# Persistent homological Quillen-McCord theorem

Vitalii Guzeev  Faculty of Mathematics, National Research University Higher
School of Economics, Russian Federation<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

The Quillen-McCord theorem (aka Quillen fiber lemma) gives a sufficient
condition on a map between classifying spaces of posetal categories to be a
homotopy equivalence. Jonathan Ariel Barmak in his paper [J. Comb. Theory Ser.
A 118, 8 (November 2011), 2445–2453.] gives an elementary topological proof
and proves a homological version of the theorem.

Following his scheme of the proof, we formulate and prove the homological
Quillen-McCord theorem, stable with respect to interleaving distances. To
formulate the theorem and apply the scheme, we introduce persistence objects
as objects in appropriate functor categories, describe a-la barcode
decompositions of persistence posets, and prove several results, e.g. order
extension principle for objects in Fun(I, Pos) and approximate triviality of
left derived functors of approximately trivial objects in Fun(I, R-Mod).

Since the given proof gives explicit Lipschitz constant for the map of
persistence classifying spaces, we expect this result to be useful in TDA for
reducing complexity of experimental data.

The article was submitted as the graduation thesis of the author at the HSE
University.

##  1\. Introduction

Computation of the homotopy type of an arbitrary CW complex is an open
problem. It is theoretically possible to compute homology groups in all cases,
but in practice, such computations are limited by resources. Complexity
generally grows with a dimension of a complex. Hence there is an optimization
problem — given CW complex $B$, construct CW complex $A$ such that
$\operatorname{dim}(A)<\operatorname{dim}(B)$ and $H_{\star}(A)\cong
H_{\star}(B)$.

The Alexandrov-Čech theorem is proven to be a useful tool in applications [1,
2]. One can associate to a given covering $\mathcal{U}$ a partially ordered
set of sets $U\in\mathcal{U}$ and all their intersections, ordered by
inclusion. The classifying space of this poset (to be precise, its posetal
category) is a barycentric subdivision of geometric realization of the nerve
of $\mathcal{U}$, hence these two spaces are homeomorphic. Given this
observation, one can formulate the theorem in terms of classifying spaces.

Jonathan Barmak proves [5] the homological version of the Quillen-McCord
theorem (also known as Quillen fiber lemma or Quillen’s theorem A for posets).
It can be stated as follows.

######  Theorem.

Assume $X,Y$ are finite posets, $f:X\to Y$ is an order-preserving map, $R$ is
a PID.
If $H_{i}(\mathcal{B}(f^{-1}(Y_{\leqslant y})),R)=0$ for any $i$ and $y\in Y$,
then $\mathcal{B}f$ induces isomorphisms of all homology groups with
coefficients in $R$ on $BX$ and $BY$.

Provided an algorithm for the construction of $X$ and a map by $Y$, this
theorem may provide a partial solution to the stated optimization problem, in
particular, for nerves of coverings.

Coverings we operate may come from a series of observations, for instance,
from a long experiment. While considering the series as a whole, we are
interested in the persistence of nerves. To operate experimental data some
stable version of the theorem is required.

In this paper, we formulate and prove the persistent homological version of
the Quillen-McCord theorem.

######  Theorem.

Assume $X,Y$ are persistence posets of finite type indexed by a very good
monoid $I$, $f:X\to Y$ is an order-preserving map. Let $m$ be the number of
elements of $Y$ and $R$ be a PID. Assume that the persistence topological
space $\mathcal{B}(f^{-1}(Y_{\leqslant y}))$ is $\varepsilon$-acyclic over $R$
for any $y=(\ldots,y_{i},\ldots)\in Y$. Then $BX$ and $BY$ are
$4m\varepsilon$-interleaved over $R$.

The terms used in the statement are explained in sections 2 and 3.

The paper is structured as follows.

  1. (2)

In preliminaries we give an outline of well-known notions used throughout the
paper.

    1. (2)1.

