In fact, $\mathbb{I}_{[s,t]}$ spectacularly fails to be locally compact. This
follows from the surprising observation that $\mathbb{I}_{[s,t]}$ has no non-
trivial compact sub-modules, since there are no injective maps
$\mathbb{I}_{[p,q)}\hookrightarrow\mathbb{I}_{[s,t]}$. The fact that such a
module exists reflects the non-Noetherian properties of the ring
$\mathbb{Q}[X]$ of polynomials with exponents in $\mathbb{R}_{+}$. Indeed,
from the commutative algebra perspective $M=\mathbb{I}_{[s,t]}$ has a free
presentation

$\bigoplus_{n}X^{t+1/n}\mathbb{Q}[X]\to X^{s}\mathbb{Q}[X]\twoheadrightarrow
M$

where $\bigoplus_{n}X^{t+1/n}\mathbb{Q}[X]$ corresponds to the ideal $\langle
X^{t+1/n}\mid n\in\mathbb{N}\rangle$ in $\mathbb{R}_{+}$-graded
$\mathbb{Q}[X]$-modules, which is clearly not finitely generated.

Given that cofibrant implies locally compact (above degree 0), a natural
question is whether the converse is true. Over the tame objects, which are a
sub-class of locally compact persistence complexes, the converse holds.

######  Lemma 6.25.

If $i:\mathbb{X}\to\mathbb{Y}$ is an injective map of tame
$\mathbb{X},\mathbb{Y}\in(p\mathsf{Ch}^{*}_{\mathbb{Q}})^{tame}$, then $i$ is
a cofibration in the interval sphere model structure.

###### Proof.

Tame persistence modules form an abelian category [CGL21], so the space of
cocycles $Z\mathbb{X}=\text{Ker}\,(d:\mathbb{X}\to\mathbb{X}[1])$ is tame. The
map reduces to an injective map $i:Z\mathbb{X}\to Z\mathbb{Y}$ on cocycles.
Since $Z\mathbb{X}$ and $Z\mathbb{Y}$ are tame, the quotient is tame and thus
has a graded interval decompostion
$Z\mathbb{Y}/i(Z\mathbb{X})=\oplus_{\gamma}\mathbb{I}_{[s_{\gamma},t_{\gamma})}^{k_{\gamma}}$
at the level of graded persistence modules. For each interval pick generators
$y_{s_{\gamma}}\in Z\mathbb{Y}/i(Z\mathbb{X})(s_{\gamma})$ and note that
$y_{t_{\gamma}}$ must be in $i(Z\mathbb{X})$ by the interval decomposition. We
add missing cocycles into $i(\mathbb{X})$ via the pushout

${{\bigoplus_{\gamma\in\Gamma}\mathbb{S}_{[s_{\gamma},t_{\gamma})}^{k_{\gamma}+1}}}$${{i(\mathbb{X})}}$${{\bigoplus_{\gamma\in\Gamma}\mathbb{D}_{s_{\gamma}}^{k_{\gamma}+1}}}$${{i(\mathbb{X})\cup
Z\mathbb{Y}}}$$\scriptstyle{(0,y_{t_{\gamma}})}$$\scriptstyle{y_{s_{\gamma}}}$$\scriptstyle{\lrcorner}$

which is clearly still tame. Since quotients of tame objects by tame
subobjects are tame, assign an interval decomposition

$\mathbb{Y}/\big{(}i(\mathbb{X})\cup
Z\mathbb{Y}\big{)}=\bigoplus_{\delta\in\Delta}\mathbb{I}_{[s_{\delta},t_{\delta})}^{k_{\delta}}$

at the level of graded persistence modules. Assigning generators
$y_{s_{\delta}}\in\mathbb{Y}$ to each interval, we have that
$dy_{s_{\delta}}\in Z\mathbb{Y}$ by the cochain condition and that
$y_{t_{\delta}}\in i(\mathbb{X})\cup Z\mathbb{Y}$ by the interval
decomposition. We add in the rest of $\mathbb{Y}$ via the pushout

${{\bigoplus_{\delta\in\Delta}\mathbb{S}_{[s_{\delta},t_{\delta})}^{k_{\delta}+1}}}$${{i(\mathbb{X})\cup
Z\mathbb{Y}}}$${{\bigoplus_{\delta\in\Delta}\mathbb{D}_{s_{\delta}}^{k_{\delta}+1}}}$${\mathbb{Y}}$$\scriptstyle{(dy_{s_{\delta}},y_{t_{\delta}})}$$\scriptstyle{y_{s_{\delta}}}$$\scriptstyle{\lrcorner}$

This realises $i:\mathbb{X}\to\mathbb{Y}$ as a cofibration via the composition
$\mathbb{X}\xrightarrow{\cong}i(\mathbb{X})\rightarrowtail i(\mathbb{X})\cup
Z\mathbb{Y}\rightarrowtail\mathbb{Y}.$ ∎

######  Corollary 6.26.

If $\mathbb{X}\in p\mathsf{Ch}^{*}_{\mathbb{Q}}$ is tame then $\mathbb{X}$ is
cofibrant in the interval sphere model structure.

######  Example 6.27.

The open interval $\mathbb{I}_{(s,t)}\otimes S^{k}\in
p\mathsf{Ch}^{*}_{\mathbb{Q}}$ for $k\geq 1$ is an example of a cofibrant
object which is locally compact but not tame. Such an object can be
constructed via a sequence of attachments along
$\mathbb{S}^{k+1}_{[s+\epsilon_{i},t)}\hookrightarrow\mathbb{D}^{k+1}_{s+\epsilon_{i}}$
for any descreasing sequence $(\epsilon_{i}\mid
i\in\mathbb{N},\epsilon_{i}\leq t-s)$ that converges to $0$. We conjecture
that locally compact fully characterises the connected cofibrant objects, but
leave this open for future work.

######  Example 6.28 (Cofibrancy in Degree $0$).

Cofibrant objects in the interval sphere model structure need not be interval
decomposable. Indeed, consider the complex
$\mathbb{I}^{0}_{[0,0]}=\mathbb{I}_{[0,0]}\otimes S^{0}$ with $\mathbb{Q}$ in
degree $0$ at time $0$ and zero elsewhere. This complex is the colimit of the
sequence

$\mathbb{I}_{[0,1)}^{0}\to\mathbb{I}_{[0,\frac{1}{2})}^{0}\to\cdots\to\mathbb{I}_{[0,\frac{1}{n})}^{0}\to\mathbb{I}_{[0,\frac{1}{n+1})}^{0}\to\cdots$

in which each map fits into a pushout square

${\mathbb{S}_{[\frac{1}{n+1},\frac{1}{n})}^{0}}$${\mathbb{I}_{[0,\frac{1}{n})}^{0}}$${\mathbb{D}_{\frac{1}{n+1}}^{0}}$${\mathbb{I}_{[0,\frac{1}{n+1})}^{0}.}$${\ulcorner}$

This description exhibits $\mathbb{I}_{[0,0]}^{0}$ as an $\mathbb{I}$-cell
complex, thus as a cofibrant object in the interval sphere model structure on
$p\mathsf{Ch}^{*}$. However, this complex is not interval decomposable.

