtain

$\widehat{T}^{n}(M)=\\!\\!\\!\\!\varinjlim_{k\in\mathbb{N}_{0},k\geq
n}\big{(}S^{n-k}T^{k}(M),\Phi_{k,k+1}^{n}(M)\big{)}=\varinjlim_{k\in\mathbb{N}_{0}}\big{(}S^{-k}T^{n+k}(M),\Phi_{n+k,n+k+1}^{n}(M)\big{)}\,.$
(3.4)

As $T^{\bullet}$ is a cohomological functor and left satellite functors are
additive, each functor $S^{-k}T^{n+k}$ is additive. Since
$\varinjlim_{\mathcal{D}}$ is additive, $\widehat{T}^{n}$ is an additive
functor. Every short exact sequence $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ in the category $\mathcal{C}$ yields a long exact sequence in
the cohomological functors $T^{\bullet}$ and $T^{\bullet}\langle m\rangle$ for
any $m\in\mathbb{N}_{0}$ by Axiom 2.2. Hence we obtain a long exact sequence
in
$\widehat{T}^{\bullet}=\varinjlim_{\mathcal{D},m\in\mathbb{N}_{0}}T^{\bullet}\langle
m\rangle$ and in particular, a connecting homomorphism
$\widehat{\delta}^{\bullet}$ only if the direct limit functor
$\varinjlim_{\mathcal{D},m\in\mathbb{N}_{0}}$ is exact.

More specifically, to demonstrate that
$(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$ is a cohomological
functor it suffices to show that $\widehat{\delta}^{\bullet}$ is a natural
transformation and how a long exact sequence in $\widehat{T}^{\bullet}$ arises
from a short exact sequence $0\rightarrow
A\xrightarrow{f}B\xrightarrow{g}C\rightarrow 0$ in $\mathcal{C}$. As for every
$m\leq n\in\mathbb{N}_{0}$ we have a morphism of cohomological functors
$\Phi_{m,n}^{\bullet}$, the following diagram commutes for every
$k\in\mathbb{Z}$:

${T^{k}\langle m\rangle(C)}$${T^{k+1}\langle m\rangle(A)}$${T^{k}\langle
n\rangle(C)}$${T^{k+1}\langle n\rangle(A)}$$\scriptstyle{\delta^{k}\langle
m\rangle}$$\scriptstyle{\Phi_{m,n}^{k}(C)}$$\scriptstyle{\Phi_{m,n}^{k+1}(A)}$$\scriptstyle{\delta^{k}\langle
n\rangle}$

As $\Phi_{m,n}^{k}(-)$ (resp. $\Phi_{m,n}^{k+1}(-)$) is a functor and
$\delta^{k}\langle m\rangle$ (resp. $\delta^{k+1}\langle n\rangle$) is
natural, the connecting homomorphism
$\widehat{\delta}^{k}=\varinjlim_{\mathcal{D},n\in\mathbb{N}_{0}}\delta^{k}\langle
n\rangle:\,\widehat{T}^{k}(C)\rightarrow\widehat{T}^{k+1}(A)$ is natural by
the above square. Using again that $\Phi_{m,n}^{\bullet}$ is a morphism of
cohomological functors, also the following diagram commutes:

${{}\dots}$${T^{k-1}\langle m\rangle(C)}$${T^{k}\langle
m\rangle(A)}$${T^{k}\langle m\rangle(B)}$${T^{k}\langle
m\rangle(C)}$${{}\dots}$${{}\dots}$${T^{k-1}\langle
n\rangle(C)}$${T^{k}\langle n\rangle(A)}$${T^{k}\langle
n\rangle(B)}$${T^{k}\langle
n\rangle(C)}$${{}\dots}$$\scriptstyle{\delta^{k-1}\langle
m\rangle}$$\scriptstyle{\Phi_{m,n}^{k-1}(C)}$$\scriptstyle{T^{k}\langle
m\rangle(f)}$$\scriptstyle{\Phi_{m,n}^{k}(A)}$$\scriptstyle{T^{k}\langle
m\rangle(g)}$$\scriptstyle{\Phi_{m,n}^{k}(B)}$$\scriptstyle{\delta^{k}\langle
m\rangle}$$\scriptstyle{\Phi_{m,n}^{k}(C)}$$\scriptstyle{\delta^{k-1}\langle
n\rangle}$$\scriptstyle{T^{k}\langle n\rangle(f)}$$\scriptstyle{T^{k}\langle
n\rangle(g)}$$\scriptstyle{\delta^{k}\langle n\rangle}$

Since the above constitutes a direct system of long exact sequences and
countable direct limits $\varinjlim_{\mathcal{D}}$ are exact, we obtain the
corresponding long exact sequence in
$(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$

${\dots}$${\widehat{T}^{k-1}(C)}$${\widehat{T}^{k}(A)}$${\widehat{T}^{k}(B)}$${\widehat{T}^{k}(C)}$${\dots}$$\scriptstyle{\widehat{\delta}^{k-1}}$$\scriptstyle{\widehat{T}^{k}(f)}$$\scriptstyle{\widehat{T}^{k}(g)}$$\scriptstyle{\widehat{\delta}^{k}}$

######  Lemma 3.2.

The cohomological functor $(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$
constructed via satellite functors is a Mislin completion of
$(T^{\bullet},\delta^{\bullet})$.

######  Proof.

As for G. Mislin’s satellite functor construction, the proof is an adaptation
of material found on page 296 of his paper [28]. Let $P$ be projective
projective object in $\mathcal{C}$. If we choose the short exact sequence
$0\rightarrow\big{(}0\rightarrow P\rightarrow P\big{)}\rightarrow 0$, then we
see that $S^{-1}T^{n}(P)=0$ and thus $S^{-k}T^{n+k}(P)=0$ for any $k\geq 1$.
In particular, $(\widehat{T}^{\bullet},\delta^{\bullet})$ vanishes on
projectives. Let us define a canonical morphism
$\Phi^{\bullet}:(T^{\bullet},\delta^{\bullet})\rightarrow(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$
such that any morphism
$\Psi^{\bullet}:(T^{\bullet},\delta^{\bullet})\rightarrow(U^{\bullet},\zeta^{\bullet})$
to a cohomological functor vanishing on projectives factors uniquely as

$\Psi^{\bullet}:(T^{\bullet},\delta^{\bullet})\xrightarrow{\Phi^{\bullet}}(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})\rightarrow(U^{\bullet},\zeta^{\bullet})\,.$

By Lemma 3.1 there is a morphism
$\Phi_{m}^{\bullet}:(T^{\bullet},\delta^{\bullet})\rightarrow(T^{\bullet}\langle
m\rangle,\delta^{\bullet}\langle m\rangle)$ for any $m\in\mathbb{N}_{0}$ and
denote by $\widehat{\Phi}_{m}^{\bullet}:(T^{\bullet}\langle
m\rangle,\delta^{\bullet}\langle
m\rangle)\rightarrow(\widehat{T}^{\bullet},\widehat{\delta^{\bullet}})$ the
morphism to the direct limit. Defining
$\Phi^{\bullet}:=\widehat{\Phi}_{m}^{\bullet}\circ\Phi_{m}^{\bullet}:(T^{\bullet},\delta^{\bullet})\rightarrow(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$
is then independent of $m\in\mathbb{N}_{0}$ since

$\widehat{\Phi}_{m}^{\bullet}\circ\Phi_{m}^{\bullet}=(\widehat{\Phi}_{n}^{\bullet}\circ\Phi_{m,n}^{\bullet})\circ\Phi_{m}^{\bullet}=\widehat{\Phi}_{n}^{\bullet}\circ(\Phi_{m,n}^{\bullet}\circ\Phi_{m}^{\bullet})=\widehat{\Phi}_{n}^{\bullet}\circ\Phi_{n}^{\bullet}$

for any $m\leq n$. We can also define a morphism
$\Psi_{m}^{\bullet}:(T^{\bullet}\langle m\rangle,\delta^{\bullet}\langle
m\rangle)\rightarrow(U^{\bullet},\zeta^{\bullet})$ by setting
$\Psi_{m}^{n}=\Psi^{n}$ for any $m\leq n$. Using Lemma 3.1 we extend
$\Psi_{m}^{\bullet}$ uniquely to a morphism of cohomological functors. Given
that $\Phi_{m}^{n}=\mathrm{id}_{\mathcal{D}}$ for $m\leq n\in\mathbb{Z}$, we
have $\Psi^{n}=\Psi_{m}^{n}\circ\Phi_{m}^{n}$ and Lemma 3.1 implies that

$\Psi^{\bullet}=\Psi_{m}^{\bullet}\circ\Phi_{m}^{\bullet}:(T^{\bullet},\delta^{\bullet})\rightarrow(T^{\bullet}\langle
m\rangle,\delta^{\bullet}\langle
m\rangle)\rightarrow(U^{\bullet},\zeta^{\bullet})\,.$

Deduce analogously
$\Psi_{m}^{\bullet}=\Psi_{n}^{\bullet}\circ\Phi_{m,n}^{\bullet}$ for any
$m\leq n\in\mathbb{N}_{0}$. We apply the universal property of direct limits
to obtain the unique morphism
$\widehat{\Psi}^{\bullet}:(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})\rightarrow(U^{\bullet},\zeta^{\bullet})$
such that
$\Psi_{m}^{\bullet}=\widehat{\Psi}^{\bullet}\circ\widehat{\Phi}_{m}^{\bullet}$.
Therefore

$\Psi^{\bullet}=\Psi_{m}^{\bullet}\circ\Phi_{m}^{\bullet}=\widehat{\Psi}^{\bullet}\circ\widehat{\Phi}_{m}^{\bullet}\circ\Phi_{m}^{\bullet}=\widehat{\Psi}^{\bullet}\circ\widehat{\Phi}^{\bullet}:(T^{\bullet},\delta^{\bullet})\rightarrow(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})\rightarrow(U^{\bullet},\zeta^{\bullet})\,.$

We want to show that this factorisation is unique. So assume that
$\Psi^{\bullet}=X^{\bullet}\circ\widehat{\Phi}^{\bullet}$ is another
factorisation. Thus

$\widehat{\Psi}^{\bullet}\circ\widehat{\Phi}_{m}^{\bullet}\circ\Phi_{m}^{\bullet}=\widehat{\Psi}^{\bullet}\circ\widehat{\Phi}^{\bullet}=X^{\bullet}\circ\widehat{\Phi}^{\bullet}=X^{\bullet}\circ\widehat{\Phi}_{m}^{\bullet}\circ\Phi_{m}^{\bullet}\,.$

As $\Phi_{m}^{n}=\mathrm{id}_{\mathcal{D}}$ for $m\leq n\in\mathbb{N}_{0}$,
$\widehat{\Psi}^{n}\circ\widehat{\Phi}_{m}^{n}=X^{n}\circ\widehat{\Phi}_{m}^{n}$
and

$\Psi_{m}=\widehat{\Psi}^{\bullet}\circ\widehat{\Phi}_{m}^{\bullet}=X^{\bullet}\circ\widehat{\Phi}_{m}^{\bullet}:(T^{\bullet}\langle
m\rangle,\delta^{\bullet}\langle
m\rangle)\rightarrow(U^{\bullet},\zeta^{\bullet})$

by Lemma 3.1. Since $\widehat{\Psi}^{\bullet}$ is the unique morphism such
that
$\Psi_{m}^{\bullet}=\widehat{\Psi}^{\bullet}\circ\widehat{\Phi}_{m}^{\bullet}$
for any $m\in\mathbb{N}_{0}$, we conclude that
$X^{\bullet}=\widehat{\Psi}^{\bullet}$. $\square$

Let us provide an explicit formula for the connecting homomorphisms of the
satellite functor construction. For this, we require H. Cartan and S.
Eilenberg’s extension of the concept of left satellite functors to natural
transformations from [7, Corollary 5.3] that we present in the following. As
before, their considerations for modules over a ring pass to abelian
categories. Let $\varphi:F\rightarrow G$ be a natural transformation between
two half-exact functors $F,G:\mathcal{C}\rightarrow\mathcal{D}$, meaning that
for any short exact sequence $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ in $\mathcal{C}$ the induced sequences in $\mathcal{D}$

$F(A)\rightarrow F(B)\rightarrow F(C)\text{ and }G(A)\rightarrow
G(B)\rightarrow G(C)$

are exact [25, p. 205]. For $C\in\mathrm{obj}(\mathcal{C})$, let $0\rightarrow
M\rightarrow P\rightarrow C\rightarrow 0$ be a short exact sequence with $P$
projective. This induces the commutative diagram

${F(M)}$${F(P)}$${F(C)}$${G(M)}$${G(P)}$${G(C)}$$\scriptstyle{\varphi(M)}$$\scriptstyle{\varphi(P)}$$\scriptstyle{\varphi(C)}$

In the same manner as we have defined morphisms
$S^{-1}T^{n}(f):S^{-1}T^{n}(C^{\prime})\rightarrow S^{-1}T^{n}(C)$ between
first left satellite functors, we denote by
$S^{-1}\varphi(C):S^{-1}F(C)\rightarrow S^{-1}G(C)$ the morphism induced from
$\varphi(M):F(M)\rightarrow G(M)$ between the corresponding kernels.
Pictorially

${S^{-1}F(C)}$${F(M)}$${S^{-1}G(C)}$${G(M)}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{S^{-1}\varphi(C)}$$\scriptstyle{\varphi(M)}$$\scriptstyle{\varepsilon^{-1}}$

Write $S^{0}{\varphi}:=\varphi$ and

$S^{-k}\varphi:=S^{-1}(S^{-k+1}\varphi):S^{-k}F\rightarrow S^{-k}G\,,$

for any $k\in\mathbb{N}$. We term this the $k^{\text{th}}$ left satellite
transformation of $\varphi$. Now remember that any connecting homomorphism
$\delta^{n}:T^{n}\rightarrow T^{n+1}$ is a natural transformation. We have
seen in Subsection 2 that it factors through
$\underline{\delta}^{n}:T^{n}(M)\rightarrow S^{-1}T^{n+1}(M)$ for any
$M\in\mathrm{obj}(\mathcal{C})$. Then P. H. Kropholler’s account [25, p.
206–208] suggests the following result.