First preliminary subsection is devoted to the notion of the interleaving
distance (following [7]) which fits as a required measure of similarity
between series. Usability of this notion is guaranteed by persistence theorem
[6]. We try to keep some level of generality, in particular, we work with a
generalized version of the theorem proven in [8] and formulate it. This
generality is significant for applications and allows us to operate both
discrete and continuous models of time in an experiment.

    2. (2)2.

Second subsection gives a classical formulation of the Quillen-McCord theorem
and necessary definitions. To bind the theorem to an abstract context in which
the general Quillen A theorem [9] is used, we derive it from the general
theorem. Additionally, we outline Barmak’s ([5]) proofs of classical and
homological versions of the theorem.

  2. (3)

In the apparatus section we systematize and sometimes introduce various
results forming the toolchain to prove the target theorem. Its necessary set
of definitions is close to well-established in a field (for instance, [12]),
but the whole toolchain is independent.

    1. (3)1.

At first, we reformulate several preliminary definitions in terms of functor
categories and transfer some notions, e.g. classifying space of a category and
a covering, to appropriate functor categories.

    2. (3)2.

In the second section we formulate and prove the linear extension principle
for functors to the category of posets. We give both finite and general
versions with an AC-dependent proof of the latter.

    3. (3)3.

At the end of the section we give some technical stability results concerning
objects of appropriate functor categories.

  3. (4)

Finally, we formulate and prove the main result — stable (w.r.t. interleaving
distances) homological Quillen-McCord theorem. We adapt a proof given in ([5])
using the developed apparatus.

##  2\. Preliminaries

###  2.1. Persistence modules and interleaving distance

We start with the definition of a simplicial set.

######  Definition 2.1.

The simplex category $\Delta$ is a category of non-empty totally-ordered sets
of finite length with order-preserving functions as morphisms.

######  Definition 2.2.

Simplicial set is a contravariant functor from $\Delta$ to $Set$.

######  Proposition 2.3.

Simplicial sets form the category $sSet$.

The real initial definition, representing a series of simplicial sets, is the
following.

######  Definition 2.4.

Persistence simplicial set is a family of simplicial sets
$C_{0}\xrightarrow{f_{0}}C_{1}\xrightarrow{f_{1}}C_{2}\xrightarrow{f_{2}}\ldots$
where $f_{i}$ are natural transformations. We call the maps
$f=(\ldots,f_{i},\ldots)$ the structure maps of a persistence simplicial set.

######  Definition 2.5.

Let $R$ be a ring. Persistence module over $R$ is a family of $R$-modules
$M_{i}$ with homomorphisms $\phi_{i}:M^{i}\to M^{i+1}$ as the structure maps.
We denote the composition of structure maps between $M_{i}$ and $M_{j}$ by
$\phi_{ij}$.

An example of a persistence module is given by homology modules of persistence
simplicial set $C_{\star}$ (persistent homology). We set
$H_{i}^{j}(C_{\star}):=H_{i}(C_{j})$, the maps $\phi_{j}$ are induced by
$f_{i}$.

######  Definition 2.6.

Persistence simplicial set (module) is of finite type over $R$ if all its
simplicial sets (modules) have a finite number of non-empty images (are
finitely generated as $R$-modules) and all $f_{i}$ ($\phi_{i}$) are
isomorphisms for $i>m$ for some $m$.

######  Definition 2.7.

A persistence module is of finitely presented type over $R$ if all its modules
are finitely presented as $R$-modules and all $f_{i}$ ($\phi_{i}$) are
isomorphisms for $i>m$ for some $m$.

We shall generally use the terms of modules in this section.

One can study maps of persistence modules.

######  Definition 2.8.

Let $M$ and $N$ be persistence modules. The family $f$ of maps $f_{i}:M_{i}\to
N_{i}$ is called a morphism from $M$ to $N$ if all $f_{i}$s commute with
structure maps.