###  6.4. Tame minimal models in $p\mathsf{Ch}^{*}_{\mathbb{Q}}$

In many respects persistent cohomology functions as the persistent minimal
model in the category $(p\mathsf{Ch}^{*}_{\mathbb{Q}})^{\mathsf{tame}}$. In
[Roi93], the author presents a definition of minimal model for an arbitrary
model category. Over $\mathbb{Q}$, the minimal model of $X\in\mathsf{Ch}^{*}$
is given by a weak-equivalence $m:HX\to X$ [Roi93, Prop. 2.1] where $HX$ is
the cohomology of $X$ with trivial coboundary.

We can translate Definition 5.5 of persistent minimal models in
$p\mathsf{CGDA}_{\mathbb{Q}}$ to the category of
$p\mathsf{Ch}^{*}_{\mathbb{Q}}$ as follows. A tame minimal model of
$\mathbb{X}\in(p\mathsf{Ch}^{*}_{\mathbb{Q}})^{\mathsf{tame}}$ consists of

  1. (1)

a minimal model $m(r):H\mathbb{X}(r)\to\mathbb{X}(r)$ for each $r$ in the
discretization of $\mathbb{X}$,

  2. (2)

a minimal model

${{H\mathbb{X}(r)}}$${{H\mathbb{X}(r+1)}}$${{\mathbb{X}(r)}}$${{\mathbb{X}(r+1)}}$$\scriptstyle{m(r+1)}$$\scriptstyle{\mathbb{X}(r<r+1)}$$\scriptstyle{m(r)}$$\scriptstyle{h(r)}$

of the map $\mathbb{X}(r<r+1)$ for each $r<r+1$ such the vertical maps agree
with $(1)$.

Thus, by gluing along the maps $m(r)$ we see that persistent cohomology
$H\mathbb{X}$ is the minimal model of $\mathbb{X}$. Since $H\mathbb{X}$ is
tame, it is degree wise interval decomposable. It is clear that we can
reconstruct $H\mathbb{X}$ as a sequence of pushouts

${{\mathbb{S}^{k+1}(H^{k}\mathbb{X})}}$${{H^{\leq
k-1}\mathbb{X}}}$${0}$${{\mathbb{D}^{k+1}(H^{k}\mathbb{X})}}$${{H^{\leq
k}\mathbb{X}}}$${{H^{k}\mathbb{X}}}$$\scriptstyle{0}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$

along degree-$k$ persistence modules $H^{k}\mathbb{X}$. Note the structural
similarities with Theorem 5.13.

In $p\mathsf{Ch}^{*}_{\mathbb{Q}}$ and $p\mathsf{CGDA}_{\mathbb{Q}}$, the cell
decompositions we have given in degree $k$ are in bijective correspondence
with the intervals in $k$-th persistent cohomology and rational homotopy
groups respectively (See Theorem 7.5, part 3). A significant difference is
that the attaching map can be non-trivial in $p\mathsf{CGDA}_{\mathbb{Q}}$
minimal models, which prevents the minimal model from being the free-algebra
$\bigotimes_{k}\Lambda H^{k}\mathbb{C}_{m_{k-1}}$ over the degree $k$
generators. This is reflected at the level of copersistent spaces by non-
trival Postnikov invariants, as we will see later.

###  6.5. Model structure on persistent CDGAs

####  6.5.1. Transfer of the interval sphere model structure

In this section we show that the model structure constructed above on
$p\mathsf{Ch}^{*}_{\mathbb{Q}}$ can be transferred to the category
$p\mathsf{CGDA}_{\mathbb{Q}}$ of persistent CDGAs. This will be done by means
of the free-forgetful adjunction

${\Lambda:p\mathsf{Ch}^{*}_{\mathbb{Q}}}$${p\mathsf{CGDA}_{\mathbb{Q}}:U}$${\scriptscriptstyle\boldsymbol{\bot}}$

where the free algebra functor $\Lambda:\mathsf{Ch}^{*}\to\mathsf{CGDA}$ is
extended to persistent objects in the obvious way; when $\mathbb{X}\in
p\mathsf{Ch}^{*}_{\mathbb{Q}}$ and $s\geq 0$ is a real number, we define

$(\Lambda\mathbb{X})(s)=\Lambda\mathbb{X}(s)$

and the structure maps are given by $(\Lambda\mathbb{X})(s\leq
t)=\Lambda(\mathbb{X}(s\leq t))$ for all $0\leq s\leq t$. In other words,
$\Lambda\mathbb{X}$ is just the composite functor
$\Lambda\circ\mathbb{X}:\mathbb{R}_{+}\to\mathsf{Ch}^{*}_{\mathbb{Q}}\to\mathsf{CGDA}_{\mathbb{Q}}$.

The category $p\mathsf{CGDA}_{\mathbb{Q}}$ is locally finitely presentable,
hence the small object argument can be applied to any set of maps. Its compact
objects are the persistent CDGAs $\mathbb{A}$ such that $\mathbb{A}(t)$ is a
$\mathbb{Q}$-algebra of finite presentation for every $t\in\mathbb{R}_{+}$. We
denote by $U^{-1}\mathbb{W}$ the class of maps $f:\mathbb{A}\to\mathbb{B}$ in
$p\mathsf{CGDA}_{\mathbb{Q}}$ such that $Uf\in\mathbb{W}$. In other words,
$f(s)$ is a quasi-isomorphism for every $s\in\mathbb{R}_{+}$. We also write

$\displaystyle\Lambda\mathbb{I}$
$\displaystyle=\big{\\{}\>\Lambda\mathbb{S}^{k}_{[s,t)}\to\Lambda\mathbb{D}^{k}_{s}\>|\>0\leq
s\leq t\>\big{\\}}$ $\displaystyle\Lambda\mathbb{J}$
$\displaystyle=\big{\\{}\>\Lambda\mathbb{D}^{k}_{t}\to\Lambda\mathbb{D}^{k}_{s}\>|\>0\leq
s\leq t\>\big{\\}}$

for the images of $\mathbb{I},\mathbb{J}\subseteq
p\mathsf{Ch}^{*}_{\mathbb{Q}}$ under the functor $\Lambda$.

######  Theorem 6.29.

There is a combinatorial model structure on $p\mathsf{CGDA}_{\mathbb{Q}}$ for
which $U^{-1}\mathbb{W}$ is the class of weak equivalences and the sets
$\Lambda\mathbb{I},\Lambda\mathbb{J}$ are respectively the generating
cofibrations and trivial cofibrations.