######  Lemma 3.3.

For any $n\in\mathbb{Z}$, $\underline{\delta}^{n}:T^{n}\rightarrow
S^{-1}T^{n+1}$ is a natural transformation. In case that $0\rightarrow
A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact sequence in
$\mathcal{C}$, we take the connecting homomorphism
$\varepsilon^{-k}:S^{-k}T^{n+k}(C)\rightarrow S^{-k+1}T^{n+k}(A)$ for
$k\in\mathbb{N}_{0}$. Then the $n^{\text{th}}$ connecting homomorphism of the
satellite functor construction is given by

$\widehat{\delta}^{n}=\varinjlim_{k\in\mathbb{N}}\varepsilon^{-k}:\widehat{T}^{n}(C)\rightarrow\widehat{T}^{n+1}(A)$

where the morphisms $\varepsilon^{-k}$ are connected over
$S^{-k}\underline{\delta}^{n+k}:S^{-k}T^{n+k}\rightarrow S^{-k-1}T^{n+k+1}$.

######  Proof.

The proof is based on determining all terms of each morphism of cohomological
functors $\Phi_{k,k+1}^{\bullet}:T^{\bullet}\langle k\rangle\rightarrow
T^{\bullet}\langle k+1\rangle$ that give rise to the satellite functor
construction. We already know that
$\Phi_{k,k+1}^{n}=\mathrm{id}_{\mathcal{D}}$ for $k<n$. To describe
$\Phi_{k,k+1}^{k}$, consider a short exact sequence $0\rightarrow M\rightarrow
P\rightarrow C\rightarrow 0$ with $P$ projective. We note that
$\Phi_{k,k+1}^{\bullet}$ is meant to commute with the connecting homomorphisms
of $T^{\bullet}\langle k\rangle$ and $T^{\bullet}\langle k+1\rangle$, in
particular with the respective $k^{\text{th}}$ connecting homomorphisms

$\delta^{k}\langle k\rangle=\delta^{k}:T^{k}(C)\rightarrow T^{k+1}(A)\text{
and }\delta^{k}\langle k+1\rangle=\varepsilon^{-1}:S^{-1}T^{k+1}(C)\rightarrow
T^{k+1}(A)\,.$

Since $(T^{\bullet},\delta^{\bullet})$ is a cohomological functor, we obtain
from Diagram 3.2 the commutative diagram

${{}\dots}$${T^{k}(P)}$${T^{k}(C)}$${T^{k+1}(M)}$${T^{k+1}(P)}$${{}\dots}$${{}\dots}$${T^{k}(B)}$${T^{k}(C)}$${T^{k+1}(A)}$${T^{k+1}(B)}$${{}\dots}$$\scriptstyle{T^{k}(h)}$$\scriptstyle{\delta^{k}}$$\scriptstyle{\mathrm{id}_{T^{k}(C)}}$$\scriptstyle{T^{k+1}(h^{\ast})}$$\scriptstyle{T^{k+1}(h)}$$\scriptstyle{\delta^{k}}$
(3.5)

Recalling that $S^{-1}T^{k+1}(C)$ is a kernel and
$\varepsilon^{-1}:S^{-1}T^{k+1}(C)\rightarrow T^{k+1}(M)$ its canonical
monomorphism, we can write the middle square as

${T^{k}(C)}$${S^{-1}T^{k+1}(C)}$${T^{k+1}(M)}$${T^{k}(C)}$${T^{k+1}(A)}$$\scriptstyle{\underline{\delta}^{k}}$$\scriptstyle{\mathrm{id}_{T^{k}(C)}}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{T^{k+1}(h^{\ast})}$$\scriptstyle{\delta^{n}}$

By definition, $\delta^{k}\langle
k+1\rangle=\varepsilon^{-1}:S^{-1}T^{k+1}(C)\xrightarrow{\varepsilon^{-1}}T^{k+1}(M)\xrightarrow{T^{k+1}(h^{\ast})}T^{k+1}(A)$.
Tilting the above diagram, we obtain the desired commutative diagram

${T^{k}(C)}$${T^{k+1}(A)}$${S^{-1}T^{k+1}(C)}$${T^{k+1}(A)}$$\scriptstyle{\delta^{k}}$$\scriptstyle{\underline{\delta}^{k}}$$\scriptstyle{\mathrm{id}_{T^{k+1}(A)}}$$\scriptstyle{\varepsilon^{-1}}$
(3.6)

We conclude that $\underline{\delta}^{k}=\Phi_{k,k+1}^{k}:T^{k}\rightarrow
S^{-1}T^{k+1}$ is a natural transformation by the proof of Lemma 3.1 found in
[7, p. 46–47]. By the construction of left satellite transformations, we infer
that $\Phi_{k,k+1}^{n}=S^{n-k}\underline{\delta}^{k}:S^{n-k}T^{k}\rightarrow
S^{n-k+1}T^{k+1}$ for any $n<k$. In particular, the square

${S^{n-k}T^{k}(C)}$${S^{-k+1}T^{n}(A)}$${S^{n-k-1}T^{k+1}(C)}$${S^{n-k}T^{k+1}(A)}$$\scriptstyle{\varepsilon^{n-k}}$$\scriptstyle{S^{n-k}\underline{\delta}^{k}}$$\scriptstyle{S^{n-k+1}\underline{\delta}^{k}}$$\scriptstyle{\varepsilon^{n-k-1}}$
(3.7)

commutes. In the direct limit, these squares give rise to the connecting
homomorphism $\widehat{\delta}^{n}$ and the formula in the lemma follows from
reindexing as in Equation 3.4. $\square$

Lastly, we provide an explicit formula for the induced morphisms in the
satellite functor construction.

######  Proposition 3.4.

Let $f:A\rightarrow B$ be a morphism in $\mathcal{C}$. Then for any
$n\in\mathbb{Z}$ the commuting square

${S^{-k}T^{n+k}(A)}$${S^{-k}T^{n+k}(B)}$${S^{-k-1}T^{n+k+1}(A)}$${S^{-k-1}T^{n+k+1}(B)}$$\scriptstyle{S^{-k}T^{n+k}(f)}$$\scriptstyle{S^{-k}\underline{\delta}^{n+k}}$$\scriptstyle{S^{-k}\underline{\delta}^{n+k}}$$\scriptstyle{S^{-k-1}T^{n+k+1}(f)}$

give rise to
$\widehat{T}^{n}(f)=\varinjlim_{\mathcal{D},k\in\mathbb{N}_{0}}S^{-k}T^{n+k}(f):\widehat{T}(A)\rightarrow\widehat{T}(B)$.

######  Proof.

This follows from Equation 3.4 and the description of the cohomological
functors $\Phi_{n+k,n+k+1}^{\bullet}$ found in the proof of Lemma 3.3.
$\square$

##  4 The resolution and the satellite functor construction

The main goal of this section is to prove that the resolution construction
gives rise to Mislin completions of cohomological functors. First, we show in
Subsection 4.1 that the cohomology groups resulting from the satellite functor
construction are isomorphic to those from the resolution construction. In
Subsection 4.2, we provide explicit formulae for the induced morphisms and
connecting homomorphisms of the resolution construction. Then, in Subsection
4.3, we prove that the isomorphisms between the cohomology groups extend to
isomorphisms of cohomological functors. This sets the ground for Subsection
4.4 where we demonstrate that there are countably many distinct constructions
of Mislin completions.

###  4.1 Isomorphism of cohomology groups

To set up notation for the resolution construction, let
$M\in\mathrm{obj}(\mathcal{C})$, $(M_{n})_{n\in\mathbb{N}_{0}}$ be a
projective resolution of $M$ and $\widetilde{M}_{k}$ be the $k^{\text{th}}$
syzygy of $M_{\bullet}$ for $k\in\mathbb{N}_{0}$. Keeping our notation from
last section, we denote for $n\in\mathbb{Z}$ the $n^{\text{th}}$ term of the
satellite functor construction by $\widehat{T}^{n}(M)$. If the connecting
homomorphisms $\delta^{n+k}:T^{n+k}(\widetilde{M}_{k})\rightarrow
T^{n+k+1}(\widetilde{M}_{k+1})$ are taken as in Subsection 2, we write the
$n^{\text{th}}$ term of the resolution construction as
$T_{Res}^{n}(M):=\varinjlim_{k\in\mathbb{N}_{0}}\big{(}T^{n+k}(\widetilde{M}_{k}),\delta^{n+k}\big{)}$.
However it depends on a choice of projective resolution a priori. This issue
is resolved by demonstrating

######  Lemma 4.1.

For any $n\in\mathbb{Z}$ there is an isomorphism
$\omega_{n}(M):\widehat{T}^{n}(M)\rightarrow T_{Res}^{n}(M)$ in $\mathcal{D}$.

A proof of this lemma can be extracted from [28, p. 299] for modules over
rings and can be found in [19, p. 20] for unenriched relative Ext-functors. In
the following, we present an analogous proof for the cohomological functor
$T^{\bullet}:\mathcal{C}\rightarrow\mathcal{D}$. The manner in which we
conduct it allows us in Subsection 4.3 to extend the resulting isomorphism to
an isomorphism of cohomological functors.

######  Proof.

The proof consists of a two step endeavour where we introduce the relevant
notation throughout its course. In the first step, we construct an isomorphism

$\eta_{n}(M):\varinjlim_{k\in\mathbb{N}}(S^{-1}T^{n+k}(\widetilde{M}_{k-1}),\Sigma^{-1}\delta^{n+k})\rightarrow
T_{Res}^{n}(M)$

and in the second step we extend it to an isomorphism
$\omega_{n}(M):\widehat{T}^{n}(M)\rightarrow T_{Res}^{n}(M)$.

Step 1 Defining an isomorphism
$\eta_{n}(M):\varinjlim_{k\in\mathbb{N}}(S^{-1}T^{n+k}(\widetilde{M}_{k-1}),\Sigma^{-1}\delta^{n+k})\rightarrow
T_{Res}^{n}(M)$
By Axiom 2.2, the short exact sequence
$0\rightarrow\widetilde{M}_{k+1}\xrightarrow{\iota_{k+1}}M_{k}\xrightarrow{\pi_{k}}\widetilde{M}_{k}\rightarrow
0$ induces a long exact sequence in the cohomological functor
$(T^{\bullet},\delta^{\bullet})$ one of whose parts is

$T^{n+k}(\widetilde{M}_{k})\xrightarrow{\delta^{n+k}}T^{n+k+1}(\widetilde{M}_{k+1})\xrightarrow{T^{n+k+1}(\iota_{k+1})}T^{n+k+1}(M_{k})\xrightarrow{T^{n+k+1}(\pi_{k})}T^{n+k+1}(\widetilde{M}_{k})\,.$

Because $M_{k}$ is projective, it follows from the exactness of the above and
from the definition of left satellite functors that

$\mathrm{Im}(\delta^{n+k})=\mathrm{Ker}(T^{n+k+1}(\iota_{k+1}))=S^{-1}T^{n+k+1}(\widetilde{M}_{k})\,.$

This allows us to think of
$\underline{\delta}^{n+k}:T^{n+k}(\widetilde{M}_{k})\rightarrow
S^{-1}T^{n+k+1}(\widetilde{M}_{k})$ as being obtained by restricting the
codomain of $\delta^{n+k}$ onto the subobject
$S^{-1}T^{n+k+1}(\widetilde{M}_{k})$ of $T^{n+k+1}(\widetilde{M}_{k+1})$.
Because $\varepsilon^{-1}:S^{-1}T^{n+k}(\widetilde{M}_{k-1})\rightarrow
T^{n+k}(\widetilde{M}_{k})$ is the canonical monomorphism from a kernel, we
can interpret

$\Sigma^{-1}\delta^{n+k}:=\underline{\delta}^{n+k}\circ\varepsilon^{-1}:\;\;S^{-1}T^{n+k}(\widetilde{M}_{k-1})\rightarrow
T^{n+k}(\widetilde{M}_{k})\rightarrow S^{-1}T^{n+k+1}(\widetilde{M}_{k})$

as arising by restricting the domain of $\delta^{n+k}$ onto
$S^{-1}T^{n+k}(\widetilde{M}_{k-1})$ and its codomain onto
$S^{-1}T^{n+k+1}(\widetilde{M}_{k})$. Using this and Diagram 3.6 we obtain the
commuting diagram

${S^{-1}T^{n+k}(\widetilde{M}_{k-1})}$${T^{n+k}(\widetilde{M}_{k})}$${S^{-1}T^{n+k+1}(\widetilde{M}_{k})}$${T^{n+k+1}(\widetilde{M}_{k+1})}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{\Sigma^{-1}\delta^{n+k}}$$\scriptstyle{\underline{\delta}^{n+k}}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\varepsilon^{-1}}$
(4.1)

Since the above squares give rise to a direct system of morphisms, we define

$\eta_{n}(M)=\varinjlim\varepsilon^{-1}:\varinjlim_{k\in\mathbb{N}}(S^{-1}T^{n+k}(\widetilde{M}_{k-1}),\Sigma^{-1}\delta^{n+k})\rightarrow
T_{Res}^{n}(M)\,.$

If $N_{n+k}$ is the cokernel of the monomorphism
$\varepsilon^{-1}:S^{-1}T^{n+k}(\widetilde{M}_{k-1})\rightarrow
T^{n+k}(\widetilde{M}_{k})$, then it follows from its universal property that
we can extend Diagram 4.1 by a morphism $m_{n+k}:N_{n+k}\rightarrow N_{n+k+1}$
in a commutative manner as

${0}$${S^{-1}T^{n+k}(\widetilde{M}_{k-1})}$${T^{n+k}(\widetilde{M}_{k})}$${N_{n+k}}$${0}$${0}$${S^{-1}T^{n+k+1}(\widetilde{M}_{k})}$${T^{n+k+1}(\widetilde{M}_{k+1})}$${N_{n+k+1}}$${0}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{\Sigma^{-1}\delta^{n+k}}$$\scriptstyle{q_{n+k}}$$\scriptstyle{\underline{\delta}^{n+k}}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{m_{n+k}}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{q_{n+k+1}}$
(4.2)

However, note that

$m_{n+k}\circ
q_{n+k}=q_{n+k+1}\circ\varepsilon^{-1}\circ\underline{\delta}^{n+k}:T^{n+k}(\widetilde{M}_{k})\rightarrow
N_{n+k+1}$

is the zero morphism. Since $q_{n+k}$ is surjective, we deduce that
$m_{n+k}=0$. Given that these diagrams form a direct system of short exact
sequences and countable direct limits in $\mathcal{D}$ are exact, we obtain a
short exact sequence

$0\rightarrow\varinjlim_{k\in\mathbb{N}}(S^{-1}T^{n+k}(\widetilde{M}_{n+k}),\Sigma^{-1}\delta^{n+k})\xrightarrow{\eta_{n}(M)}T_{Res}^{n}(M)\rightarrow
0\rightarrow 0\,.$

Hence, $\eta_{n}(M)$ is an isomorphism.