######  Definition 2.9.

Let $M$ and $N$ be persistence modules, $f$ is a collection of maps
$f_{i}:M_{i}\to N_{i+\varepsilon}$. Then if all $f_{i}$ commute with structure
maps, $f$ is called an $\varepsilon$-morphism.

There is a general notion of interleaving distance between persistence
modules.

######  Definition 2.10.

We denote by $\operatorname{Id}_{\varepsilon}:M\to M$ the shift of persistence
module, defined by compositions of structure maps $M_{i}\to M_{i+\varepsilon}$
for all $i$.

######  Definition 2.11.

Persistence modules $M$ and $N$ are called $\varepsilon$-interleaved
($M\stackrel{{\scriptstyle\varepsilon}}{{\sim}}N$) if there exists a pair of
$\varepsilon$-morphisms $(\phi:M\to N,\;\psi:N\to M)$ called an
$\varepsilon$-interleaving such that
$\phi\circ\psi=\operatorname{Id}_{2\varepsilon}:N\to N$ and
$\psi\circ\phi=\operatorname{Id}_{2\varepsilon}:M\to M$.

######  Remark 2.12.

It follows that $M\stackrel{{\scriptstyle\varepsilon}}{{\sim}}N$ implies
$M\stackrel{{\scriptstyle\alpha}}{{\sim}}N$ for any $\alpha>\varepsilon$ since
for an $\varepsilon$-interleaving $(\phi,\psi)$ we have an
$\alpha$-interleaving
$(\operatorname{Id}_{\alpha-\varepsilon}\circ\phi,\operatorname{Id}_{\alpha-\varepsilon}\circ\psi)$.

######  Definition 2.13.

The $\varepsilon$-interleaving induces an extended pseudometric on a set of
persistence modules. This pseudometric is defined as
$d(X,Y)=min\\{\varepsilon\in
I\;|\;X\stackrel{{\scriptstyle\varepsilon}}{{\sim}}Y\\}$. This pseudometric is
called interleaving distance. [7, Definition 2.12]

There is a well-known theorem.

######  Theorem 1.

[6, Theorem 3.1]
The category of persistence modules of finite type over Noetherian ring with
unity $R$ is equivalent to the category of graded finitely generated
$R[t]$-modules.

It is proven in [8]. The authors provide a generalization that is more
suitable for our needs.

######  Definition 2.14.

Let $(G,\star)$ be a commutative monoid and $g_{1},g_{2}\in G$. We say that
$g_{1}\preceq g_{2}$ if $\exists h\in G:\;h\star g_{1}=g_{2}$, $h$ is not a
neutral element.

######  Definition 2.15.

Let $(G,\star)$ be a monoid. Consider $H\subset G$. An element $m\in G$ is
called a common multiple of $H$ if $h\preceq m$ for any $h\in H$. A common
multiple $m$ of $H$ is called partially least if there is no common multiple
$m_{1}$ of $H$ such that $m_{1}\preceq m$.

######  Definition 2.16.

following [8, Definition 11] Monoid $(G,\star)$ is called good if the
following hold:

  * •

$(G,\star)$ is commutative;

  * •

$g_{1}\star g_{2}=g_{1}\star g_{3}$ implies $g_{2}=g_{3}$ (cancellation);

  * •

$g_{1}\preceq g_{2}$ and $g_{2}\preceq g_{1}$ imply $g_{1}=g_{2}$ (anti-
symmetricity);

  * •

For any finite $H\subseteq G$ there exists at most finitely many partially
least common multiples (property of being weak plcm).

######  Definition 2.17.

[8, Definition 12] Let $R$ be a ring and $G$ be a good monoid. (Generalized)
persistence module is a family of $R$-modules $M^{i}$ for $i\in G$ with
homomorphisms $\phi_{ij}:M^{i}\to M^{j}$ for $i\preceq j$, satisfying identity
relation $\phi_{ii}=\operatorname{Id}$ and composition relation
$\phi_{ij}\circ\phi_{jk}=\phi_{ik}$ for any $i\preceq j\preceq k\in G$ as the
structure maps.