###### Proof.

It suffices to check that the conditions for the recognition theorem of D. Kan
[Hir03, Theorem 11.3.2] are satisfied. For this it is enough to show that

$\text{cell}(\Lambda\mathbb{J})\subseteq U^{-1}\mathbb{W}$

where $\text{cell}(\Lambda\mathbb{J})\subseteq p\mathsf{CGDA}$ denotes the
class of relative $\Lambda\mathbb{J}$-cell complexes. Since
$\Lambda\mathbb{J}\subseteq U^{-1}\mathbb{W}$, it suffices to show that the
class $U^{-1}\mathbb{W}$ is closed under cobase change and transfinite
composition. The latter condition is clear since the forgetful functor $U$
preserves filtered colimits. We thus need to prove that the class
$\Lambda\mathbb{J}$ is closed under cobase change. To this end consider a
pushout square

${{\Lambda\mathbb{D}^{k}_{t}}}$${\mathbb{A}}$${{\Lambda\mathbb{D}^{k}_{s}}}$${\mathbb{B}}$${\ulcorner}$

in $p\mathsf{CGDA}_{\mathbb{Q}}$. One sees immediately that
$\mathbb{A}(u)\to\mathbb{B}(u)$ is an isomorphism when $u<s$ or $t\leq u$.
When $s\leq u<t$, we find
$\mathbb{B}(u)\cong\mathbb{A}(u)\otimes\Lambda(a_{u},da_{u})$ so that the
inclusion $\mathbb{A}(u)\subseteq\mathbb{B}(u)$ is a quasi-isomorphism. ∎

####  6.5.2. Minimal Models are Cell Complexes

The definition of Sullivan and of minimal Sullivan algebras bears a strong
analogy with cell complexes and CW-complexes in topology. The role of cell
inclusions $\partial\Delta^{n}\to\Delta^{n}$ in topology is played by the
elementary Hirsch extensions $\Lambda S^{n}\to\Lambda D^{n}$ in the world of
CDGAs. The formalism of model categories renders precise this analogy: if $X$
is a simply connected space of finite type, a minimal model $\mathcal{M}_{X}$
for $X$ is a cofibrant replacement of $\mathcal{A}_{PL}(X)$ in the projective
model struture on $\mathsf{CGDA}$. The algebra $\mathcal{M}_{X}$ is in fact a
cell complex with respect to the set

$\Lambda I=\big{\\{}\>\Lambda S^{n}\to\Lambda D^{n}\>|\>n\geq 0\>\big{\\}}$

of generating cofibrations from Section 6.1. Denoting by

$\mathbb{Q}\subseteq\mathcal{M}_{2}\subseteq\mathcal{M}_{3}\subseteq\mathcal{M}_{4}\subseteq\cdots\>\subseteq\bigcup_{k\geq
0}\mathcal{M}_{k}=\mathcal{M}_{X}$

the skeletal filtration of $\mathcal{M}_{X}$, every inclusion
$\mathcal{M}_{k-1}\subseteq\mathcal{M}_{k}$ fits in a pushout square

${\Lambda S^{k+1}(V_{k})}$${\mathcal{M}_{k-1}}$${\Lambda
D^{k+1}(V_{k})}$${\mathcal{M}_{k}}$${\ulcorner}$

for some choice of $\mathbb{Q}$-vector space $V_{k}$. Any choice of such
cellular structure on $\mathcal{M}_{X}$ yields for each $k\geq 1$ the well
known isomorphisms

$V_{k}\cong\pi^{\mathbb{Q}}_{k}(X)$

which gives a correspondence between the set of $k$-cell of $\mathcal{M}_{X}$
and a basis for the $\mathbb{Q}$-vector space $\pi_{k}^{\mathbb{Q}}(X)$. A
guiding principle for our construction of the model structure of Theorem 6.29
on persistent CDGAs was the ability to extend this connection to the
persistent setting.

The whole setup has now been made for this correspondence to follow. Let
$\mathbb{X}\in p^{*}\mathsf{sSet}$ be a tame copersistent space whose
components $\mathbb{X}(s)$ are simply connected and of finite type for every
$s\geq 0$. The persistent CDGA $\mathcal{A}_{PL}(\mathbb{X})$ admits a minimal
model $\mathbb{M}_{\mathbb{X}}$ which can be build constructively by Theorem
5.13. We denote by

$\mathbb{Q}\subseteq\mathbb{M}_{2}\subseteq\mathbb{M}_{3}\subseteq\mathbb{M}_{4}\subseteq\cdots\>\subseteq\mathsf{colim}_{k}\>\mathbb{M}_{k}=\mathbb{M}_{\mathbb{X}}$

its skeletal filtration. It was shown in Theorem 5.13 that each inclusion
$\mathbb{M}_{k-1}\subseteq\mathbb{M}_{k}$ sits in a pushout square

${\Lambda\mathbb{S}^{k+1}(\mathbb{V}_{k})}$${\mathbb{M}_{k-1}}$${\Lambda\mathbb{D}^{k+1}(\mathbb{V}_{k})}$${\mathbb{M}_{k}}$${\ulcorner}$

for some choice of persistence module $\mathbb{V}_{k}$, exhibiting
$\mathbb{M}_{k}$ as a persistent Hirsch extension
$\mathbb{M}\langle\mathbb{V}_{k}\rangle$. The data of the persistence modules
$(\mathbb{V}_{k})_{k\geq 2}$, together with the attaching maps
$d:\mathbb{V}_{k}\to\mathbb{M}_{k-1}[1]$ is what we call a cell structure on
the persistent CDGA $\mathbb{M}_{\mathbb{X}}$.

Any such cell structure exhibits $\mathbb{M}_{\mathbb{X}}$ as a
$\Lambda\mathbb{I}$-cell complex, where

$\Lambda\mathbb{I}=\big{\\{}\>\Lambda\mathbb{S}^{k}_{[s,t)}\to\Lambda\mathbb{D}^{k}_{s}\>\>|\>\>k\geq
0\>\>,\>\>0\leq s\leq t\leq\infty\>\big{\\}}$

is the set of persistent cells introduced in Section 6.2. Moreover, since the
indecomposables $Q^{k}(\mathbb{M}_{X})$ correspond to the generators
$\mathbb{V}_{k}$ of the $k$-th persistent Hirsch extension, Proposition 5.6
ensures that any choice of cell structure on $\mathbb{M}_{\mathbb{X}}$ yield
an isomorphism of persistence modules

$\mathbb{V}_{k}\cong\pi^{\mathbb{Q}}_{k}(\mathbb{X}).$

This successfully carries the correspondence between cells of the minimal
model and generators of the rational homotopy groups to the persistent
setting.

###  6.6. Model structure on copersistent spaces

####  6.6.1. Spatial Realisation

The algebra of polynomial forms on a space provides a connection between
topology and algebra. There is a construction going the other way which we
briefly recall below.

The functor
$\mathcal{A}_{PL}:\mathsf{sSet}^{\mathsf{op}}\to\mathsf{CGDA}_{\mathbb{Q}}$
which takes a simplicial set to its algebra of polynomial forms admits a left
adjoint, constructed by the usual nerve-realization procedure. If
$\mathcal{A}$ is a CDGA, its spatial realization $\langle\mathcal{A}\rangle$
is the simplicial set whose $n$-simplices are

$\langle\mathcal{A}\rangle_{n}=\mathsf{Hom}_{\mathsf{CGDA}}(\mathcal{A},\Omega_{n})$

with faces and degeneracies induced by those of the simplicial object
$\Omega:\mathbf{\Delta}^{\mathsf{op}}\to\mathsf{CGDA}$. The following
proposition describes how spatial realizations relates spheres and disks in
CGDAs with their topological analogues.