Step 2 Defining an isomorphism $\omega_{n}(M):\widehat{T}^{n}(M)\rightarrow
T_{Res}^{n}(M)$
The way we construct this isomorphism is by connecting the terms of the direct
system
$(S^{-k}T^{n+k}(M),S^{-k}\underline{\delta}^{n+k})_{k\in\mathbb{N}_{0}}$ to
the ones of $(T^{n+k}(\widetilde{M}_{k}),\delta^{n+k})_{k\in\mathbb{N}_{0}}$.
This requires a larger commutative diagram for which we introduce all the
necessary morphisms. By the previous step, we can extend Diagram 3.7 to the
commutative diagram

${S^{-1}T^{n+k+2}(\widetilde{M}_{k+1})}$${T^{n+k+2}(\widetilde{M}_{k+2})}$${S^{-2}T^{n+k+3}(\widetilde{M}_{k+1})}$${S^{-1}T^{n+k+3}(\widetilde{M}_{k+2})}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{\Sigma^{-1}\delta^{n+k+2}}$$\scriptstyle{S^{-1}\underline{\delta}^{n+k+2}}$$\scriptstyle{\underline{\delta}^{n+k+2}}$$\scriptstyle{\varepsilon^{-2}}$

Since $\varepsilon^{-2}:S^{-2}T^{n+k+3}(\widetilde{M}_{k+1})\rightarrow
S^{-1}T^{n+k+3}(\widetilde{M}_{k+2})$ is the monomorphism of a kernel, we can
think of
$S^{-1}\underline{\delta}^{n+k+2}:S^{-1}T^{n+k+2}(\widetilde{M}_{k+1})\rightarrow
S^{-2}T^{n+k+3}(\widetilde{M}_{k+1})$ as being obtained by restricting the
codomain of $\Sigma^{-1}\delta^{n+k+2}$ onto the subobject
$S^{-2}T^{n+k+3}(\widetilde{M}_{k+1})$ of
$S^{-1}T^{n+k+3}(\widetilde{M}_{k+2})$. Furthermore, we can interpret the
morphism

$\displaystyle\Sigma^{-2}\delta^{n+k+2}$
$\displaystyle:=S^{-1}\underline{\delta}^{n+k+2}\circ\varepsilon^{-2}:$
$\displaystyle S^{-2}T^{n+k+2}(\widetilde{M}_{k})\rightarrow
S^{-1}T^{n+k+2}(\widetilde{M}_{k+1})\rightarrow
S^{-2}T^{n+k+3}(\widetilde{M}_{k+1})$

as arising from restricting the domain of $\Sigma^{-1}\delta^{n+k+2}$ onto
$S^{-2}T^{n+k+2}(\widetilde{M}_{k})$ and its codomain onto
$S^{-2}T^{n+k+3}(\widetilde{M}_{k+1})$. As for left satellite functors, we may
define $\Sigma^{0}{\delta^{n+k}}:=\delta^{n+k}$ and

$\displaystyle\Sigma^{-l}\delta^{n+k+l}$
$\displaystyle:=S^{-l+1}\underline{\delta}^{n+k+l}\circ\varepsilon^{-l}:$
(4.3) $\displaystyle S^{-l}T^{n+k+l}(\widetilde{M}_{k})\rightarrow
S^{-l+1}T^{n+k+l}(\widetilde{M}_{k+1})\rightarrow
S^{-l}T^{n+k+l+1}(\widetilde{M}_{k+1})$

for any $l\in\mathbb{N}$. We can think of $\Sigma^{-l}\delta^{n+k+l}$ as being
obtained by restricting the domain of $\Sigma^{-l+1}\delta^{n+k+l-1}$ onto
$S^{-l}T^{n+k+l}(\widetilde{M}_{k})$ and its codomain onto
$S^{-l}T^{n+k+l+1}(\widetilde{M}_{k+1})$. We can again extend Diagram 3.7 to
another commutative diagram of the form

${S^{-l}T^{n+k+l}(\widetilde{M}_{k})}$${S^{-l+1}T^{n+k+l}(\widetilde{M}_{k+1})}$${S^{-l-1}T^{n+k+l+1}(\widetilde{M}_{k})}$${S^{-l}T^{n+k+l+1}(\widetilde{M}_{k+1})}$$\scriptstyle{\varepsilon^{-l}}$$\scriptstyle{\Sigma^{-l}\delta^{n+k+l}}$$\scriptstyle{S^{-l}\underline{\delta}^{n+k+l}}$$\scriptstyle{S^{-l+1}\underline{\delta}^{n+k+l}}$$\scriptstyle{\varepsilon^{-l-1}}$
(4.4)

By [7, Section III.2], for every $2\leq l\in\mathbb{N}$ and
$k\in\\{0,\dots,l-1\\}$ the short exact sequence
$0\rightarrow\widetilde{M}_{k+1}\rightarrow
M_{k}\rightarrow\widetilde{M}_{k}\rightarrow 0$ gives rise to a long exact
sequence one of whose parts is

$S^{-l}T^{n+k+l}(M_{k})\rightarrow
S^{-l}T^{n+k+l}(\widetilde{M}_{k})\xrightarrow{\varepsilon^{-l}}S^{-l+1}T^{n+k+l}(\widetilde{M}_{k+1})\rightarrow
S^{-l+1}T^{n+k+l}(M_{k})\,.$

By construction, left satellite functors vanish on projective objects such as
$M_{k}$. Hence the morphism
$\varepsilon^{-l}:S^{-l}T^{n+k+l}(\widetilde{M}_{k})\rightarrow
S^{-l+1}T^{n+k+l}(\widetilde{M}_{k+1})$ is an isomorphism. Define the
isomorphisms

$\displaystyle\widetilde{o}^{k}:=\varepsilon^{-3}\circ{}\dots\circ\varepsilon^{-k+1}\circ\varepsilon^{-k}:\;\;{}$
$\displaystyle S^{-k}T^{n+k}(M)\rightarrow S^{-2}T^{n+k}(\widetilde{M}_{k-2})$
$\displaystyle\text{and}\quad
o^{k}:=\varepsilon^{-2}\circ\widetilde{o}^{k}:\;\;{}$ $\displaystyle
S^{-k}T^{n+k}(M)\rightarrow S^{-1}T^{n+k}(\widetilde{M}_{k-1})\,.$ (4.5)

We can form the following diagram with all morphisms we have introduced thus
far:

${{\scriptstyle T^{n}(M)}}$${{\scriptstyle S^{-1}T^{n+1}(M)}}$${{\scriptstyle
T^{n+1}(\widetilde{M}_{1})}}$${{\scriptstyle
S^{-2}T^{n+2}(M)}}$${{\scriptstyle
S^{-1}T^{n+2}(\widetilde{M}_{1})}}$${{\scriptstyle
T^{n+2}(\widetilde{M}_{2})}}$${{\scriptstyle
S^{-3}T^{n+3}(M)}}$${{\scriptstyle
S^{-2}T^{n+3}(\widetilde{M}_{1})}}$${{\scriptstyle
S^{-1}T^{n+3}(\widetilde{M}_{2})}}$${{\scriptstyle
T^{n+3}(\widetilde{M}_{3})}}$${{\scriptstyle
S^{-4}T^{n+4}(M)}}$${{\scriptstyle
S^{-2}T^{n+4}(\widetilde{M}_{2})}}$${{\scriptstyle
S^{-1}T^{n+4}(\widetilde{M}_{3})}}$${{\scriptstyle
T^{n+4}(\widetilde{M}_{4})}}$${{}{\scriptstyle\vdots}{}}$${{}{\scriptstyle\vdots}{}}$${{}{\scriptstyle\vdots}{}}$${{}{\scriptstyle\vdots}{}}$${{\scriptstyle
S^{-k}T^{n+k}(M)}}$${{\scriptstyle
S^{-2}T^{n+k}(\widetilde{M}_{k-2})}}$${{\scriptstyle
S^{-1}T^{n+k}(\widetilde{M}_{k-1})}}$${{\scriptstyle
T^{n+k}(\widetilde{M}_{k})}}$${{\scriptstyle
S^{-k-1}T^{n+k+1}(M)}}$${{\scriptstyle
S^{-2}T^{n+k+1}(\widetilde{M}_{k-1})}}$${{\scriptstyle
S^{-1}T^{n+k+1}(\widetilde{M}_{k})}}$${{\scriptstyle
T^{n+k+1}(\widetilde{M}_{k+1})}}$${{}{\scriptstyle\vdots}{}}$${{}{\scriptstyle\vdots}{}}$${{}{\scriptstyle\vdots}{}}$${{}{\scriptstyle\vdots}{}}$$\scriptstyle{\scriptscriptstyle\delta^{n}}$$\scriptstyle{\scriptscriptstyle\underline{\delta}^{n}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-1}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-1}\delta^{n+1}}$$\scriptstyle{\scriptscriptstyle
S^{-1}\underline{\delta}^{n+1}}$$\scriptstyle{\scriptscriptstyle\delta^{n+1}}$$\scriptstyle{\scriptscriptstyle\underline{\delta}^{n+1}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-2}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-2}\delta^{n+2}}$$\scriptstyle{\scriptscriptstyle
S^{-2}\underline{\delta}^{n+2}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-1}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-1}\delta^{n+2}}$$\scriptstyle{\scriptscriptstyle
S^{-1}\underline{\delta}^{n+2}}$$\scriptstyle{\scriptscriptstyle\delta^{n+2}}$$\scriptstyle{\scriptscriptstyle\underline{\delta}^{n+2}}$$\scriptstyle{\scriptscriptstyle\widetilde{o}^{3}}$$\scriptstyle{\scriptscriptstyle
S^{-3}\underline{\delta}^{n+3}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-2}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-2}\delta^{n+3}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-1}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-1}\delta^{n+3}}$$\scriptstyle{\scriptscriptstyle
S^{-1}\underline{\delta}^{n+3}}$$\scriptstyle{\scriptscriptstyle\delta^{n+3}}$$\scriptstyle{\scriptscriptstyle\underline{\delta}^{n+3}}$$\scriptstyle{\scriptscriptstyle\widetilde{o}^{4}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-2}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-1}}$$\scriptstyle{\scriptscriptstyle\widetilde{o}^{k}}$$\scriptstyle{\scriptscriptstyle
o^{k}}$$\scriptstyle{\scriptscriptstyle
S^{-k}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-2}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-2}\delta^{n+k}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-1}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-1}\delta^{n+k}}$$\scriptstyle{\scriptscriptstyle
S^{-1}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptscriptstyle\delta^{n+k}}$$\scriptstyle{\scriptscriptstyle\underline{\delta}^{n+k}}$$\scriptstyle{\scriptscriptstyle\widetilde{o}^{k+1}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-2}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-1}}$
(4.6)

We need to argue why this diagram commutes before we can demonstrate how it
yields the desired isomorphism. The triangles at the very right side
containing the morphisms $\delta^{n+k}:T^{n+k}(\widetilde{M}_{k})\rightarrow
T^{n+k+1}(\widetilde{M}_{k+1})$ commute by Diagram 3.6. All other triangles
commute due to Diagram 4.4. Finally, due to Diagram 3.7 and 4.4 the following
diagram commutes for any $k\geq 3$.

${{\scriptstyle S^{-k}T^{n+k}(M)}}$${{\scriptstyle
S^{-k+1}T^{n+k}(\widetilde{M}_{1})}}$${{}{\scriptstyle\dots}{}}$${{\scriptstyle
S^{-k-1}T^{n+k+1}(M)}}$${{\scriptstyle
S^{-k}T^{n+k+1}(\widetilde{M}_{1})}}$${{}{\scriptstyle\dots}{}}$${{}{\scriptstyle\dots}{}}$${{\scriptstyle
S^{-2}T^{n+k}(\widetilde{M}_{k-2})}}$${{\scriptstyle
S^{-1}T^{n+k}(\widetilde{M}_{k-1})}}$${{}{\scriptstyle\dots}{}}$${{\scriptstyle
S^{-3}T^{n+k+1}(\widetilde{M}_{k-2})}}$${{\scriptstyle
S^{-2}T^{n+k+1}(\widetilde{M}_{k-1})}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-k}}$$\scriptstyle{\scriptscriptstyle
o^{k}}$$\scriptstyle{\scriptscriptstyle
S^{-k}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-k+1}}$$\scriptstyle{\scriptscriptstyle
S^{-k+1}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-k-1}}$$\scriptstyle{\scriptscriptstyle\widetilde{o}^{k+1}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-k}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-3}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-2}}$$\scriptstyle{\scriptscriptstyle\Sigma^{-2}\delta^{n+k}}$$\scriptstyle{\scriptscriptstyle
S^{-2}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptscriptstyle
S^{-1}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-4}}$$\scriptstyle{\scriptscriptstyle\varepsilon^{-3}}$
(4.7)

We conclude from it that
$\Sigma^{-2}\delta^{n+k}\circ\widetilde{o}^{k}=\widetilde{o}^{k-1}\circ
S^{-k}\underline{\delta}^{n+k}$. In particular, Diagram 4.6 commutes as all
its stand-alone squares also do so.