######  Theorem 2.

[8, Theorem 21]
Let $R$ be a ring with unity and $G$ be a good monoid. Then the category of
finitely presented graded $R[G]$-modules is isomorphic to the category of
$G$-indexed persistence modules over $R$ of finitely presented type.

######  Remark 2.18.

Cancellative commutative monoid with $\preceq$ being a total order (totally
ordered) is good.

######  Definition 2.19.

We call cancellative commutative totally ordered monoids very good.

######  Example 2.20.

Consider the monoid of non-negative real numbers $\mathbb{R}_{\geqslant 0}$
with addition as a monoidal operation. The addition of real numbers is
cancellative and $\preceq$ is a total order. This monoid is very good.

For contrast, we can consider $\mathbb{R}$. Since $g_{1}=h+g_{2}$ implies
$g_{2}=(-h)+g_{1}$ $\preceq$ is trivial and $R$ is not a good monoid in the
sense of the given definition.

We can examine how notions related to interleaving distance look in a category
of graded modules under additional constraints on the indexing set.

######  Definition 2.21.

[7, Definition 2.7]
Let $M$ and $N$ be graded $R[G]$-modules, $f:M\to N$ be a homomorphism of
modules. Then $f$ is called $\varepsilon$-morphism if $f(M^{j})\subset
N^{j+\varepsilon}$.

Let $R$ be a ring with unity and $G$ be a very good monoid. Then for any
graded $R[G]$-module $M$ and any $\varepsilon\in\mathbb{G}$ there exists a
$\varepsilon$-morphism $\operatorname{Id}_{\varepsilon}:M\to M$. For instance,
if the monoid $G$ is the monoid of non-negative integers, the
$\operatorname{Id}_{\varepsilon}$ is a multiplication by $t^{\varepsilon}$,
where $t$ is a generator of a polynomial algebra [7, Example 2.8].

This example can be generalized as follows.

######  Proposition 2.22.

[8, Following the Equation 4] Let $M$ be an $R[G]$-module. The
$\varepsilon$-morphism $\operatorname{Id}_{\varepsilon}:M\to M$ is a
multiplication by a fixed element $m\in R[G]^{\varepsilon}$, where grading on
$R[G]$ is given by $G$.

######  Proposition 2.23.

[7, Proposition 2.13] Condition
$M\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$ is equivalent to condition
$m^{2}M=0$.

######  Lemma 2.24.

Let $0\to M\to L\to N\to 0$ be a short exact sequence of graded modules. Then
the following properties hold.

  * •

If $M\stackrel{{\scriptstyle\varepsilon_{1}}}{{\sim}}0$ and
$N\stackrel{{\scriptstyle\varepsilon_{2}}}{{\sim}}0$ then
$L\stackrel{{\scriptstyle\varepsilon_{1}+\varepsilon_{2}}}{{\sim}}0$. [7,
Proposition 4.6]

  * •

If $L\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$ then
$M\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$ and
$N\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$.

  * •

If $M\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$ then
$L\stackrel{{\scriptstyle 2\varepsilon}}{{\sim}}N$. [7, Proposition 4.1]

  * •

If $N\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$ then
$M\stackrel{{\scriptstyle 2\varepsilon}}{{\sim}}L$. [7, Proposition 4.1]

The second statement of the lemma requires proof. We give it for non-negative
integers first for compatibility with the referenced result.

Proof. Denote non-trivial maps in s.e.s as $i$ and $q$.

Then $i(t^{2\varepsilon}a)=t^{2\varepsilon}i(a)=0$ for any $a\in M$. The
homomorphism $i$ is injective. Hence by Proposition 2.23 we have
$M\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$.