######  Proposition 6.30.

For every $n\geq 2$, there are $\mathbb{Q}$-equivalences

$\langle\Lambda S^{n}\rangle\simeq
K(\mathbb{Q},n)\>\>\>\>\>\>\>\>\>\>\>\>\>\>\text{and}\>\>\>\>\>\>\>\>\>\>\>\>\>\>\langle\Lambda
D^{n}\rangle\simeq*.$

As advertised above, the spatial realization functor is part of an adjoint
pair

${\langle-\rangle:\mathsf{CGDA}}$${\mathsf{sSet}^{\mathsf{op}}:\mathcal{A}_{PL}}$${\scriptscriptstyle\boldsymbol{\bot}}$

which can be used to compare the homotopy theory of CDGA to that of spaces.
Recall that there is a model structure on $\mathsf{sSet}$ whose

  1. (1)

cofibrations are monomorphisms,

  2. (2)

weak equivalences are $H^{*}(-;\mathbb{Q})$-isomorphisms, ie. maps that induce
isomorphisms on singular cohomology with $\mathbb{Q}$-coefficients.

This model structure is a left Bousfield localization of the Kan model
structure on simplicial sets. We call it the $\mathbb{Q}$-model structure and
call its fibrations and weak equivalences $\mathbb{Q}$-fibrations and
$\mathbb{Q}$-equivalences respectively. The following result, first proved by
Bousfield-Gugenjeim in [BG76], provides the basis for the use of CDGAs in
rational homotopy theory.

######  Theorem 6.31 ([BG76], Section 8).

The functors $\langle-\rangle$ and $\mathcal{A}_{PL}$ form a Quillen
adjunction

${\langle-\rangle:\mathsf{CGDA}}$${\mathsf{sSet}^{\mathsf{op}}:\mathcal{A}_{PL}}$${\scriptscriptstyle\boldsymbol{\bot}}$

where $\mathsf{CGDA}$ is endowed with the projective model structure and
$\mathsf{sSet}^{op}$ with the (opposite of) the $\mathbb{Q}$-model structure.
This adjunction restricts to a Quillen equivalence between the full
subcategories of simply-connected CDGAs of finite type $\mathsf{CGDA}^{\geq
2,ft}\subseteq\mathsf{CGDA}$ and that of simply-connected spaces of finite
type $\mathsf{sSet}_{\geq 2}^{ft}\subseteq\mathsf{sSet}$.

######  Remark 6.32.

The theorem stated above contains all the properties of the spatial
realization functor that will be necessary to our later developments. In
particular, given a Hirsch extension $\mathcal{A}\subseteq\mathcal{A}\langle
V\rangle$, the induced map of simplicial sets

$\big{\langle}\mathcal{A}\langle
V\rangle\big{\rangle}\to\langle\mathcal{A}\rangle$

is a $\mathbb{Q}$-fibration, which is trivial in case the Hirsch extension was
a quasi-isomorphism. Moreover, if $\mathcal{M}_{X}$ is a minimal model for a
simply-connected space $X$, the composite map
$X\to\langle\mathcal{A}_{PL}(X)\rangle\to\langle\mathcal{M}_{X}\rangle$ is a
$\mathbb{Q}$-equivalence. Since the space $\langle\mathcal{M}_{X}\rangle$ is a
rational space (its homotopy groups are $\mathbb{Q}$-vector spaces), it
follows that this map is in fact a $\mathbb{Q}$-localization.

####  6.6.2. Copersistent spaces

In this subsection we establish a persistent variant of the Quillen pair
described above, setting the ground for the study of persistent rational
homotopy theory. Recall that we denote by

$p^{*}\mathsf{sSet}=\mathsf{Fun}(\mathbb{R}_{+}^{\mathsf{op}},\mathsf{sSet})\cong\mathsf{Fun}(\mathbb{R}_{-},\mathsf{sSet})$

the category of copersistent spaces. Similarly as above, the functors
$\langle-\rangle,\mathcal{A}_{PL}$ can be extended to persistent CDGAs and
copersistent spaces respectively. This yields the following contravariant
adjunction:

${\langle-\rangle:p\mathsf{CGDA}}$${(p^{*}\mathsf{sSet})^{\mathsf{op}}:\mathcal{A}_{PL}.}$${\scriptscriptstyle\boldsymbol{\bot}}$

The category $p^{*}\mathsf{sSet}=\mathsf{Fun}(\mathbb{R}_{-},\mathsf{sSet})$
of copersistent spaces admits the projective $\mathbb{Q}$-model structure.
This structure is described as follows: a map $f:\mathbb{X}\to\mathbb{Y}$ in
$p^{*}\mathsf{sSet}$ is:

  1. (1)

a fibration if $f(s):\mathbb{X}(s)\to\mathbb{Y}(s)$ is a
$\mathbb{Q}$-fibration for every $s\leq 0$,

  2. (2)

a weak equivalence if $f(s):\mathbb{X}(s)\to\mathbb{Y}(s)$ is a
$\mathbb{Q}$-equivalence for every $s\leq 0$.

######  Proposition 6.33.

The contravariant adjunction

${\langle-\rangle:p\mathsf{CGDA}}$${(p^{*}\mathsf{sSet})^{\mathsf{op}}:\mathcal{A}_{PL}}$${\scriptscriptstyle\boldsymbol{\bot}}$

is a Quillen pair when $(p^{*}\mathsf{sSet})^{\mathsf{op}}$ is endowed with
the opposite of the projective $\mathbb{Q}$-model structure, and
$p\mathsf{CGDA}$ is given the interval sphere model structure of Theorem 6.29.

###### Proof.

It suffices to check that $\langle-\rangle$ sends the generating (trivial)
cofibrations in $p\mathsf{CGDA}$ to (trivial) $\mathbb{Q}$-fibrations in
$p^{*}\mathsf{sSet}$. Given a map $f\in\Lambda\mathbb{I}$ of the form
$\Lambda\mathbb{S}^{k}_{[s,t)]}\to\Lambda\mathbb{D}^{k}_{s}$ its component
$\langle f\rangle(u)=\langle f(u)\rangle$ is the map induced on spatial
realizations by the Hirsch extension

$\Lambda S^{k}\subseteq\Lambda D^{k}$

when $s\leq u<t$ and an isomorphism otherwise. These maps are all
$\mathbb{Q}$-fibrations as a consequence of Theorem 6.31. Similarly, the
components of any $f\in\Lambda\mathbb{J}$ are Hirsch extensions and quasi-
isomorphisms, hence map to trivial $\mathbb{Q}$-fibrations under spatial
realization. ∎

##  7\. Application: Postnikov barcodes

In classical rational homotopy theory, Postnikov towers of spaces are dual to
the sequence of Hirsch extensions in the construction of their minimal models.
In this section, we will show that our level-wise construction of the
persistent minimal model is Eckmann-Hilton dual to a copersistent version of
the Postnikov tower. This construction permits us to decompose the (rational)
Postnikov invariant222In some texts, the Postnikov invariant is called the
$K$-invariant. We use the terminology to disambiguate it from our use of $k$
as the degree. of a copersistent space into a ‘Postnikov barcode’ of
intervals.