We see in Diagram 4.6 that the right most column corresponds to the direct
system $(T^{n+k}(\widetilde{M}_{k}),\delta^{n+k})_{k\in\mathbb{N}_{0}}$ while
the top diagonal together with the left most column forms the direct system
$(S^{-k}T^{n+k}(M),S^{-k}\underline{\delta}^{n+k})_{k\in\mathbb{N}_{0}}$. The
right most squares in the diagram recover the isomorphism

$\eta_{n}(M)=\varinjlim\varepsilon^{-1}:\varinjlim_{k\in\mathbb{N}}(S^{-1}T^{n+k}(\widetilde{M}_{k-1}),\Sigma^{-1}\delta^{n+k})\rightarrow
T_{Res}^{n}(M)\,.$

For $k\geq 3$, all the remaining squares yield together squares of the form

${S^{-k}T^{n+k}(M)}$${S^{-1}T^{n+k}(\widetilde{M}_{k-1})}$${S^{-k-1}T^{n+k+1}(M)}$${S^{-1}T^{n+k+1}(\widetilde{M}_{k})}$$\scriptstyle{o^{k}}$$\scriptstyle{S^{-k}\underline{\delta}^{n+k}}$$\scriptstyle{\Sigma^{-1}\delta^{n+k}}$$\scriptstyle{o^{k+1}}$

As the morphisms $o^{k}$ are isomorphisms and countable direct limits in
$\mathcal{D}$ are exact, this gives rise to the isomorphism

$\varinjlim
o^{k}:\widehat{T}^{n}(M)\rightarrow\varinjlim_{k\in\mathbb{N}}(S^{-1}T^{n+k}(\widetilde{M}_{k-1}),\Sigma^{-1}\delta^{n+k})\,.$

Due to the universal property of direct limits, we observe that $\varinjlim
o^{k}\circ\varinjlim\varepsilon^{-1}=\varinjlim o^{k}\circ\varepsilon^{-1}$.
Because $\eta_{n}(M)=\varinjlim\varepsilon^{-1}$ is already an isomorphism,

$\omega_{n}(M)=\varinjlim
o^{k}\circ\varepsilon^{-1}:\widehat{T}^{n}(M)\rightarrow T_{Res}^{n}(M)$

is our desired isomorphism. Since the morphisms $o^{k}\circ\varepsilon^{-1}$
span the entire width of Diagram 4.6, every single morphism was needed to
construct this isomorphism. $\square$

###  4.2 Induced morphisms and connecting homomorphisms

The first step to turn $\omega_{n}(M):\widehat{T}^{n}(M)\rightarrow
T_{Res}^{n}(M)$ into an isomorphism of cohomological functor is to show that
it is natural. Let $f:M\rightarrow N$ be a morphism in $\mathcal{C}$ and
$M_{\bullet}$, $N_{\bullet}$ be projective resolutions of $M$, $N$. Set
$\widetilde{f}_{0}:=f:\widetilde{M}_{0}\rightarrow\widetilde{N}_{0}$ and
assume that
$\widetilde{f}_{k+1}:\widetilde{M}_{k}\rightarrow\widetilde{N}_{k}$ has been
defined for $k\geq 0$. Let us denote by
$\pi_{k}:M_{k}\rightarrow\widetilde{M}_{k}$ the epimorphism from the
projective $M_{k}$ onto $\widetilde{M}_{k}$ and by
$\iota_{k+1}:\widetilde{M}_{k+1}\rightarrow M_{k}$ the monomorphism from the
$(k+1)^{\text{st}}$ syzygy $\widetilde{M}_{k+1}$ into $M_{k}$. We adopt an
analogous convention for $\widetilde{N}_{k+1}$, $N_{k}$ and
$\widetilde{N}_{k}$. Then we can lift $\widetilde{f}_{k}$ to fit in the
commutative diagram

${0}$${\widetilde{M}_{k+1}}$${M_{k}}$${\widetilde{M}_{k}}$${0}$${0}$${\widetilde{N}_{k+1}}$${N_{k}}$${\widetilde{N}_{k}}$${0}$$\scriptstyle{\iota_{k+1}}$$\scriptstyle{\widetilde{f}_{k+1}}$$\scriptstyle{\pi_{k}}$$\scriptstyle{f_{k}}$$\scriptstyle{\widetilde{f}_{k}}$$\scriptstyle{\iota_{k+1}}$$\scriptstyle{\pi_{k}}$
(4.8)

since $M_{0}$ is a projective object and
$\iota_{k+1}:\widetilde{N}_{k+1}\rightarrow N_{k}$ can be taken to be the
kernel of $\pi_{k}$.

######  Proposition 4.2.

Let $f:M\rightarrow N$ and
$(\widetilde{f}_{k}:\widetilde{M}_{k}\rightarrow\widetilde{N}_{k})_{k\in\mathbb{N}_{0}}$
be a sequence of morphism as arising from Diagram 4.8. As
$(T^{\bullet},\delta^{\bullet})$ is a cohomological functor, we denote the
morphism in the direct limit of the commuting squares

${T^{n+k}(\widetilde{M}_{k})}$${T^{n+k}(\widetilde{N}_{k})}$${T^{n+k+1}(\widetilde{M}_{k+1})}$${T^{n+k+1}(\widetilde{N}_{k+1})}$$\scriptstyle{T^{n+k}(\widetilde{f}_{k})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{T^{n+k+1}(\widetilde{f}_{k+1})}$

by
$T_{Res}^{n}(f):=\varinjlim_{k\in\mathbb{N}_{0}}T^{n+k}(\widetilde{f}_{k})$.
Then the square

${\widehat{T}^{n}(M)}$${T_{Res}^{n}(M)}$${\widehat{T}^{n}(N)}$${T_{Res}^{n}(N)}$$\scriptstyle{\omega_{n}(M)}$$\scriptstyle{\widehat{T}^{n}(f)}$$\scriptstyle{T_{Res}^{n}(f)}$$\scriptstyle{\omega_{n}(N)}$

commutes. In particular, every term $T_{Res}^{n}(-)$ forms an additive functor
and the homomorphisms $\omega_{n}:\widehat{T}^{n}\rightarrow T_{Res}^{n}$ a
natural isomorphism.

######  Proof.

Recall that
$\widehat{T}^{n}(f)=\varinjlim_{k\in\mathbb{N}_{0}}S^{k}T^{n+k}(f)$ by
Proposition 3.4. Let $k\in\mathbb{N}_{0}$, $l\in\mathbb{N}$ and write
$\Lambda=n+k+l$. Consider the diagram

${{\scriptstyle S^{-l}T^{\Lambda}(\widetilde{M}_{k})}}$${{\scriptstyle
S^{-l+1}T^{\Lambda}(\widetilde{M}_{k+1})}}$${{\scriptstyle
S^{-l}T^{\Lambda}(\widetilde{N}_{k})}}$${{\scriptstyle
S^{-l+1}T^{\Lambda}(\widetilde{N}_{k+1})}}$${{\scriptstyle
S^{-l-1}T^{\Lambda+1}(\widetilde{M}_{k})}}$${{\scriptstyle
S^{-l}T^{\Lambda+1}(\widetilde{M}_{k+1})}}$${{\scriptstyle
S^{-l-1}T^{\Lambda+1}(\widetilde{N}_{k})}}$${{\scriptstyle
S^{-l}T^{\Lambda+1}(\widetilde{N}_{k+1})}}$$\scriptstyle{\scriptstyle
S^{-l}T^{\Lambda}(\widetilde{f}_{k})}$$\scriptstyle{\scriptstyle\varepsilon^{-l}}$$\scriptstyle{\scriptstyle
S^{-l}\underline{\delta}^{\Lambda}}$$\scriptstyle{\scriptstyle
S^{-l+1}T^{\Lambda}(\widetilde{f}_{k+1})}$$\scriptstyle{\scriptstyle
S^{-l+1}\underline{\delta}^{\Lambda}}$$\scriptstyle{\scriptstyle\varepsilon^{-l}}$$\scriptstyle{\scriptstyle
S^{-l}\underline{\delta}^{\Lambda}}$$\scriptstyle{\scriptstyle
S^{-l-1}T^{\Lambda+1}(\widetilde{f}_{k})}$$\scriptstyle{\scriptstyle\varepsilon^{-l-1}}$$\scriptstyle{\scriptstyle
S^{-l+1}T^{\Lambda+1}(\widetilde{f}_{k+1})}$$\scriptstyle{\scriptstyle\varepsilon^{-l-1}}$$\scriptstyle{\scriptstyle
S^{-l+1}\underline{\delta}^{\Lambda}}$

The front and back side commute by Diagram 4.4. The left hand and right hand
side commute since $S^{-l}\underline{\delta}^{\Lambda}$ and
$S^{-l+1}\underline{\delta}^{\Lambda}$ are natural transformations by Lemma
3.3. Lastly, the top and bottom side commute by Equation 3.3. Remember that we
have defined the morphisms
$o^{k}=\varepsilon^{-2}\circ\varepsilon^{-3}\circ{}\dots\circ\varepsilon^{-k}$
in the proof of Lemma 4.1. Because $\delta^{n+k}$ is also a natural
transformation, we conclude from Diagram 4.6 and the above cube that the
diagram

${{\scriptstyle S^{-k}T^{n+k}(M)}}$${{\scriptstyle
T^{n+k}(\widetilde{M}_{k})}}$${{\scriptstyle
S^{-k}T^{n+k}(N)}}$${{\scriptstyle
T^{n+k}(\widetilde{N}_{k})}}$${{\scriptstyle
S^{-k-1}T^{n+k+1}(M)}}$${{\scriptstyle
T^{n+k+1}(\widetilde{M}_{k+1})}}$${{\scriptstyle
S^{-k-1}T^{n+k+1}(N)}}$${{\scriptstyle
T^{n+k+1}(\widetilde{N}_{k+1})}}$$\scriptstyle{\scriptstyle
S^{-k}T^{n+k}(f)}$$\scriptstyle{\scriptstyle\varepsilon^{-1}\circ\;o^{k}}$$\scriptstyle{\scriptstyle
S^{-k}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptstyle
T^{n+k}(\widetilde{f}_{k})}$$\scriptstyle{\scriptstyle\delta^{n+k}}$$\scriptstyle{\scriptstyle\varepsilon^{-1}\circ\;o^{k}}$$\scriptstyle{\scriptstyle
S^{-k}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptstyle
S^{-k-1}T^{n+k+1}(f)}$$\scriptstyle{\scriptstyle\varepsilon^{-1}\circ\;o^{k+1}}$$\scriptstyle{\scriptstyle
T^{n+k+1}(\widetilde{f}_{k+1})}$$\scriptstyle{\scriptstyle\varepsilon^{-1}\circ\;o^{k+1}}$$\scriptstyle{\scriptstyle\delta^{n+k}}$

is commutative. Referring again to the proof of Lemma 4.1, we recall that we
have defined $\omega_{n}=\varinjlim_{k\in\mathbb{N}}\varepsilon^{-1}\circ
o^{k}:\widehat{T}^{n}\rightarrow T_{Res}^{n}$. Therefore, the bottom square of
the proposition commutes and $T_{Res}^{n}(f)$ is well defined in the sense
that it does not depend on the choice of sequence
$(\widetilde{f}_{k}:\widetilde{M}_{k}\rightarrow\widetilde{N}_{k})_{k\in\mathbb{N}_{0}}$.
Because every term $\widehat{T}^{n}$ is an additive functor, any $T_{Res}^{n}$
is so too. Or more directly, for any $g:M\rightarrow N$ and any correspondign
sequence
$(\widetilde{g}_{k}:\widetilde{M}_{k}\rightarrow\widetilde{N}_{k})_{k\in\mathbb{N}_{0}}$
we see that

$T_{Res}^{n}(f+g)=\varinjlim_{k\in\mathbb{N}_{0}}T^{n+k}(\widetilde{f}_{k}+\widetilde{g}_{k})=\varinjlim_{k\in\mathbb{N}_{0}}\big{(}T^{n+k}(\widetilde{f}_{k})+T^{n+k}(\widetilde{g}_{k})\big{)}=T_{Res}^{n}(f)+T_{Res}^{n}(g)\,.$

$\square$

In order to turn $T_{Res}^{\bullet}$ into a cohomological functor, we need to
construct connecting homomorphisms. For this let
$0\rightarrow\widetilde{A}_{k}\rightarrow\widetilde{B}_{k}\rightarrow\widetilde{C}_{k}\rightarrow
0$ be a short exact sequence in $\mathcal{C}$. By the Horseshoe Lemma [42, p.
37], there is a commutative diagram

${0}$${0}$${0}$${0}$${\widetilde{A}_{k+1}}$${\widetilde{B}_{k+1}}$${\widetilde{C}_{k+1}}$${0}$${0}$${A_{k}}$${B_{k}}$${C_{k}}$${0}$${0}$${\widetilde{A}_{k}}$${\widetilde{B}_{k}}$${\widetilde{C}_{k}}$${0}$${0}$${0}$${0}$$\scriptstyle{\widetilde{f}_{k+1}}$$\scriptstyle{\widetilde{g}_{k+1}}$$\scriptstyle{f_{k}}$$\scriptstyle{g_{k}}$$\scriptstyle{\widetilde{f}_{k}}$$\scriptstyle{\widetilde{g}_{k}}$
(4.9)

where $A_{k}$, $B_{k}$ and $C_{k}$ are projective and all rows and columns are
exact. Then one can generalise Proposition III.4.1 of H. Cartan and S.
Eilenberg’s book [7] from modules over a ring to abelian categories either by
turning all diagram chases into diagrams involving the respective kernels,
cokernels and morphism or by invoking the Freyd-Mitchell Embedding Theorem
that can be found in [42, p. 25]. This says that the square

${T^{n+k}(\widetilde{C}_{k})}$${T^{n+k+1}(\widetilde{A}_{k})}$${T^{n+k+1}(\widetilde{C}_{k+1})}$${T^{n+k+2}(\widetilde{A}_{k+1})}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\delta^{n+k+1}}$$\scriptstyle{\delta^{n+k+1}}$
(4.10)

anticommutes for any $n\in\mathbb{Z}$ and $k\in\mathbb{N}_{0}$, meaning that
the composite morphisms in the abelian group
$\mathrm{Hom}_{\mathcal{D}}(T^{n+k}(\widetilde{C}_{k}),T^{n+k+2}(\widetilde{A}_{k+1}))$
have opposite signs.