On the other side, we have $0=q(t^{2\varepsilon}a)=t^{2\varepsilon}q(a)$ where
$q$ is surjective. Hence $N\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$. ∎

Proposition 2.22 allows to generalise the lemma by changing $t^{\varepsilon}$
to $m$.

By equivalence of categories, the lemma holds for persistence modules over
rings with unity indexed by very good monoids.

###  2.2. The Quillen-McCord theorem

######  Definition 2.25.

Let $X$ be a set and $\mathcal{P}(X)$ be its powerset. A set
$S\subset\mathcal{P}(X)$ is called a simplicial complex if $W\in S$ for any
$V\in S$ and $W\subset V$. An element of a simplicial complex of cardinality
$n$ is called $n$-simplex.

######  Proposition 2.26.

Simplicial complexes form the category $sCpx$.

######  Definition 2.27.

The join $A\star B$ of simplicial complexes $A$ and $B$ is the simplicial
complex with simplices — all possible unions of simplices $a\in A$ and $b\in
B$.

Let $A$ be a simplicial complex.

######  Definition 2.28.

The star $\operatorname{st}(x)$ of simplex $x\in A$ is the minimal by
inclusion simplicial complex containing all simplices $a\in A$ such that there
exists an inclusion $x\hookrightarrow a$.

######  Definition 2.29.

Link $\operatorname{lk}(x)$ of simplex $x\in A$ is defined as follows:
$\operatorname{lk}(x)=\\{v\in\operatorname{st}(x)|\;x\cap v=\varnothing\\}$.

######  Proposition 2.30.

For $0$-simplex $x$ it holds that
$\operatorname{st}(x)=\operatorname{lk}(x)\star x$.

######  Definition 2.31.

Functor $\left[\bullet\right]:sCpx\to Top$ which maps $n$-simplices to
standard geometric $n$-simplices is called the standard geometric realization
of a simplicial complex.

######  Definition 2.32.

Functor $\left|\bullet\right|:sSet\to Top$ which maps $n$-simplices to
$n$-cells and face maps to attachment maps is called the geometric realization
of a simplicial set.

######  Definition 2.33.

Join of topological spaces $A$ and $B$ is defined as follows: $A\star B\
:=A\sqcup_{p_{0}}(A\times B\times[0,1])\sqcup_{p_{1}}B$, where $p$ are
projections of the cylinder $A\times B\times[0,1]$ onto faces.

The next proposition gives motivation for the definition of a join of
simplicial complexes.

######  Proposition 2.34.

$\left|A\star B\right|=\left|A\right|\star\left|B\right|$. Hence if $x$ is a
$0$-simplex, then $\left|\operatorname{st}(x)\right|$ is a cone over
$\left|\operatorname{lk}(x)\right|$.

######  Definition 2.35.

There is a standard set of definitions, accompanying the definition of a
simplicial set. Let $S$ be a simplicial set. Then

  1. (1)

images $S([n])$ of sets of cardinality $n$ are called $n$-simplices;

  2. (2)

images of injective maps $S([n]\to[n+1])$ are called face maps;

  3. (3)

images of surjective maps $S([n+1]\to[n]$ are called degeneracy maps.

Details of the explicit construction can be found in [4]

######  Definition 2.36.

Let $\mathcal{C}$ be a small category. Then we can functorially (w.r.t. to
category $Cat$) assign to it a simplicial set $\mathcal{N}(\mathcal{C})$
called nerve of a category to it.

Construction goes as follows:

  * •

we assign to each object of $\mathcal{C}$ a $0$-simplex and to each morphism
in $\mathcal{C}$ a $1$-simplex with order following corresponding arrow in a
category;

  * •

then we take the set of all morphisms as an alphabet and write all the words
in it such that we can move from the first letter to the last following arrows
in a category. We assign an $l$-simplex to a word of length $l$. Each
commutative triangle $f,g,h=f\circ g$ in $\mathcal{C}$ gives rise to a
morphism between these words — it replaces $f\circ g$ with $h$ and serves as a
face map. Replacements $f\to f\circ Id$ represent degeneracy maps.