###  7.1. Postnikov Towers

For a more comprehensive introduction to rational Postnikov towers see
[Ber12]. A (rational) Postnikov tower of a space $X$ is given by a diagram

${{X_{0}}}$${{X_{1}}}$${{X_{2}}}$${\cdots}$${X}$${X}$${X}$${\cdots}$

such that

  * •

the map $X\rightarrow X_{k}$ is a $k$-rational equivalence;

  * •

$\pi^{\mathbb{Q}}_{i}(X_{k})=0$ for all $i>k$.

In the case that $X$ is a simply connected space, $X$ admits a rational
Postnikov tower made of principal fibrations. That is, it can be realised as a
pullback fitting into the following diagram

${K(\pi^{\mathbb{Q}}_{k}(X),k)}$${X_{k}}$${PK(\pi^{\mathbb{Q}}_{k}(X),k+1)\simeq*}$${*}$${X_{k-1}}$${K(\pi^{\mathbb{Q}}_{k}(X),k+1)}$${\lrcorner}$${\lrcorner}$$\scriptstyle{\sigma}$

Recall the classical result that there is a natural isomorphism

$[X_{k-1},K(\pi_{k}(X),k+1)]\xrightarrow{\cong}H^{k+1}(X_{k-1};\pi_{k}(X))$
(12)

between the set of based homotopy classes of maps
$[X_{k-1},K(\pi^{\mathbb{Q}}_{k}(X),k+1)]$ and the cohomology
$H^{k+1}(X_{k-1};\pi^{\mathbb{Q}}_{k}(X))$ in $\pi^{\mathbb{Q}}_{k}(X)$. This
isomorphism is constructed by taking the fundamental class $[\alpha]\in
H^{k+1}(K(\pi^{\mathbb{Q}}_{k}(X),k+1);\pi^{\mathbb{Q}}_{k}(X))$ and applying
the induced map $H^{k+1}(\sigma)([\alpha])$. The resultant class
$H^{k+1}(\sigma)([\alpha])$ is called the $(k-1)$-th (rational) Postnikov
invariant. Along with the homotopy groups $\pi^{\mathbb{Q}}_{i}(X)$, they can
be used to completely recover the (rational) homotopy type of $X$.

###  7.2. Copersistent Eilenberg-MacLane Spaces

In this section we define copersistent rational Eilenberg-MacLane spaces. Much
like the classical case, these are dual to interval sphere via spatial
realisation.

######  Definition 7.1.

A (connected) copersistent rational Eilenberg-MacLane space
$K(\mathbb{I}_{[s,t)},k)$ over an inteval $\mathbb{I}_{[s,t)}$ is a
copersistent space given by

  1. (1)

$K(\mathbb{I}_{[s,t)},k)(i)\simeq\begin{cases}K(\mathbb{Q},k)&s\leq i<t\\\
K(\mathbb{Q},0)&\text{else}\end{cases}$

  2. (2)

$K(\mathbb{I}_{[s,t)},k)(j\geq i)$ is an isomorphism for $s\leq i\leq j<t$.

The above is a tame copersistent space. By construction, the only non-trivial
persistent rational homotopy groups of this object are
$\pi_{k}^{\mathbb{Q}}(K(\mathbb{I}_{[s,t)},k))\cong\mathbb{I}_{[s,t)}^{\vee}$.
Products of persistent rational Eilenberg-MacLane spaces are defined point-
wise and are also tame when the number of products taken is finite. The
interval spheres in CDGA correspond to CDGA models of copersistent rational
Eilenberg-MacLane spaces.

######  Lemma 7.2.

The spatial realisation of the CDGA interval sphere
$\langle\Lambda\mathbb{S}_{[s,t)}^{k}\rangle$ is a copersistent rational
Eilenberg-MacLane space $K(\mathbb{I}_{[s,t)},k)$.

###### Proof.

Proposition 6.30 implies that there is a rational equivalence
$\langle\Lambda\mathbb{S}^{k}_{[s,t)}\rangle(i)=\langle\Lambda
S^{k}\rangle\simeq K(\mathbb{Q},k)$ for all $s\leq i<t$. Further, for all
$s\leq i\leq j<t$, since $\Lambda\mathbb{S}^{k}_{[s,t)}(i\leq j)$ is the
identity then $\langle\Lambda\mathbb{S}^{k}_{[s,t)}\rangle(i\leq j)$ is the
identity as well. ∎

For a persistent vector space $\mathbb{V}$ with interval decomposition
$\oplus_{\gamma}\mathbb{I}_{[s_{\gamma},t_{\gamma})}$, we use the notation
$K(\mathbb{V},k)$ to denote
$\prod_{\gamma}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k)$. Since the interval
decomposition is unique up to isomorphism, $K(\mathbb{V},k)$ is also unique up
to isomorphism.

######  Corollary 7.3.

Let $\mathbb{V}=\bigoplus_{\gamma}\mathbb{I}_{[s_{\gamma},t_{\gamma})}$ be a
persistent graded vector space. Then
$\langle\Lambda\mathbb{S}^{k}(\mathbb{V})\rangle$ is a $K(\mathbb{V},k)$.

###### Proof.

Firstly, note that
$\langle\Lambda\mathbb{S}^{k}(\mathbb{V})\rangle=\langle\bigotimes_{\gamma}\Lambda\mathbb{S}^{k}_{[s_{\gamma},t_{\gamma})}\rangle$.
The result then follows from Lemma 7.2 in combination with the fact that
tensor products of persistent CDGAs are pointwise coproducts, products of
persistent (contravariant) spaces are pointwise products and that
$\langle-\rangle$ is a left adjoint so takes coproducts to products. ∎

###  7.3. Copersistent Postnikov Towers

Classically, minimal models are dual to Postnikov towers via spatial
realisation. Using our construction from the previous sections, we show that
this duality also holds in the persistent case.

######  Definition 7.4.

Let $\mathbb{X}$ be a copersistent space. A copersistent Postnikov tower over
$\mathbb{X}$ is the data in the diagram

${{\mathbb{X}_{0}}}$${{\mathbb{X}_{1}}}$${{\mathbb{X}_{2}}}$${\cdots}$${\mathbb{X}}$${\mathbb{X}}$${\mathbb{X}}$${\cdots}$

such that for each $r$ we have a Postnikov tower

${{\mathbb{X}_{0}(r)}}$${{\mathbb{X}_{1}(r)}}$${{\mathbb{X}_{2}(r)}}$${\cdots}$${{\mathbb{X}(r)}}$${{\mathbb{X}(r)}}$${{\mathbb{X}(r)}}$${\cdots}$

at the level of spaces.