######  Definition 4.3.

For a short exact sequence $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ in $\mathcal{C}$ we define the (proposed) connecting
homomorphism of $T_{Res}^{\bullet}$ to be

$\widetilde{\delta}^{n}:=\varinjlim_{k\in\mathbb{N}_{0}}(-1)^{k}\delta^{n+k}:T_{Res}^{n}(C)\rightarrow
T_{Res}^{n+1}(A)$

where the morphisms $(-1)^{k}\delta^{n+k}$ are connected over the morphisms
$\delta^{n+k}$ and $\delta^{n+k+1}$ from Diagram 4.10.

###  4.3 Isomorphisms of cohomological functors

The key step in establishing that
$(T_{Res}^{\bullet},\widetilde{\delta}^{\bullet})$ is a cohomological functor
and proving that $\omega_{\bullet}:\widehat{T}^{\bullet}\rightarrow
T_{Res}^{\bullet}$ extends to an isomorphism of cohomological functors is

######  Lemma 4.4.

Denote by $\widehat{\delta}^{\bullet}$ the connecting homomorphism from the
satellite functor construction $\widehat{T}^{\bullet}$. Then for any short
exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ in
$\mathcal{C}$ and any $n\in\mathbb{Z}$ the diagram

${\widehat{T}^{n}(C)}$${T_{Res}^{n}(C)}$${\widehat{T}^{n+1}(A)}$${T_{Res}^{n+1}(A)}$$\scriptstyle{\omega_{n}}$$\scriptstyle{\widehat{\delta}^{n}}$$\scriptstyle{\widetilde{\delta}^{n}}$$\scriptstyle{\omega_{n+1}}$

commutes.

######  Proof.

The proof is based on showing that the diagram on the next page is
commutative. The front side is a copy of Diagram 4.6 as well as the back side
up to its commuting right most column. The right hand side of the diagram
gives rise to $\widetilde{\delta}^{n}:T_{Res}^{n}(C)\rightarrow
T_{Res}^{n+1}(A)$ in the direct limit and the top and left hand side to
$\widehat{\delta}^{n}:\widehat{T}^{n}(C)\rightarrow\widehat{T}^{n+1}(A)$. In
particular, all squares and triangles from these sides commute. Using the
cohomological functors defined in Equation 3.3 we see that the horizontal
squares commute by [7, Proposition III.4.1].

(4.11)

To investigate the vertical squares running from back to front, we can insert
commuting triangles of the form

${S^{-l}T^{n+k}(\widetilde{M}_{k-l})}$${S^{-l-1}T^{n+k+1}(\widetilde{M}_{k-l})}$${S^{-l}T^{n+k+1}(\widetilde{M}_{k-l+1})}$$\scriptstyle{\Sigma^{-l}\delta^{n+k}}$$\scriptstyle{S^{-l}\underline{\delta}^{n+k}}$$\scriptstyle{\varepsilon^{-l-1}}$

into the front and back side by Diagram 4.6. These give rise to prisms of the
form

${{\scriptstyle S^{-l+1}T^{n+k}(\widetilde{A}_{k-l})}}$${{\scriptstyle
S^{-l}T^{n+k}(\widetilde{C}_{k-l})}}$${{\scriptstyle
S^{-l}T^{n+k+1}(\widetilde{A}_{k-l})}}$${{\scriptstyle
S^{-l+1}T^{n+k+1}(\widetilde{A}_{k-l+1})}}$${{\scriptstyle
S^{-l-1}T^{n+k+1}(\widetilde{C}_{k-l})}}$${{\scriptstyle
S^{-l}T^{n+k+1}(\widetilde{C}_{k-l+1})}}$$\scriptstyle{\scriptstyle\Sigma^{-l+1}\delta^{n+k}}$$\scriptstyle{\scriptstyle
S^{-l+1}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptstyle(-1)^{k-l}\varepsilon^{-l}}$$\scriptstyle{\scriptstyle\Sigma^{-l}\delta^{n+k}}$$\scriptstyle{\scriptstyle\varepsilon^{-l}}$$\scriptstyle{\scriptstyle(-1)^{k-l}\varepsilon^{-l-1}}$$\scriptstyle{\scriptstyle\varepsilon^{-l-1}}$$\scriptstyle{\scriptstyle
S^{-l}\underline{\delta}^{n+k}}$$\scriptstyle{\scriptstyle(-1)^{k-l+1}\varepsilon^{-l}}$

where their right hand square correspond to the vertical squares in Diagram
4.11. These squares commute because the bottom square of the prism does so and
the top left square by Diagram 3.7. In particular, Diagram 4.11 commutes.

Denote by $\chi=\varinjlim\delta^{n+k+1}:T_{Res}^{n+1}(A)\rightarrow
T_{Res}^{n+1}(A)$ the morphism resulting from the right most column of the
back side of Diagram 4.11 and write
$\Psi_{n+1}^{k}:T^{n+k+1}(\widetilde{A}_{k})\rightarrow T_{Res}^{n+1}(A)$ for
the morphism to the direct limit. Since every square in this column can be
written as

${T^{n+k+1}(\widetilde{A}_{k})}$${T^{n+k+2}(\widetilde{A}_{k+1})}$${T^{n+k+2}(\widetilde{A}_{k+1})}$${T^{n+k+3}(\widetilde{A}_{k+2})}$$\scriptstyle{\delta^{n+k+1}}$$\scriptstyle{\delta^{n+k+1}}$$\scriptstyle{\delta^{n+k+2}}$$\scriptstyle{\delta^{n+k+2}}$$\scriptstyle{\mathrm{id}}$
(4.12)

we deduce that the diagram

${T^{n+k+2}(\widetilde{A}_{k+1})}$${T_{Res}^{n+1}(A)}$${T^{n+k+2}(\widetilde{A}_{k+1})}$${T^{n+k+3}(\widetilde{A}_{k+2})}$$\scriptstyle{\delta^{n+k+2}}$$\scriptstyle{\Psi_{n+1}^{k+1}}$$\scriptstyle{\delta^{n+k+2}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\Psi_{n+1}^{k+2}}$
(4.13)

commutes. Observing that
$(T^{n+k+1}(\widetilde{A}_{k}),\delta^{n+k+1})_{k\in\mathbb{N}}$ is cofinal in
$(T^{n+k+1}(\widetilde{A}_{k}),\delta^{n+k+1})_{k\in\mathbb{N}_{0}}$, we
conclude from the previous two diagrams that
$\mathrm{id}=\chi:T_{Res}^{n+1}(A)\rightarrow T_{Res}^{n+1}(A)$.

Let us consider the top and left most edge in the front side of Diagram 4.11
yielding $\widehat{T}^{n}(C)$. Going over the front side yields the morphism
$\omega_{n}:\widehat{T}^{n}(C)\rightarrow T_{Res}^{n}(C)$. Through the right
hand side we obtain $\widetilde{\delta}^{n}:T_{Res}^{n}(C)\rightarrow
T_{Res}^{n+1}(A)$. Thus, going around these sides yields
$\widetilde{\delta}^{n}\circ\omega_{n}$. Instead, if we go over the left hand
side first, we obtain the connecting homomorphism
$\widehat{\delta}^{n}:\widehat{T}^{n}(C)\rightarrow\widehat{T}^{n+1}(A)$.
Going subsequently over the back side yields first
$\omega_{n+1}:\widehat{T}^{n+1}(A)\rightarrow T_{Res}^{n+1}(A)$ and then
$\chi:T_{Res}^{n+1}(A)\rightarrow T_{Res}^{n+1}(A)$. Hence, going around those
sides, we obtain $\chi\circ\omega_{n+1}\circ\widehat{\delta}^{n}$.Because
Diagram 4.11 commutes, we conclude that
$\widetilde{\delta}^{n}\circ\omega_{n}=\chi\circ\omega_{n+1}\circ\widehat{\delta}^{n}$.
$\square$

######  Theorem 4.5.

The resolution construction $(T_{Res}^{\bullet},\widetilde{\delta}^{\bullet})$
forms a cohomological functor and
$\omega_{\bullet}:(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})\rightarrow(T_{Res}^{\bullet},\widetilde{\delta}^{\bullet})$
is an isomorphism of cohomological functors.

######  Proof.

First, we show that the proposed connecting homomorphisms
$\widetilde{\delta}^{\bullet}$ are natural. For this let $n\in\mathbb{Z}$ and
let

${0}$${A}$${B}$${C}$${0}$${0}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$${0}$$\scriptstyle{f}$$\scriptstyle{\overline{\varphi}^{\ast}}$$\scriptstyle{g}$$\scriptstyle{\overline{\varphi}}$$\scriptstyle{\varphi}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$

be a commutative diagram in $\mathcal{C}$ with exact rows.Consider the diagram

${\widehat{T}^{n+1}(C)}$${T_{Res}^{n+1}(C)}$${\widehat{T}^{n}(A)}$${T_{Res}^{n}(A)}$${\widehat{T}^{n+1}(C^{\prime})}$${T_{Res}^{n+1}(C^{\prime})}$${\widehat{T}^{n}(A^{\prime})}$${T_{Res}^{n}(A^{\prime})}$$\scriptstyle{\omega_{n+1}}$$\scriptstyle{\widehat{T}^{n+1}(\overline{\varphi}^{\ast})}$$\scriptstyle{T_{Res}^{n+1}(\overline{\varphi}^{\ast})}$$\scriptstyle{\widehat{\delta}^{n}}$$\scriptstyle{\omega_{n}}$$\scriptstyle{\widetilde{\delta}^{n}}$$\scriptstyle{T_{Res}^{n}(G{,}\varphi)}$$\scriptstyle{\omega_{n+1}}$$\scriptstyle{\widehat{\delta}^{n}}$$\scriptstyle{\omega_{n}}$$\scriptstyle{\widehat{T}^{n}(\varphi)}$$\scriptstyle{\widetilde{\delta}^{n}}$

By Proposition 4.2, the isomorphism $\omega_{n}$ (resp. $\omega_{n+1}$) is
natural and thus the front and back squares of the cube commute. According to
Lemma 4.4, the top and bottom squares of the cube commute. As
$(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$ is a cohomological
functor, the left hand side commutes. Due to this and the fact that
$\omega_{n}:\widehat{T}^{n}(A)\rightarrow T_{Res}^{n}(A)$ is an isomorphism,
we infer that also the right hand side commutes. Hence
$(T_{Res}^{\bullet},\widetilde{\delta}^{\bullet})$ satisfies Axiom 2.1 of a
cohomological functor. Using this together with the fact
$\omega_{n}:\widehat{T}^{n}\rightarrow T_{Res}^{n}$ is a natural
transformation for every $n\in\mathbb{Z}$, we see that the diagram

${{}\dots}$${T_{Res}^{n}(A)}$${T_{Res}^{n}(B)}$${T_{Res}^{n}(C)}$${T_{Res}^{n+1}(A)}$${{}\dots}$${{}\dots}$${\widehat{T}^{n}(A)}$${\widehat{T}^{n}(B)}$${\widehat{T}^{n}(C)}$${\widehat{T}^{n+1}(A)}$${{}\dots}$$\scriptstyle{T_{Res}^{n}(f)}$$\scriptstyle{\omega_{n}}$$\scriptstyle{T_{Res}^{n}(g)}$$\scriptstyle{\omega_{n}}$$\scriptstyle{\widetilde{\delta}^{n}}$$\scriptstyle{\omega_{n}}$$\scriptstyle{\omega_{n+1}}$$\scriptstyle{\widehat{T}^{n}(G{,}f)}$$\scriptstyle{\widehat{T}^{n}(G{,}g)}$$\scriptstyle{\widehat{\delta}^{n}}$

commutes. Using again that every $\omega_{n}:\widehat{T}^{n}\rightarrow
T_{Res}^{n}$ is an isomorphism, we infer by the Five Lemma [7, Proposition
I.1.1] that every image and kernel in the bottom row is isomorphic to the
corresponding image or kernel in the top row. Because
$(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$ is a cohomological
functor, the bottom row forms a long exact sequence and thus the top row does
so too. In particular, $(T_{Res}^{\bullet},\widetilde{\delta}^{\bullet})$
satisfies Axiom 2.2 and is a cohomological functor. From this we conclude that
$\omega_{\bullet}:\widehat{T}^{\bullet}\rightarrow T_{Res}^{\bullet}$ is an
isomorphism of cohomological functors. $\square$

###  4.4 Countably many distinct constructions of Mislin completions

Based on the material of this section thus far, we can demonstrate that there
are countably many distinct constructions of Mislin completions.