This collection of data forms a simplicial set, details can be found in [3].

######  Definition 2.37.

Geometric realization $B\mathcal{C}$ of $\mathcal{N}(\mathcal{C})$ is called
classifying space of $\mathcal{C}$.

We denote the composition of nerve and geometric realization as $\mathcal{B}$.
It is a composition of functors, hence a functor. By definition
$\mathcal{B}(\mathcal{C})=B\mathcal{C}$ and we prefer notation
$\mathcal{B}(f)$ to $Bf$.

######  Definition 2.38.

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor and $d$ — object in
$\mathcal{D}$. Then comma category $d\downarrow F$ is a category with objects
— pairs $(s,i_{s})$ of objects in $\mathcal{C}$ and morphisms $i_{s}:d\to
F(s)$ and morphisms — morphisms $g$ in $\mathcal{C}$ such that triangle
$i_{s},i_{g(s)},F(g)$ is commutative.

######  Theorem 3.

[9, Theorem A]
If $F:C\to D$ is a functor such that the classifying space $B(d\downarrow F)$
of the comma category $d\downarrow F$ is contractible for any object $d\in D$,
then $F$ induces a homotopy equivalence $BC\to BD$.

The nerve construction on a posetal category yields a simplicial set with the
following good properties: the image of a set of reorderings on any set in
$\Delta$ contains unique simplex and subsets of its simplices are also
simplices. These simplicial sets form a subcategory $ssCpx$ of $sSet$.

There is a functor $F$ from this subcategory to the category of simplicial
complexes, that turns a simplicial set to a set of its images.

######  Proposition 2.39.

Let $X$ be an object of $ssCpx$. Then $|X|$ is homeomorphic to the geometric
realization of simplicial complex $[F(X)]$.

This proposition allows us to operate simplicial complexes instead of
simplicial sets and avoid general definitions of join, star, and link for
simplicial sets.

Application of Quillen’s A theorem to posets yields the following theorem (we
identify poset with its posetal category).

######  Theorem 4.

the Quillen-McCord theorem
Assume $X,Y$ are finite posets, $f:X\to Y$ is an order-preserving map.
If $\forall y\in Y\;\mathcal{B}(f^{-1}(Y_{\leqslant y}))$ is contractible,
then $\mathcal{B}f$ is a homotopy equivalence between $BX$ and $BY$.

######  Theorem 5.

Homological Quillen-McCord theorem [5, Corollary 5.5]
Assume $X,Y$ are finite posets, $f:X\to Y$ is an order-preserving map, $R$ is
a PID.
If $\forall y\in Y\;H_{i}(\mathcal{B}(f^{-1}(Y_{\leqslant y})),R)=0$ for any
$i$, $\mathcal{B}f$ induces isomorphisms of all homology groups of $BX$ and
$BY$ with coefficients in $R$.

Proofs of both theorems are used in our arguments and we recall them in brief.

####  2.2.1. Barmak’s proof of the Quillen-McCord theorem

######  Proposition 2.40.

Variation of [5, Lemma 2.2] Let $f,g:X\to Y$ be order-preserving maps between
finite posets such that $\forall x\;f(x)\leqslant g(x)$. Then $\mathcal{B}(f)$
is homotopy-equivalent to $\mathcal{B}(g)$.

######  Proposition 2.41.

Note that $\operatorname{lk}(F(\mathcal{N}(x)))=F(\mathcal{N}(X_{>x}))\star
F(\mathcal{N}(X_{<x}))$. Therefore
$\left|\operatorname{lk}(\mathcal{N}(x))\right|=\mathcal{B}(X_{>x})\star\mathcal{B}(X_{<x})$.

The crucial observation is the existence of the following covering for any
$x\in X$.