This implies that the arrows $\mathbb{X}_{k-1}\twoheadleftarrow\mathbb{X}_{k}$
are fibrations in the $\mathbb{Q}$-projective model structure on copersistent
spaces. We say a copersistent Postnikov tower is principal if each of the maps
$\mathbb{X}_{k}(r)\twoheadleftarrow\mathbb{X}_{k+1}(r)$ are principal
fibrations for all $r,k$. For $\mathcal{M}_{X}$ a minimal model of $X$, the
adjoint map $X\rightarrow\langle\mathcal{M}_{X}\rangle$ of
$\mathcal{M}_{X}\xrightarrow{\simeq}\mathcal{A}_{PL}(X)$ from Theorem 6.31 is
a natural $\mathbb{Q}$-equivalence. This implies that we can extend it a
natural $\mathbb{Q}$-equivalence of copersistent spaces
$\mathbb{X}\rightarrow\langle\mathbb{M}\rangle$ where $\mathbb{M}$ is the
persistent minimal model of $\mathbb{X}$.

######  Theorem 7.5.

Let $\mathbb{X}$ be a tame, simply-connected copersistent space and
$\mathbb{M}_{2}\rightarrowtail\mathbb{M}_{3}\rightarrowtail\ldots$ be the
filtered minimal model over $\mathcal{A}_{PL}(\mathbb{X})$ given in Theorem
5.13, with tame cones $\\{\mathbb{C}_{m_{k}}\mid k\in\mathbb{N}\\}$. Then
spatial realisation induces a principal persistent Postnikov tower for
$\langle\mathbb{M}\rangle=\langle\mathsf{colim}\mathbb{M}_{k}\rangle$ where
the $k$-th section is obtained in the following pullback diagram

${{K(\pi_{k}^{\mathbb{Q}}(\mathbb{X})^{\vee},k)}}$${{\langle\mathbb{M}_{k}\rangle}}$${{\langle\mathbb{D}^{k+1}(\pi_{k}^{\mathbb{Q}}(\mathbb{X})^{\vee})\rangle}}$${{*}}$${{\langle\mathbb{M}_{k-1}\rangle}}$${{K(\pi_{k}^{\mathbb{Q}}(\mathbb{X})^{\vee},k+1)}}$$\scriptstyle{\sigma}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$
(13)

for all $k$. Moreover, the vertical maps in the above diagram are fibrations
in the projective $\mathbb{Q}$-model structure on copersistent spaces.

###### Proof.

Spatial realisation takes pointwise $k$-equivalences between Sullivan algebras
to pointwise $k$-rational equivalences between spaces. Since the
$k$-equivalences of persistent objects are determined pointwise, applying
spatial realisation to the filtered minimal model

${\mathbb{Q}}$${\mathbb{M}_{2}}$${\mathbb{M}_{3}}$${\cdots}$${\mathbb{M}}$${\mathbb{M}}$${\mathbb{M}}$${\cdots}$

yields a Postnikov tower

${{*}}$${{\langle\mathbb{M}_{2}\rangle}}$${{\langle\mathbb{M}_{3}\rangle}}$${\cdots}$${{\langle\mathbb{M}\rangle}}$${{\langle\mathbb{M}\rangle}}$${{\langle\mathbb{M}\rangle}}$${\cdots}$

Note that spatial realisation takes cofibrations in the interval sphere model
structure on persistent CDGAs to fibrations in copersistent spaces by
Proposition 6.33. Applying spatial realisation to Diagram 7 and applying
Corollary 7.3 yields

${{K(H^{k}\mathbb{C}_{m_{k-1}},k)}}$${{\langle\mathbb{M}_{k}\rangle}}$${{\langle\mathbb{D}^{k+1}(H^{k}\mathbb{C}_{m_{k-1}})\rangle}}$${{*}}$${{\langle\mathbb{M}_{k-1}\rangle}}$${{K(H^{k}\mathbb{C}_{m_{k-1}},k+1)}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$
(14)

which shows that the persistent rational homotopy groups of
$\langle\mathbb{M}\rangle$ are $H^{k}\mathbb{C}_{m_{k-1}}^{\vee}$. Since
$\langle\mathbb{M}\rangle\simeq_{\mathbb{Q}}\mathbb{X}$, it follows that
$H^{k}\mathbb{C}_{m_{k-1}}\cong\pi_{k}^{\mathbb{Q}}(\mathbb{X})^{\vee}$. ∎

If $\pi_{k}^{\mathbb{Q}}(\mathbb{X})^{\vee}$ has the interval decomposition
$H^{k}(\mathbb{C}_{m_{k-1}})\cong\bigoplus_{\gamma}\mathbb{I}_{[s_{\gamma},t_{\gamma})}$,
the classifying map factors as a product

$\langle\mathbb{M}_{k-1}\rangle\xrightarrow{\prod_{\gamma}\sigma_{\gamma}}\prod_{\gamma}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1)$

of maps between copersistent spaces. This provides much more structure than a
persistent Postnikov tower; it enables us to analyse $\sigma$ by analysing its
individual components. This shows - somewhat unexpectedly - that the interval
decomposition can be transferred from the persistence module
$H^{k}(\mathbb{C}_{m_{k-1}})$ to the category of copersistent spaces via
adjunctions from $p\mathsf{Ch}^{*}_{\mathbb{Q}}$ and
$p\mathsf{CGDA}_{\mathbb{Q}}$.

###  7.4. Postnikov Barcodes

In this section we explain how the rational Postnikov invariants of a tame
copersistent space split into a barcode of intervals. We show that trivial
Postnikov barcode implies that a copersistent space is rationally equivalent
to a product of copersistent Eilenberg-MacLane spaces and that the barcode can
be computed algebraically at the level of persistent CDGAs.

####  7.4.1. Copersistent Fundamental Classes

The Universal Coefficient Theorem states that cohomology of a space with
coefficients in $\mathbb{Q}^{n}$ is given by

$H^{k}(X;\mathbb{Q}^{n})=\mathsf{Hom}_{\mathbb{Q}}(H_{k}(X;\mathbb{Q}),\mathbb{Q}^{n})=\bigoplus_{n}\mathsf{Hom}_{\mathbb{Q}}(H_{k}(X;\mathbb{Q}),\mathbb{Q})=\bigoplus_{n}H^{k}(X;\mathbb{Q}).$

Since this correspondence is natural, the morphism $H^{k}(f;\mathbb{Q}^{n})$
in cohomology with $\mathbb{Q}^{n}$-coefficients induced by a map of space
$f:Y\to X$ respects the direct sum decomposition

$H^{k}(f;\mathbb{Q}^{n})=\bigoplus_{n}H^{k}(f;\mathbb{Q}):H^{k}(X;\mathbb{Q}^{n})\to
H^{k}(Y;\mathbb{Q}^{n}).$ (15)

Let
$K(\mathbb{I}_{[s,t)},k+1)=\langle\Lambda\mathbb{S}^{k+1}_{[s_{\gamma},t_{\gamma})}\rangle$.
The  generating fundamental class $\gamma_{s_{\gamma}}$ at time $s_{\gamma}$
is the isomorphism

$\gamma_{s_{\gamma}}:H_{k+1}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1)(s_{\gamma})\to\pi_{k+1}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1)(s_{\gamma})\cong\mathbb{Q}$

given by the inverse of the Hurewicz homomorphism. The Universal Coefficient
Theorem implies that $\gamma_{s_{\gamma}}$ corresponds to a cohomology class

$\gamma_{s_{\gamma}}\in
H^{k+1}\big{(}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1);\mathbb{Q}\big{)}(s_{\gamma}).$

Since the structure maps are identities between $s_{\gamma}$ and $t_{\gamma}$,

$\gamma_{p}=H^{k+1}\big{(}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1)(p>s_{\gamma});\mathbb{Q}\big{)}(\gamma_{s_{\gamma}})$

is the fundamental class at time $p$ for $s_{\gamma}\leq p<t_{\gamma}$.