######  Lemma 4.6.

Let $a=(a_{k})_{k\in\mathbb{N}}\in\\{0,1\\}^{\mathbb{N}}$ be a sequence taking
values in $0$ and $1$. Define $(P(a)_{k})_{k\in\mathbb{N}_{0}}$ by
$P(a)_{0}:=0$ and $P(a)_{k}:=\sum_{i=1}^{k}a_{i}$ and
$(D(a)_{k})_{k\in\mathbb{N}_{0}}$ by $D(a)_{0}:=0$ and
$D(a)_{k}:=\sum_{i=1}^{k}(1-a_{i})$. For any $M\in\mathrm{obj}(\mathcal{C})$
and $n\in\mathbb{Z}$ form the direct system
$\big{(}S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}}),\delta_{a}^{n+k}\big{)}_{k\in\mathbb{N}_{0}}$
where

${\scriptstyle\delta_{a}^{n+k}}:=\begin{cases}{\scriptstyle\Sigma^{-P(a)_{k}}\delta^{n+k}:\>S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}})\rightarrow
S^{-P(a)_{k}}T^{n+k+1}(\widetilde{M}_{D(a)_{k}+1})}&{\scriptstyle\text{if
}P(a)_{k+1}=P(a)_{k}}\\\ {\scriptstyle
S^{-P(a)_{k}}\underline{\delta}^{n+k}:\>S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}})\rightarrow
S^{-P(a)_{k}-1}T^{n+k+1}(\widetilde{M}_{D(a)_{k}})}&{\scriptstyle\text{if
}P(a)_{k+1}=P(a)_{k}+1}\end{cases}$

and the morphisms $\Sigma^{-P(a)_{k}}\delta^{n+k}$ are taken as in Equation
4.3. Write

$T_{a}^{n}(M):=\varinjlim_{\mathcal{D},k\in\mathbb{N}_{0}}S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}})\,.$

Then there exists an isomorphism $\omega_{a,n}:T_{a}^{n}(M)\rightarrow
T_{Res}^{n}(M)$.

######  Proof.

For any $k\in\mathbb{N}_{0}$ we define the morphism

$\widehat{o}_{a}^{k}:=\begin{cases}\varepsilon^{-1}\circ{}\dots\circ\varepsilon^{-P(a)_{k}}:S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}})\rightarrow
T^{n+k}(\widetilde{M}_{k})&\text{if }P(a)_{k}\geq 1\\\
\qquad\qquad\qquad\quad\mathrm{id}:T^{n+k}(\widetilde{M}_{k})\rightarrow
T^{n+k}(\widetilde{M}_{k})&\text{if }P(a)_{k}=0\end{cases}$

similarly to the morphism $o^{k}$ from Equation 4.5. By Diagram 4.6 we infer
that the square

${S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}})}$${T^{n+k}(\widetilde{M}_{k})}$${S^{-P(a)_{k+1}}T^{n+k+1}(\widetilde{M}_{D(a)_{k+1}})}$${T^{n+k+1}(\widetilde{M}_{k+1})}$$\scriptstyle{\widehat{o}_{a}^{k}}$$\scriptstyle{\delta_{a}^{n+k}}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\widehat{o}_{a}^{k+1}}$

commutes. We conclude as in the proof of Lemma 4.1 that
$\omega_{a,n}:=\varinjlim_{k\in\mathbb{N}_{0}}\widehat{o}_{a}^{k}$ is an
isomorphism. $\square$

We can reiterate the proof of Proposition 4.2 to demonstrate

######  Proposition 4.7.

Let $f:M\rightarrow N$ be a morphism in $\mathcal{C}$ and
$(\widetilde{f}_{k}:\widetilde{M}_{k}\rightarrow\widetilde{N}_{k})_{k\in\mathbb{N}_{0}}$
be a sequence of morphism as in Diagram 4.8. Then the squares

${S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}})}$${S^{-P(a)_{k}}T^{n+k}(\widetilde{N}_{D(a)_{k}})}$${S^{-P(a)_{k+1}}T^{n+k+1}(\widetilde{M}_{D(a)_{k+1}})}$${S^{P(a)_{k+1}}T^{n+k+1}(\widetilde{N}_{D(a)_{k+1}})}$$\scriptstyle{S^{-P(a)_{k}}T^{n+k}(\widetilde{f}_{D(a)_{k}})}$$\scriptstyle{\delta_{a}^{n+k}}$$\scriptstyle{\delta_{a}^{n+k}}$$\scriptstyle{S^{-P(a)_{k+1}}T^{n+k+1}(\widetilde{f}_{D(a)_{k+1}})}$

commute. If we define
$T_{a}^{n}(f):=\varinjlim_{k\in\mathbb{N}_{0}}S^{-P(a)_{k}}T^{n+k}(\widetilde{f}_{D(a)_{k}})$,
then the square

${T_{a}^{n}(M)}$${T_{Res}^{n}(M)}$${T_{a}^{n}(N)}$${T_{Res}^{n}(N)}$$\scriptstyle{\omega_{a,n}(M)}$$\scriptstyle{T_{a}^{n}(f)}$$\scriptstyle{T_{Res}^{n}(f)}$$\scriptstyle{\omega_{a,n}(N)}$

is also commutative. In particular, every term $T_{a}^{n}(-)$ forms an
additive functor and the homomorphisms
$\omega_{a,n}:\widehat{T}^{n}\rightarrow T_{Res}^{n}$ a natural isomorphism.

######  Lemma 4.8.

If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact
sequence, define for any $k\in\mathbb{N}_{0}$ the morphism

$\displaystyle\delta_{a,k}^{n+k}$
$\displaystyle:=\varepsilon^{-P(a)_{k}-1}\circ
S^{-P(a)_{k}}\underline{\delta}^{n+k}:$ $\displaystyle
S^{-P(a)_{k}}T^{n+k}(\widetilde{C}_{D(a)_{k}})\rightarrow
S^{-P(a)_{k}-1}T^{n+k+1}(\widetilde{C}_{D(a)_{k}})\rightarrow
S^{-P(a)_{k}}T^{(n+1)+k}(\widetilde{A}_{D(a)_{k}})$

Then the squares

${S^{-P(a)_{k}}T^{n+k}(\widetilde{C}_{D(a)_{k}})}$${T^{n+k}(\widetilde{C}_{k})}$${S^{-P(a)_{k}}T^{(n+1)+k}(\widetilde{A}_{D(a)_{k}})}$${T^{(n+1)+k}(\widetilde{A}_{k})}$$\scriptstyle{\widehat{o}_{a}^{k}}$$\scriptstyle{\delta_{a}^{n+k}}$$\scriptstyle{\delta^{n+k}}$$\scriptstyle{\widehat{o}_{a}^{k+1}}$
(4.14)

commute whence we define
$\widetilde{\delta}_{a}^{n}:=\varinjlim_{k\in\mathbb{N}_{0}}\delta_{a,k}^{n+k}:T_{a}^{n}(C)\rightarrow
T_{a}^{n+1}(A)$. Therefore, the square

${T_{a}^{n}(C)}$${T_{Res}^{n}(C)}$${T_{a}^{n+1}(A)}$${T_{Res}^{n+1}(A)}$$\scriptstyle{\omega_{a,n}}$$\scriptstyle{\widetilde{\delta}_{a}^{n}}$$\scriptstyle{\widetilde{\delta}^{n}}$$\scriptstyle{\omega_{a,n+1}}$
(4.15)

is also commutative, $(T_{a}^{\bullet},\widetilde{\delta}_{a}^{\bullet})$
forms a cohomological functor and
$\omega_{a,\bullet}:(T_{a}^{\bullet},\widetilde{\delta}_{a}^{\bullet})\rightarrow(T_{Res}^{\bullet},\widetilde{\delta}^{\bullet})$
is an isomorphism of cohomological functors.

######  Proof.

Regarding the Diagram 4.14, the term
$S^{-P(a)_{k}}T^{n+k}(\widetilde{C}_{D(a)_{k}})$ lies in the front side of
Diagram 4.11. Through the morphism
$\widehat{o}_{a}^{k}:S^{-P(a)_{k}}T^{n+k}(\widetilde{C}_{D(a)_{k}})\rightarrow
T^{n+k}(\widetilde{C}_{k})$ we reach the very right hand side of the latter
diagram. The morphism $\delta^{n+k}:T^{n+k}(\widetilde{C}_{k})\rightarrow
T^{(n+1)+k}(\widetilde{A}_{k})$ leads us to the back side and we move by
$\mathrm{id}:T^{(n+1)+k}(\widetilde{A}_{k})\rightarrow
T^{(n+1)+k}(\widetilde{A}_{k})$ from Diagram 4.12 across a square in the very
right hand column of the back side. On the other had, the morphism
$\delta_{a,k}^{n+k}:S^{-P(a)_{k}}T^{n+k}(\widetilde{C}_{D(a)_{k}})\rightarrow
S^{-P(a)_{k}}T^{(n+1)+k}(\widetilde{A}_{D(a)_{k}})$ takes us first down the
front side of Diagram 4.11 and then over to the back side. Lastly, we reach
the left hand side of the right most column of the back side by the morphism
$\widehat{o}_{a}^{k+1}:S^{-P(a)_{k}}T^{(n+1)+k}(\widetilde{A}_{D(a)_{k}})\rightarrow
T^{(n+1)+k}(\widetilde{A}_{k})$. Since Diagram 4.11 commutes, we conclude that

$\mathrm{id}\circ\delta^{n+k}\circ\widehat{o}_{a}^{k}=\widehat{o}_{a}^{k+1}\circ\delta_{a,k}^{n+k}:\;S^{-P(a)_{k}}T^{n+k}(\widetilde{C}_{D(a)_{k}})\rightarrow
S^{-P(a)_{k}}T^{(n+1)+k}(\widetilde{A}_{E_{k}})\,.$ (4.16)

Therefore, Diagram 4.14 commutes. Using again that Diagram 4.11 is
commutative, we observe that the squares of Diagram 4.14 form a direct system
in whose direct limit we obtain Diagram 4.15. The remainder of the lemma is
proved identically to Theorem 4.5. $\square$

######  Corollary 4.9.

Let us denote by $e\in\\{0,1\\}^{\mathbb{N}}$ the sequence all whose terms are
set to $0$ and by $f\in\\{0,1\\}^{\mathbb{N}}$ the sequence all whose terms
are set to $1$. Then
$(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})=(T_{f}^{\bullet},\widetilde{\delta}_{f}^{\bullet})$
and
$(T_{Res}^{\bullet},\widetilde{\delta}^{\bullet})=(T_{e}^{\bullet},\widetilde{\delta}_{e}^{\bullet})$.

######  Proof.

By definition, $\widehat{T}^{n}=T_{f}^{n}$ and $T_{Res}^{n}=T_{e}^{n}$ for
every $n\in\mathbb{Z}$. In the same manner as we have concluded by Diagram
4.12 and 4.13 that the morphism $\chi$ in the proof of Lemma 4.4 agrees with
the identity, we see that

$\widetilde{\delta}_{f}^{n}=\varinjlim_{k\in\mathbb{N}_{0}}\delta_{f,k}^{n+k}=\varinjlim_{k\in\mathbb{N}}\varepsilon^{-k-1}\circ
S^{-k}\underline{\delta}^{n+k}=\varinjlim_{k\in\mathbb{N}}\varepsilon^{-k}=\widehat{\delta}^{n}$

where $\varepsilon^{-k-1}\circ
S^{-k}\underline{\delta}^{n+k}:S^{-k}T^{n+k}\rightarrow S^{-k}T^{(n+1)+k}$ and
$\varepsilon^{-k}:S^{-k}T^{n+k}\rightarrow S^{-k+1}T^{n+k}$. By Equation 4.16,
we observe that $\widetilde{\delta}_{e}^{n}=\widetilde{\delta}^{n}$. $\square$

######  Lemma 4.10.

For $a,b\in\\{0,1\\}^{\mathbb{N}}$ we say that the constructions of the Mislin
completions $(T_{a}^{\bullet},\widetilde{\delta}_{a}^{\bullet})$ and
$(T_{b}^{\bullet},\widetilde{\delta}_{b}^{\bullet})$ agree if there is an
infinite (cofinal) subset $\Lambda\subseteq\mathbb{N}$ such that
$S^{-P(a)_{\lambda}}T^{n+{\lambda}}(\widetilde{M}_{D(a)_{\lambda}})=S^{-P(b)_{\lambda}}T^{n+{\lambda}}(\widetilde{M}_{D(b)_{\lambda}})$
for every $\lambda\in\Lambda$, $n\in\mathbb{Z}$ and
$M\in\mathrm{obj}(\mathcal{C})$. The basic idea is that the constructions
agree whenever their terms

$T_{a}^{n}(M)=\varinjlim_{k\in\mathbb{N}_{0}}S^{-P(a)_{k}}T^{n+k}(\widetilde{M}_{D(a)_{k}})\text{
and
}T_{b}^{n}(M)=\varinjlim_{k\in\mathbb{N}_{0}}S^{-P(b)_{k}}T^{n+k}(\widetilde{M}_{D(b)_{k}})$

are isomorphic in a very straightforward manner. If this is not the case, we
say that the constructions of
$(T_{a}^{\bullet},\widetilde{\delta}_{a}^{\bullet})$ and
$(T_{b}^{\bullet},\widetilde{\delta}_{b}^{\bullet})$ are distinct. Then there
are countably many distinct constructions of Mislin completions of the form
$(T_{a}^{\bullet},\widetilde{\delta}_{a}^{\bullet})$.