(2.1)
$\mathcal{B}(X)=\mathcal{B}(X\setminus\\{x\\})\cup\left|\operatorname{st}(\mathcal{N}(x))\right|.\\\
$

######  Lemma 2.42.

Let $X$ be a finite poset and for $x\in X$ either $\mathcal{B}(X_{>x})$ or
$\mathcal{B}(X_{<x})$ is contractible. Then embedding
$\mathcal{B}(X\setminus\\{x\\})\hookrightarrow\mathcal{B}(X)$ is a homotopy
equivalence.

For this paper, this lemma is the most important part of the proof of the
theorem, hence we recall its proof.

Proof. By hypothesis, the space
$\left|\operatorname{lk}(\mathcal{N}(x))\right|=\left|\operatorname{st}(\mathcal{N}(x))\right|\cap\mathcal{B}(X\setminus\\{x\\})$
is contractible. Hence its embedding to its cone
$\left|\operatorname{st}(\mathcal{N}(x))\right|$ is a homotopy equivalence by
Whitehead theorem. Being a subcomplex, it is a strong deformation retract.
Then, $\mathcal{B}(X\setminus\\{x\\})$ is a strong deformation retract of
$\mathcal{B}X=\left|\operatorname{st}(\mathcal{N}(x))\right|\cup\mathcal{B}(X\setminus\\{x\\})$
∎

######  Definition 2.43.

Variation of [5, Proposition 2.1] Let $f:X\to Y$ be an order-preserving map
between posets. Denote orders ($\leqslant$) on $X$ and $Y$ as $R_{X}$ and
$R_{Y}$. Then we define poset $M(f)=X\coprod_{f}Y$ with $R=R_{X}\cup R_{Y}\cup
R_{f}$ where $(x,y)\in R_{f}$ if and only if $(f(x),y)\in R_{Y}$.

We call this poset a mapping cylinder of $f$. There are also defined canonical
inсlusions $i_{X}:X\to M(f)$ and $i_{Y}:Y\to M(f)$.

Proof. the Quillen-McCord theorem
Let $X,Y$ be finite posets with an order-preserving map $f:X\to Y$.

Every poset has a linear extension. Let $x_{1},x_{2},\ldots,x_{n}$ be the
enumeration of $X$ in an arbitrary linear extension and
$Y^{r}=\\{x_{1},\ldots,x_{r}\\}\cup Y\subset M(f)$ for any $r$.

Consider $Y^{r}_{>x_{r}}=Y_{\geqslant f(x_{r})}$. The space
$\mathcal{B}(Y_{\geqslant f(x_{r})})$ is a cone over $\mathcal{B}(f(x_{r}))$.
It is contractible, therefore
$\mathcal{B}(Y^{r-1})\hookrightarrow\mathcal{B}(Y^{r})$ is a homotopy
equivalence by Lemma 2.42. By iteration,
$\mathcal{B}(j):\mathcal{B}(Y^{0})=\mathcal{B}(Y)\hookrightarrow\mathcal{B}(M(f))=\mathcal{B}(Y^{n})$
is homotopy equivalence between $BY$ and $M(f)$.

Then consider a linear extension of $Y$ with enumeration $y_{1},\ldots,y_{m}$
and $X^{r}=X\cup\\{y_{r+1},\ldots,y_{m}\\}\subset M(f)$. We have
$X^{r-1}_{<y_{r}}=f^{-1}(Y_{\leqslant y_{r}})$. The classifying space of the
latter is contractible by the assumption of the theorem. Hence
$\mathcal{B}(X^{r})\hookrightarrow\mathcal{B}(X^{r-1})$ is a homotopy
equivalence and by transitivity $\mathcal{B}(i_{X})$ is a homotopy equivalence
between $X$ and $M(f)$.