Let $\mathbb{V}=\oplus_{\gamma}\mathbb{I}_{[s_{\gamma},t_{\gamma})}\in
p\mathsf{Vec}_{\mathbb{Q}}$ be finitely generated. Since the Hurewicz
homomorphism commutes with products, the fundamental class at time $p$ is
given by

$\displaystyle\bigoplus_{s_{\gamma}\leq
p<t_{\gamma}}\gamma_{p}:H_{k+1}K(\mathbb{V}(p),k+1)$
$\displaystyle\to\pi_{k+1}K(\mathbb{V}(p),k+1)$
$\displaystyle\bigoplus_{s_{\gamma}\leq
p<t_{\gamma}}\gamma_{p}:\bigoplus_{s_{\gamma}\leq
p<t_{\gamma}}H_{k+1}K(\mathbb{Q},k+1)$
$\displaystyle\to\bigoplus_{s_{\gamma}\leq
p<t_{\gamma}}\pi_{k+1}K(\mathbb{Q},k+1)$

which is equivalently a cohomology class in

$H^{k+1}\Big{(}K(\mathbb{V}(p),k+1);\mathbb{V}(p)\Big{)}\cong\mathsf{Hom}_{\mathbb{Q}}(\mathbb{V}(p),\mathbb{V}(p)).$

####  7.4.2. Postnikov Barcodes

Now let $\mathbb{X}_{k-1}=\langle\mathbb{M}_{k-1}\rangle$ in the copersistent
Postnikov tower in Theorem 7.5, with classifying map

$\mathbb{X}_{k-1}\xrightarrow{\prod_{\gamma}\sigma_{\gamma}}K(\pi^{\mathbb{Q}}_{k}(\mathbb{X}),k+1)=\prod_{\gamma}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1)$

where $\gamma\in\Gamma$ is an indexing of the interval decomposition
$\pi_{k}^{\mathbb{Q}}(\mathbb{X})^{\vee}=\bigoplus_{\gamma}\mathbb{I}_{[s_{\gamma},t_{\gamma})}$
for the copersistent rational homotopy groups. Identify each bar in the
interval decomposition with a generating fundamental class
$\gamma_{s_{\gamma}}\in
H^{k+1}K(\pi_{k}^{\mathbb{Q}}(\mathbb{X}),k+1)(s_{\gamma})$.

######  Definition 7.6.

The copersistent rational Postnikov components in degree $k$ are the maps

$H^{k+1}(\sigma_{\gamma}):H^{k+1}\big{(}K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1);\mathbb{Q}\big{)}\to
H^{k+1}(\mathbb{X}_{k-1};\mathbb{Q})$

indexed by $\gamma\in\Gamma$.

Since the fundamental class at a given time $r$ is given by the sum of
generating fundamental classes $\gamma_{r}$ where $s_{\gamma}\leq
r<t_{\gamma}$, copersistent rational Postnikov components recover the
pointwise Postnikov invariants in the following way.

######  Proposition 7.7.

The point-wise, classical Postnikov invariant at any $r\in\mathbb{R}_{+}$ can
be recovered from the copersistent Postnikov components by the sum

$\bigoplus_{\gamma:s_{\gamma}\leq
r<t_{\gamma}}H^{k+1}(\sigma_{\gamma})(\gamma_{r})\in
H^{k+1}\big{(}\mathbb{X}_{k-1}(r);\pi_{k}^{\mathbb{Q}}(\mathbb{X}(r))\Big{)}.$

###### Proof.

Let $\pi_{k}^{\mathbb{Q}}(\mathbb{X}(r))=\mathbb{Q}^{n}$. Since $n$ is the
number of non-trival bars at time $r$, we have
$n=\lvert\\{\Gamma_{r}\\}\rvert=\lvert\\{\gamma\in\Gamma\mid s_{\gamma}\leq
r<t_{\gamma}\\}\rvert$. The fundamental class at time $r$ is given by

$\bigoplus_{\Gamma_{r}}\gamma_{r}\in\bigoplus_{n}H^{k+1}(K(\pi_{k}^{\mathbb{Q}}(\mathbb{X}(r)),k+1);\mathbb{Q})$

and the Postnikov invariant is given by evaluation

$\bigoplus_{\Gamma_{r}}H^{k+1}(\sigma;\mathbb{Q})(\gamma_{r})\in\bigoplus_{n}H^{k+1}(\mathbb{X}_{k-1};\mathbb{Q})$

under the morphism
$H^{k+1}(\sigma;\mathbb{Q}^{n})=\bigoplus_{n}H^{k+1}(\sigma;\mathbb{Q})$. By
definition

$H^{k+1}(\sigma;\mathbb{Q})(\gamma_{r})=H^{k+1}(\sigma_{\gamma};\mathbb{Q})(\gamma_{r})$

proving the result. ∎

Since
$H^{k+1}(K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1);\mathbb{Q})=\mathbb{I}_{[s_{\gamma},t_{\gamma})}$,
the copersistent Postnikov components are thus determined by evaluation on the
fundamental classes

$\gamma_{s_{\gamma}}\mapsto
H^{k+1}(\sigma_{\gamma})(\gamma_{s_{\gamma}})\in\text{Ker}\,H^{k+1}(\mathbb{X}_{k-1}(s_{\gamma}<t_{\gamma}))$

which must vanish by at least time $t_{\gamma}$. The non-trivial persistent
cohomology classes in the image of
$H^{k+1}(\sigma)=\bigoplus_{\gamma}H^{k+1}(\sigma_{\gamma})$ thus correspond
to a summary of all of the non-trivial Postnikov invariants across the index
parameter. We give this new rational invariant of a copersistent space the
following definition.

######  Definition 7.8.

The Postnikov barcode in degree $k$ is the interval decomposition of the
persistence module $\mathrm{im}(H^{k+1}(\sigma))$.

######  Proposition 7.9.

If the Postnikov barcode is trivial in all degrees then $\mathbb{X}$ is
rationally equivalent to a product of persistent Eilenberg-MacLane spaces
$\prod_{k}K(\pi_{k}^{\mathbb{Q}}(\mathbb{X}),k)$.

###### Proof.

For a tame copersistent space $\mathbb{X}$, if the Postnikov barcode is
trivial then the classifying maps $\sigma:\mathbb{X}_{k-1}\to
K(\pi_{k}^{\mathbb{Q}}(\mathbb{X}),k+1)$ are point-wise nullhomotopic for all
$k$. The result follows by applying Theorem 7.5. ∎

The non-triviality of the Postnikov barcode is thus a rough measure of how far
a copersistent space is from being a trivial product of copersistent
Eilenberg-MacLane space. Further, it can be computed algebraically at the
level of persistent minimal models due to the work of earlier sections.