######  Proof.

Define the binary relation ‘$\sim$’ on $\\{0,1\\}^{\mathbb{N}}$ by setting
$a\sim b$ if for every $k\in\mathbb{N}$ there is $K\geq k$ such that
$P(a)_{K}=P(b)_{K}$. If $\sim^{\prime}$ denotes the transitive closure of
$\sim$, then the proof reduces to showing that there are countably many
elements in $\\{0,1\\}^{\mathbb{N}}/\sim^{\prime}$. Hence, define for any
$m\in\mathbb{N}$ the sequences $e^{m},f^{m}\in\\{0,1\\}^{\mathbb{N}}$ by

$e_{k}^{m}:=\begin{cases}1&\text{if }k<m\\\ 0&\text{if }k\geq
m\end{cases}\qquad\text{and}\qquad f_{k}^{m}:=\begin{cases}0&\text{if }k<m\\\
1&\text{if }k\geq m\end{cases}$

Comparing with Corollary 4.9, we see that $e=e_{0}$ and $f=f_{0}$. For any
$g\in\\{0,1\\}^{\mathbb{N}}$ taking finitely many times either the value $0$
or the value $1$ there is $m\in\mathbb{N}$ such that $g\sim e_{m}$ or $g\sim
f_{m}$. For any distinct $m,p\in\mathbb{N}$ we note that
$e_{m}\nsim^{\prime}e_{p}$, $e_{m}\nsim^{\prime}f_{p}$ and
$f_{m}\nsim^{\prime}f_{p}$. To complete the proof, we demonstrate that for any
$a,b\in\\{0,1\\}^{\mathbb{N}}$ taking both the values $0$ and $1$ infinitely
often there is another sequence $c$ such that $a\sim c$ and $c\sim b$. If
$a\nsim b$, then assume that there is $K\in\mathbb{N}$ such that
$P(a)_{k}<P(b)_{k}$ for every $k\geq K$. We construct $c$ inductively as
follows. For the base case, set $c_{k}:=a_{k}$ for any $1\leq k\leq K$ and let
$c_{l}:=1$ for the subsequent $K\leq l$. Since $P(c)$ grows faster than
$P(b)$, there is $l_{1}\in\mathbb{N}$ such that $P(c)_{l_{1}}=P(b)_{l_{1}}$.
Assume for $i\in\mathbb{N}$ that we have found $k_{i}<l_{i}\in\mathbb{N}$ such
that $P(a)_{k_{i}}=P(c)_{k_{i}}$ and $P(c)_{l_{i}}=P(b)_{l_{i}}$. Setting
$c_{k}:=0$ for $k\geq l_{i}$ yields $k_{i+1}\in\mathbb{N}$ and defining
$c_{l}:=1$ for $l\geq k_{i+1}$ yields $l_{i+1}\in\mathbb{N}$. Then $a\sim
c\sim b$ as desired. $\square$

######  Notation 4.11.

In the rest of the paper, we write
$(\widehat{T}^{\bullet},\widehat{\delta}^{\bullet})$ for the Mislin completion
of a cohomological functor $(T^{\bullet},\delta^{\bullet})$ independently from
which construction it arises. When required, we specify the construction or
use a special notation for it all of which we keep consistent throughout.

##  5 The naïve and the resolution construction

In this section we prove that the naïve construction gives rise to Mislin
completions of unenriched Ext-functors by defining an isomorphism of
cohomological functors to the respective terms of the resolution construction.
For this, we rigorously perform the naïve construction in Subsection 5.1 in a
different manner than in [4] and [19]. Thereafter, we show in Subsection 5.2
that its cohomology groups are naturally isomorphic to the ones of the
resolution construction. Lastly, in Subsection 5.3, we construct a connecting
homomorphism for the naïve construction and extend the isomorphism between the
cohomology groups to an isomorphism of cohomological functors.

###  5.1 Performing the naïve construction

To start, we consider the unenriched Hom-functors

$\mathrm{Hom}_{\mathcal{C}}(-,-):\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathbf{Ab}$

which are additive in both variables. As in Definition 2.5, we turn the
corresponding unenriched Ext-functors into cohomological functors by setting
$\mathrm{Ext}_{\mathcal{C}}^{n}(A,-)=0$ for $n<0$ and any
$A\in\mathrm{obj}(\mathcal{C})$. Both A. Beligiannis and I. Reiten in [4, p.
37–38/163–165] as well as S. Guo and L. Liang in [19, p. 14–15] have
established relative homological versions of the naïve construction in great
generality. However, in order to construct Yoneda products in [16, Theorem
6.6], we require an (absolute homological) generalisation faithful to D. J.
Benson and J. F. Carlson’s original construction found in [5, p. 109], which
is why our account differs from the previous two. First, we define a bifunctor
$[-,-]_{\mathcal{C}}$ that we have already encountered in Subsection 2. Let
$\mathcal{P}_{\mathcal{C}}(A,B)\subseteq\mathrm{Hom}_{\mathcal{C}}(M,N)$ be
the subset of all morphisms factoring through a projective object. Since the
zero object $0\in\mathcal{C}$ is projective, the zero morphism is in
$\mathcal{P}_{\mathcal{C}}(A,B)$. Let $\varphi:M\rightarrow P\rightarrow N$
and $\psi:M\rightarrow Q\rightarrow N$ be morphisms factoring though
projectives. If $\Delta_{A}:A\rightarrow A\oplus A$ is the diagonal map and
$\nabla_{B}:B\oplus B\rightarrow B$ the codiagonal, then
$\varphi+\psi=\nabla_{B}\circ(\varphi\oplus\psi)\circ\Delta_{A}$ by [27, p.
253]. Factoring the map $\varphi\oplus\psi:A\oplus A\rightarrow B\oplus B$
further, we obtain

$\varphi+\psi:A\xrightarrow{\Delta_{A}}A\oplus A\rightarrow P\oplus
Q\rightarrow B\oplus B\xrightarrow{\nabla_{B}}B\,.$

As $P\oplus Q$ is a projective object [27, p. 250–251],
$\varphi+\psi\in\mathcal{P}_{\mathcal{C}}(M,N)$ and
$\mathcal{P}_{\mathcal{C}}(A,B)$ is a subgroup of
$\mathrm{Hom}_{\mathcal{C}}(A,B)$. Define
$[A,B]_{\mathcal{C}}:=\mathrm{Hom}_{\mathcal{C}}(A,B)/\mathcal{P}_{\mathcal{C}}(A,B)$.
The following proposition is not only relevant to the construction, but also
to the construction of Yoneda products in [16, Theorem 6.6].

######  Proposition 5.1.

The bifunctor
$\circ:\mathrm{Hom}_{\mathcal{C}}(B,C)\times\mathrm{Hom}_{\mathcal{C}}(A,B)\rightarrow\mathrm{Hom}_{\mathcal{C}}(A,C)$
descends to a bifunctor
$\circ:[B,C]_{\mathcal{C}}\times[A,B]_{\mathcal{C}}\rightarrow[A,C]_{\mathcal{C}}$
that is associative and additive in both variables.

######  Proof.

First we compose the composition bifunctor with the projection homomorphism
$\mathrm{Hom}_{\mathcal{C}}(A,C)\rightarrow[A,C]_{\mathcal{C}}$. Then we
quotient out
$\mathrm{Hom}_{\mathcal{C}}(B,C)\times\mathcal{P}_{\mathcal{C}}(A,B)$ in the
domain yielding
$\mathrm{Hom}_{\mathcal{C}}(B,C)\times[M,N]_{\mathcal{C}}\rightarrow[A,C]_{\mathcal{C}}$.
Lastly, we quotient out
$\mathcal{P}_{\mathcal{C}}(B,C)\times[A,B]_{\mathcal{C}}$ in the domain.
Pictorially, this can be summarised in the commutative diagram

${{\scriptstyle\mathrm{Hom}_{\mathcal{C}}(B{,}\,C)\times\mathrm{Hom}_{\mathcal{C}}(A{,}\,B)}}$${{\scriptstyle\mathrm{Hom}_{\mathcal{C}}(B{,}\,C)\times{[}A{,}\,B{]}_{\mathcal{C}}}}$${{\scriptstyle{[}B{,}\,C{]}_{\mathcal{C}}\times{[}A{,}\,B{]}_{\mathcal{C}}}}$${{\scriptstyle\mathrm{Hom}_{\mathcal{C}}(A{,}\,C)}}$${{\scriptstyle{[}A{,}\,C{]}_{\mathcal{C}}}}$$\scriptstyle{\scriptstyle-\circ-}$$\scriptstyle{\scriptstyle-\circ-}$
(5.1)

Since the original composition functor is associative as well as additive in
both variables and the new one is obtained by subsequent quotients, also the
resulting composition functor is associative as well as additive in both
variables. $\square$

If $(A_{n})_{n\in\mathbb{N}_{0}}$, $(B_{n})_{n\in\mathbb{N}_{0}}$ are
projective resolutions of $A$, $B$, then the first step of the naïve
construction is given by the following generalisation of results found in [25,
p. 204]

######  Proposition 5.2.

There exists a homomorphism
$t_{A,B}:[A,B]_{\mathcal{C}}\rightarrow[\widetilde{A}_{1},\widetilde{B}_{1}]_{\mathcal{C}}$
with the property that $t_{B,C}(-)\circ t_{A,B}(-)=t_{A,C}(-\circ-)$.

######  Proof.

Consider the following version of Diagram 4.8:

${0}$${\widetilde{A}_{1}}$${A_{0}}$${A}$${0}$${0}$${\widetilde{B}_{1}}$${B_{0}}$${B}$${0}$$\scriptstyle{\iota_{A}}$$\scriptstyle{\widetilde{f}_{1}}$$\scriptstyle{\pi_{A}}$$\scriptstyle{f_{0}}$$\scriptstyle{f}$$\scriptstyle{\iota_{B}}$$\scriptstyle{\pi_{B}}$
(5.2)

If $f_{0}^{\prime}:A_{0}\rightarrow B_{0}$ and
$\widetilde{f}_{1}^{\prime}:\widetilde{A}_{1}\rightarrow\widetilde{B}_{1}$ are
different lifts, then the diagram

${0}$${\widetilde{A}_{1}}$${A_{0}}$${A}$${0}$${0}$${\widetilde{B}_{1}}$${B_{0}}$${B}$${0}$$\scriptstyle{\iota_{A}}$$\scriptstyle{\widetilde{f}_{1}-\widetilde{f}_{1}^{\prime}}$$\scriptstyle{\pi_{A}}$$\scriptstyle{f_{0}-f_{0}^{\prime}}$$\scriptstyle{0}$$\scriptstyle{\iota_{B}}$$\scriptstyle{\pi_{B}}$

commutes. Because $\iota_{B}$ is the kernel of $\pi_{B}$, there is a morphism
$e:A_{0}\rightarrow\widetilde{B}_{1}$ making the bottom right triangle of

${0}$${\widetilde{A}_{1}}$${A_{0}}$${0}$${\widetilde{B}_{1}}$${B_{0}}$$\scriptstyle{\iota_{A}}$$\scriptstyle{\widetilde{f}_{1}-\widetilde{f}_{1}^{\prime}}$$\scriptstyle{f_{0}-f_{0}^{\prime}}$$\scriptstyle{e}$$\scriptstyle{\iota_{B}}$

commute. As the above square also commutes and $\iota_{B}$ is a monomorphism,
we observe that
$\widetilde{f}_{1}-\widetilde{f}_{1}^{\prime}=e\circ\iota_{A}$. Using that
$A_{0}$ is projective, $\widetilde{f}_{1}$ and $\widetilde{f}_{1}^{\prime}$
agree in $[\widetilde{A}_{1},\widetilde{B}_{1}]_{\mathcal{C}}$. Therefore,
there is a well define map

$s_{A,B}:\mathrm{Hom}_{\mathcal{C}}(A,B)\rightarrow[\widetilde{A}_{1},\widetilde{B}_{1}]_{\mathcal{C}},f\mapsto\widetilde{f_{1}}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{1},\widetilde{B}_{1})\,.$

We infer by construction that

$s_{A,B}(f+g)=\widetilde{f}_{1}+\widetilde{g}_{1}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{1},\widetilde{B}_{1})$

and $s_{A,B}$ is a homomorphism. If $h\in\mathrm{Hom}_{\mathcal{C}}(B,C)$,
then one can concatenate the corresponding diagrams of the above form to
obtain that $s_{B,C}(h)\circ s_{A,B}(f)=s_{A,C}(h\circ f)$. In particular, if
$f\in\mathcal{P}_{\mathcal{C}}(A,B)$, then one can choose a projective
resolution of the respective projective such that $s_{A,B}(f)$ = 0. Thus,
$s_{A,B}$ descends to the desired homomorphism
$t_{A,B}:[A,B]_{\mathcal{C}}\rightarrow[\widetilde{A}_{1},\widetilde{B}_{1}]_{\mathcal{C}}$.
Using again that $s_{B,C}(-)\circ s_{A,B}(-)=s_{A,C}(-\circ-)$, we can factor
the respective maps through quotient homomorphisms as in the commutative
diagram

${{\scriptstyle\mathrm{Hom}_{\mathcal{C}}(B{,}\,C)\times\mathrm{Hom}_{\mathcal{C}}(A{,}\,B)}}$${{\scriptstyle\mathrm{Hom}_{\mathcal{C}}(B{,}\,C)\times{[}A{,}\,B{]}_{\mathcal{C}}}}$${{\scriptstyle{[}B{,}\,C{]}_{\mathcal{C}}\times{[}A{,}\,B{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{B}_{1}{,}\,\widetilde{C}_{1}{]}\times{[}\widetilde{A}_{1}{,}\,\widetilde{B}_{1}{]}}}$${{\scriptstyle\mathrm{Hom}_{\mathcal{C}}(A{,}\,C)}}$${{\scriptstyle{[}A{,}\,C{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{A}_{1}{,}\,\widetilde{C}_{1}{]}}}$$\scriptstyle{\scriptstyle
s_{B,C}\times
s_{A,B}}$$\scriptstyle{\scriptstyle-\circ-}$$\scriptstyle{\scriptstyle
s_{B,C}\times t_{A,B}}$$\scriptstyle{\scriptscriptstyle t_{B,C}\times
t_{A,B}}$$\scriptstyle{\scriptstyle-\circ-}$$\scriptstyle{s_{A,C}}$$\scriptstyle{\scriptstyle
t_{A,C}}$

Note that Diagram 5.1 can be fitted into the left hand side of the above
rectangle. Since the quotient homomorphisms from the proof of Proposition 5.1
give rise to the surjective map

$\mathrm{Hom}_{\mathcal{C}}(B,C)\times\mathrm{Hom}_{\mathcal{C}}(A,B)\rightarrow\mathrm{Hom}_{\mathcal{C}}(B,C)\times[A,B]_{\mathcal{C}}\rightarrow[B,C]_{\mathcal{C}}\times[A,B]_{\mathcal{C}}\,,$

we conclude that the square

${{\scriptstyle{[}B{,}\,C{]}_{\mathcal{C}}\times{[}A{,}\,B{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{B}_{1}{,}\,\widetilde{C}_{1}{]}\times{[}\widetilde{A}_{1}{,}\,\widetilde{B}_{1}{]}}}$${{\scriptstyle{[}A{,}\,C{]}_{\mathcal{C}}}}$${{\scriptstyle{[}\widetilde{A}_{1}{,}\,\widetilde{C}_{1}{]}}}$$\scriptstyle{\scriptstyle
t_{B,C}\times
t_{A,B}}$$\scriptstyle{\scriptstyle-\circ-}$$\scriptstyle{\scriptstyle-\circ-}$$\scriptstyle{\scriptstyle
t_{A,C}}$

commutes, meaning that $t_{B,C}(-)\circ t_{A,B}(-)=t_{A,C}(-\circ-)$.
$\square$

######  Definition 5.3.

Let $n\in\mathbb{Z}$. Then there exists for any $k\in\mathbb{N}_{0}$ with
$n+k\geq 0$ a homomorphism
$t_{\widetilde{A}_{n+k},\widetilde{B}_{n+k}}:[\widetilde{A}_{n+k},\widetilde{B}_{k}]_{\mathcal{C}}\rightarrow[\widetilde{A}_{n+k+1},\widetilde{B}_{k+1}]_{\mathcal{C}}$.
We define the $n^{\text{th}}$ term of the naïve construction to be
$BC_{\mathcal{C}}^{n}(A,B):=\varinjlim_{\mathcal{D}}([\widetilde{A}_{n+k},\widetilde{B}_{k}]_{\mathcal{C}},t_{\widetilde{A}_{n+k},\widetilde{B}_{k}})_{k\in\mathbb{N}_{0},n+k\geq
0}$.