######  Lemma 7.10.

Let
$\tau_{\gamma}:\Lambda\mathbb{S}^{k+1}_{[s_{\gamma},t_{\gamma})}\to\mathbb{M}_{k-1}$
be the degree $k$ attaching map in a minimal model $\mathbb{M}$ of
$\mathbb{X}$ and
$\sigma_{\gamma}=\langle\tau_{\gamma}\rangle:\langle\mathbb{M}_{k-1}\rangle\to
K(\mathbb{I}_{[s_{\gamma},t_{\gamma})},k+1)$ its spatial realisation. Then
there is an isomorphism

$H^{k+1}(\tau_{\gamma})\cong H^{k+1}(\sigma_{\gamma})$

###### Proof.

The minimal model of $K(\mathbb{Q},k+1)$ is $\Lambda S^{k+1}$. For a finite
type, simply connected cofibrant CDGA $\mathcal{A}$ we have an isomorphism

$H^{k+1}(\mathcal{A})\cong[\Lambda
S^{k+1},\mathcal{A}]\cong[\langle\mathcal{A}\rangle,K(\mathbb{Q},k+1)]\cong
H^{k+1}(\langle\mathcal{A}\rangle;\mathbb{Q})$

which is natural in both variables. This yields equivalent diagrams of
persistence modules

${{[\Lambda
S^{k+1},\Lambda\mathbb{S}^{k+1}_{[s_{\gamma},t_{\gamma})}]}}$${{[\Lambda
S^{k+1},\mathbb{M}_{k-1}]}}$${{[\langle\Lambda\mathbb{S}_{[s_{\gamma},t_{\gamma})}^{k+1}\rangle,K(\mathbb{Q},k+1)]}}$${{[\langle\mathbb{M}_{k-1}\rangle,K(\mathbb{Q},k+1)]}}$$\scriptstyle{(\tau_{\gamma})_{*}}$$\scriptstyle{\cong}$$\scriptstyle{(\sigma_{\gamma})^{*}}$$\scriptstyle{\cong}$

where $\Lambda S^{k+1}$ and $K(\mathbb{Q},k+1)$ are constant functors. The
above is equivalent to the diagram

${{H^{k+1}\Lambda\mathbb{S}^{k+1}_{[s_{\gamma},t_{\gamma})}}}$${{H^{k+1}\mathbb{M}_{k-1}}}$${{H^{k+1}K(\mathbb{Q},k+1)}}$${{H^{k+1}\langle\mathbb{M}_{k-1}\rangle}}$$\scriptstyle{H^{k+1}(\tau_{\gamma})}$$\scriptstyle{\cong}$$\scriptstyle{H^{k+1}(\sigma_{\gamma})}$$\scriptstyle{\cong}$

proving the result. ∎

######  Corollary 7.11.

The Postnikov barcode $\mathrm{im}(H^{k+1})(\sigma)$ in degree $k$ is
isomorphic to the image of the cohomology of the degree $k$ attaching map
$\mathrm{im}(H^{k+1})(\tau)$ in the persistent minimal model.

######  Example 7.12.

The Postnikov barcode can be used to discriminate copersistent spaces with the
same copersistent rational homotopy groups. As an example we will define two
persistent CDGAs whose spatial realisations have the same copersistent
rational homotopy groups and different Postnikov barcodes.

Let $\Lambda\mathbb{I}^{2}_{[s,t)}$ be the minimal persistent CDGA generated
by $u_{s}$ in degree $2$ with trivial coboundary. Construct two persistent
CDGAs $\mathbb{A}$ and $\mathbb{B}$ via degree $3$ persistent Hirsch
extensions

${{\Lambda\mathbb{S}^{4}_{[s,\infty)}}}$${{\Lambda\mathbb{I}^{2}_{[s,t)}}}$${{\Lambda\mathbb{S}^{4}_{[s,\infty)}}}$${{\Lambda\mathbb{I}^{2}_{[s,t)}}}$${{\Lambda\mathbb{D}_{s}^{4}}}$${\mathbb{A}}$${{\Lambda\mathbb{D}_{s}^{4}}}$${\mathbb{B}}$$\scriptstyle{u^{2}_{s}}$$\scriptstyle{\lrcorner}$$\scriptstyle{0}$$\scriptstyle{\lrcorner}$

Both $\mathbb{A}$ and $\mathbb{B}$ share the same underlying persistent
algebra
$\Lambda\mathbb{I}_{[s,t)}^{2}\otimes\Lambda\mathbb{I}_{[s,\infty)}^{3}$ and
are distinguished by a non-trival attaching map in the construction of
$\mathbb{A}$. Since both CDGAs are minimal persistent CDGAs, the generators
correspond to the copersistent rational homotopy groups

$\pi_{*}^{\mathbb{Q}}\langle\mathbb{A}\rangle^{\vee}\cong\mathbb{I}_{[s,t)}^{2}\oplus\mathbb{I}^{3}_{[s,\infty)}\cong\pi_{*}^{\mathbb{Q}}\langle\mathbb{B}\rangle^{\vee}$

of the respective spatial realizations.

Applying Corollary 7.11 and examining the cohomology of the attaching map, we
see that $\langle\mathbb{A}\rangle$ has a non-trivial Postnikov barcode
$\mathrm{im}(H^{4}(u_{s}^{2}))=\mathbb{I}_{[s,t)}$ in degree $3$. Further,
using the classical theory of minimal models, we have rational equivalences
$\langle\mathbb{A}\rangle_{s}\simeq S^{2}_{\mathbb{Q}}$ and
$\langle\mathbb{A}\rangle_{t}\simeq S^{3}_{\mathbb{Q}}$ with the rational $2$
and $3$-spheres respectively. On the other hand $\langle\mathbb{B}\rangle$ has
a trivial Postnikov barcode, implying that we have a rational equivalence

$\langle\mathbb{B}\rangle\simeq K(\mathbb{I}_{[s,t)},2)\times
K(\mathbb{I}_{[s,\infty)},3)\simeq K(\mathbb{I}_{[s,t)},2)\times
S_{\mathbb{Q}}^{3}([s,\infty)^{op})$

where $K(\mathbb{I}_{[s,\infty)},3)$ is homotopic to the constant functor into
rational $3$-sphere $S_{\mathbb{Q}}^{3}$ over the interval $[s,\infty)^{op}$
which is trivial elsewhere. These copersistent spaces are not rationally
equivalent, since $\langle\mathbb{A}\rangle_{s}\simeq S_{\mathbb{Q}}^{2}$ is
not rationally equivalent to $\langle\mathbb{B}\rangle_{s}\simeq
K(\mathbb{Q},2)\times S_{\mathbb{Q}}^{3}$, but have the same copersistent
rational homotopy groups. The difference is, however, detected by the
Postnikov barcode, which encodes the additional information of the non-trivial
Postnikov invariants.

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