In order to define induced homomorphisms for $BC_{\mathcal{C}}^{n}(-,-)$, let
$A$, $B$, $C$ be objects in $\mathcal{C}$ and consider corresponding
projective resolutions $A_{\bullet}$, $B_{\bullet}$, $C_{\bullet}$. For a
morphism $f:B\rightarrow C$ in $\mathcal{C}$ let us construct a homomorphism
$BC_{\mathcal{C}}^{n}(A,f)$ using these projective resolutions. Lifting $f$ as
in the proof of Proposition 5.2, we obtain a sequence of morphisms
$(\widetilde{f}_{k}:\widetilde{B}_{k}\rightarrow\widetilde{C}_{k})_{k\in\mathbb{N}_{0}}$
where

$t_{\widetilde{B}_{k},\widetilde{C}_{k}}\big{(}\widetilde{f}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{B}_{k},\widetilde{C}_{k})\big{)}=\widetilde{f}_{k+1}+\mathcal{P}_{\mathcal{C}}(\widetilde{B}_{k+1},\widetilde{C}_{k+1})\,.$

Since $f=\widetilde{f}_{0}$, the element
$(\widetilde{f}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{B}_{k},\widetilde{C}_{k}))_{k\in\mathbb{N}_{0}}$
of
$([\widetilde{B}_{k},\widetilde{C}_{k}]_{\mathcal{C}})_{k\in\mathbb{N}_{0}}$
is independent of the lifts of $f$. Again by Proposition 5.2,
$t_{\widetilde{A}_{n+k},\widetilde{C}_{k}}(-\circ-)=t_{\widetilde{B}_{k},\widetilde{C}_{k}}(-)\circ
t_{\widetilde{A}_{n+k},\widetilde{B}_{k}}(-)$ and thus the square of
homomorphisms of abelian groups

${{[}\widetilde{A}_{n+k},\widetilde{B}_{k}{]}_{\mathcal{C}}}$${{[}\widetilde{A}_{n+k},\widetilde{C}_{k}{]}_{\mathcal{C}}}$${{[}\widetilde{A}_{n+k+1},\widetilde{B}_{k+1}{]}_{\mathcal{C}}}$${{[}\widetilde{A}_{n+k+1},\widetilde{C}_{k+1}{]}_{\mathcal{C}}}$$\scriptstyle{\big{(}\widetilde{f}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{B}_{k}{,}\widetilde{C}_{k})\big{)}\circ-}$$\scriptstyle{t_{\widetilde{A}_{n+k},\widetilde{B}_{k}}}$$\scriptstyle{t_{\widetilde{A}_{n+k},\widetilde{C}_{k}}}$$\scriptstyle{\big{(}\widetilde{f}_{k+1}+\mathcal{P}_{\mathcal{C}}(\widetilde{B}_{k+1}{,}\widetilde{C}_{k+1})\big{)}\circ-}$
(5.3)

commutes. Analogously, if $g:C\rightarrow A$ is another morphism, then we can
construct an element
$(\widetilde{g}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{C}_{k},\widetilde{A}_{k}))_{k\in\mathbb{N}_{0}}$
of
$([\widetilde{C}_{k},\widetilde{A}_{k}]_{\mathcal{C}})_{k\in\mathbb{N}_{0}}$
that is independent of the lifts of $g$.

######  Definition 5.4.

Taking the notation from before, we define

$BC_{\mathcal{C}}^{n}(A,f):=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
0}\big{(}\big{(}\widetilde{f}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{B}_{k},\widetilde{C}_{k})\big{)}\circ-\big{)}:BC_{\mathcal{C}}^{n}(A,B)\rightarrow
BC_{\mathcal{C}}^{n}(A,C)$

and

$BC_{\mathcal{C}}^{n}(g,B):=\varinjlim_{k\in\mathbb{N}_{0},n+k\geq
0}\big{(}-\circ\big{(}\widetilde{g}_{n+k}+\mathcal{P}_{\mathcal{C}}(\widetilde{C}_{n+k},\widetilde{A}_{n+k})\big{)}\big{)}:BC_{\mathcal{C}}^{n}(A,B)\rightarrow
BC_{\mathcal{C}}^{n}(C,B)\,.$

######  Proposition 5.5.

For every $n\in\mathbb{Z}$,
$BC_{\mathcal{C}}^{n}(-,-):\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathbf{Ab}$
forms a well defined bifunctor that is additive in both variables.

######  Proof.

We demonstrate that $BC_{\mathcal{C}}(A,B)$ is independent of choices of
projective resolutions. Hence let $A_{\bullet}$ and $A_{\bullet}^{\prime}$ be
projective resolutions of $A$. As in Definition 5.4, take
$(\widetilde{\iota}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k},\widetilde{A}_{k}^{\prime}))_{k\in\mathbb{N}_{0}}$
and
$(\widetilde{\iota}_{k}^{\prime}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k}^{\prime},\widetilde{A}_{k}))_{k\in\mathbb{N}_{0}}$
lifting $\mathrm{id}_{A}:A\rightarrow A$. Given that
$(\widetilde{\iota}_{k}^{\prime}\circ\widetilde{\iota}_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k},\widetilde{A}_{k}))_{k\in\mathbb{N}_{0}}$
and
$(\widetilde{\iota}_{k}\circ\widetilde{\iota}_{k}^{\prime}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k}^{\prime},\widetilde{A}_{k}^{\prime}))_{k\in\mathbb{N}_{0}}$
are also lifts of $\mathrm{id}_{A}$, we conclude by Proposition 5.2 that

$\iota_{k}^{\prime}\circ\iota_{k}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k},\widetilde{A}_{k})=\mathrm{id}_{\widetilde{A}_{k}}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k},\widetilde{A}_{k})\text{
and
}\iota_{k}\circ\iota_{k}^{\prime}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k}^{\prime},\widetilde{A}_{k}^{\prime})=\mathrm{id}_{\widetilde{A}_{k}^{\prime}}+\mathcal{P}_{\mathcal{C}}(\widetilde{A}_{k}^{\prime},\widetilde{A}_{k}^{\prime})$

for any $k\in\mathbb{N}_{0}$. As there is an analogous statement for any two
projective resolutions of $B$, $BC_{\mathcal{C}}^{n}(A,B)$ is unique up to
isomorphism. Knowing that composition of morphisms is associative by
Proposition 5.1, we see that

$BC_{\mathcal{C}}(A,f^{\prime}\circ f)=BC_{\mathcal{C}}(A,f^{\prime})\circ
BC_{\mathcal{C}}(A,f)\text{ and }BC_{\mathcal{C}}(g\circ
g^{\prime},B)=BC_{\mathcal{C}}(g,B)\circ BC_{\mathcal{C}}(g^{\prime},B)$

for any $f^{\prime}:C\rightarrow D$ and $g^{\prime}:D\rightarrow C$. Thus,
$BC_{\mathcal{C}}^{n}(-,-)$ a well defined bifunctor. Because composition of
morphisms is bi-additive also by Proposition 5.1 and Diagram 5.3 commutes, we
infer that

$BC_{\mathcal{C}}^{n}(A,f+f^{\prime\prime})=BC_{\mathcal{C}}^{n}(A,f)+BC_{\mathcal{C}}^{n}(A,f^{\prime\prime})\text{
and
}BC_{\mathcal{C}}^{n}(g+g^{\prime},B)=BC_{\mathcal{C}}^{n}(g,B)+BC_{\mathcal{C}}^{n}(g^{\prime\prime},B)$

for any $f,f^{\prime\prime}:B\rightarrow C$ and
$g,g^{\prime\prime}:C\rightarrow A$. In particular,
$BC_{\mathcal{C}}^{n}(-,-)$ is additive in both variables. $\square$

###  5.2 Isomorphism of cohomology groups

Let us construct a homomorphism from completed Ext-functors
$\widehat{\mathrm{Ext}}_{\mathcal{C}}^{n}(A,B)$ arising from the resolution
construction to the terms $BC_{\mathcal{C}}^{n}(A,B)$ of the naïve
construction. The idea of the construction and of the proof that it yields an
isomorphism can be retrieved from page 299 of G. Mislin’s paper [28] and are
generalised to our setting in the following. Again, this differs from S. Guo
and L. Liang’s account containing a similar construction and proof on pages
18–19 of their paper [19]. If $\partial_{n+1}:A_{n+1}\rightarrow A_{n}$
denotes a morphism of a projective resolution $A_{\bullet}$ of $A$, then

$\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+1},B))=\\{f\in\mathrm{Hom}_{\mathcal{C}}(A_{n},B)\mid
f\circ\partial_{n+1}=0\\}\,.$

Note that the cokernel of $\partial_{n+1}$ can be realised by the morphism
$\pi_{n}:A_{n}\rightarrow\widetilde{A}_{n}$ occurring in the factorisation
$\partial_{n}:A_{n}\xrightarrow{\pi_{n}}\widetilde{A}_{n}\xrightarrow{\iota_{n}}A_{n-1}$.
Thus, for every
$f\in\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+1},B))$ there is a
unique morphism $f^{\prime}:\widetilde{A}_{n}\rightarrow B$ such that
$f=f^{\prime}\circ\pi_{n}$. Therefore, there is a bijection

$\alpha_{n}(B):\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+1},B))\rightarrow\mathrm{Hom}_{\mathcal{C}}(\widetilde{A}_{n},B),\;\;f\mapsto
f^{\prime}\,.$

As $(\alpha_{n}(B)(f)+\alpha_{n}(B)(g))\circ\pi_{n}=f+g$, $\alpha_{n}(B)$ is
an isomorphisms of abelian groups. Pictorially, one can summarise the content
of this isomorphism as

${A_{n}}$${\widetilde{A}_{n}}$${B}$$\scriptstyle{\pi_{n}}$$\scriptstyle{f}$$\scriptstyle{\alpha_{n}(B)(f)}$
(5.4)

Before we proceed we need to observe that in the case $n=0$ this yields an
isomorphism

$\alpha_{0}(B):\mathrm{Ext}_{\mathcal{C}}^{0}(A,B)\rightarrow\mathrm{Hom}_{\mathcal{C}}(A,B)$

since
$\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(\partial_{0},B))=\mathrm{Ext}_{\mathcal{C}}^{0}(A,B)$.
If $n\geq 1$, assume that $f$ is an element in
$\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n},B))\leq\mathrm{Ker}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n+1},B))$.
Being of the form $f:A_{n}\xrightarrow{\partial_{n}}A_{n-1}\rightarrow B$, it
follows from the above factorisation of $\partial_{n}$ that
$\alpha_{n}(B)(f):\widetilde{A}_{n}\xrightarrow{\iota_{n}}A_{n-1}\rightarrow
B$. As $A_{n-1}$ is a projective object,
$\alpha_{n}(B)(f)\in\mathcal{P}_{R}(\widetilde{A}_{n},B)$. Thus
$\alpha_{n}(B)$ descends to a homomorphism of abelian groups

$\displaystyle\beta_{n}(B)\\!:\,\mathrm{Ext}_{\mathcal{C}}^{n}(A,B)$
$\displaystyle\rightarrow[\widetilde{A}_{n},B]_{\mathcal{C}}$ $\displaystyle
f+\mathrm{Im}(\mathrm{Hom}_{\mathcal{C}}(\partial_{n},B))$
$\displaystyle\mapsto\alpha_{n}(B)(f)+\mathcal{P}_{R}(\widetilde{A}_{n},B)\,.$

Due to the above, we remark that $\beta^{n}(B)$ is only defined for $n\geq 1$
while we already have the isomorphism $\alpha_{0}(B)$ in the case $n=0$. The
last ingredient to construct our desired isomorphism between the cohomology
groups is

######  Proposition 5.6 ([30, p. 261]).

In the category $\mathbf{Ab}$ of abelian groups, the direct limit of a direct
system $\\{M_{i},\psi_{i,j}\\}_{i\leq j\in I}$ can be given as
$\bigoplus_{i\in I}M_{i}/\sim$ where $m_{i}\in M_{i}\sim m_{j}\in M_{j}$ if
there is $k\in I$ such that $\psi_{i,k}(m_{i})=\psi_{j,k}(m_{j})\in M_{k}$. In
particular, if $\\{m_{i}\in M_{i}\\}_{i\in I}$ is a collection of nonzero
elements such that $\psi_{i,j}(m_{i})=m_{j}$ for any $i\leq j$, then the
element $(m_{i})_{i\in I}\in\varinjlim_{i\in I}M_{i}$ is nonzero.

Hence we can generalise G. Mislin’s Theorem 4.1 and its proof from [28].

######  Lemma 5.7.

For every $n\in\mathbb{Z}$ and $k\in\mathbb{N}_{0}$ with $n+k\geq 1$ the
square