Note that $i(x)\leqslant(i_{Y}\circ f)(x)$. By Proposition 2.40 the space
$\mathcal{B}(i_{X})(X)$ is homotopy-equivalent to the space
$\mathcal{B}(i_{Y}\circ f)(X)=(\mathcal{B}(i_{Y})\circ\mathcal{B}(f))(X)$.
Hence $\mathcal{B}(f)$ is the homotopy equivalence between $BX$ and $BY$. ∎

####  2.2.2. Barmak’s proof of the homological Quillen-McCord theorem

######  Proposition 2.44.

[10, Lemma 2.1] Reduced homology modules with coefficients in a principal
ideal domain $R$ satisfy the relation $H_{r+1}(A\star
B,R)\simeq\bigoplus_{i+j=r}(H_{i}(A,R)\otimes_{R}H_{j}(B,R))\oplus\bigoplus_{i+j=r-1}\operatorname{Tor}_{1}^{R}(H_{i}(A,R),H_{j}(B,R))$.

######  Lemma 2.45.

Let $X$ be a finite poset and for $x\in X$ either $H_{i}(\mathcal{B}(X_{<x}))$
or $H_{i}(\mathcal{B}(X_{>x}))$ with coefficients in a PID are equal to the
homology of a point. Then embedding
$\mathcal{B}(X\setminus\\{x\\})\hookrightarrow\mathcal{B}(X)$ induces
isomorphisms of all homology groups.

Proof.

By Proposition 2.44,
$H_{i}(\left|\operatorname{lk}(\mathcal{N}(x))\right|)=H_{i}(\mathcal{B}(X_{>x})\star\mathcal{B}(X_{<x}))$
are trivial for all indices $i$. Application of Mayer-Vietoris long exact
sequence to covering introduced in equation (2.1) yields the lemma. ∎

Proof of the theorem is similar to the proof at the end of the previous
subsection. We write it here to highlight differences. Changed parts are
written in italic.

Proof. the homological Quillen-McCord theorem
Let $X,Y$ be finite posets with an order-preserving map $f:X\to Y$.

Every poset has a linear extension. Let $x_{1},x_{2},\ldots,x_{n}$ be the
enumeration of $X$ in a fixed linear extension and
$Y^{r}=\\{x_{1},\ldots,x_{r}\\}\cup Y\subset M(f)$ for any $r$.

Consider $Y^{r}_{>x_{r}}=Y_{\geqslant f(x_{r})}$. The space
$\mathcal{B}(Y_{\geqslant f(x_{r})})$ is a cone over $\mathcal{B}(f(x_{r}))$.
It is contractible, therefore
$\mathcal{B}(Y^{r-1})\hookrightarrow\mathcal{B}(Y^{r})$ is a homotopy
equivalence by Lemma 2.42. By iteration, the map
$\mathcal{B}(j):\mathcal{B}(Y^{0})=\mathcal{B}(Y)\hookrightarrow\mathcal{B}(M(f))=\mathcal{B}(Y^{n})$
is a homotopy equivalence between $BY$ and $M(f)$.

Now consider a linear extension of $Y$ with enumeration $y_{1},\ldots,y_{m}$
and $X^{r}=X\cup\\{y_{r+1},\ldots,y_{n}\\}\subset M(f)$. We have
$X^{r-1}_{<y_{r}}=f^{-1}(Y_{\leqslant y_{r}})$. The classifying space of the
latter is acyclic over $R$ by the assumption of the theorem. Hence
$\mathcal{B}(X^{r})\hookrightarrow\mathcal{B}(X^{r-1})$ induces isomorphisms
of all homology groups and by the functoriality of homology $\mathcal{B}(i)$
induces isomorphisms of all homology groups between $X$ and $M(f)$.

Note that $i(x)\leqslant(j\circ f)(x)$. By Proposition 2.40 $\mathcal{B}(i)$
is homotopic to $\mathcal{B}(j\circ f)=\mathcal{B}(j)\circ\mathcal{B}(f)$.
Homotopic maps induce the same maps of homology modules, $j$ is a homotopy
equivalence and induces isomorphisms. Hence $\mathcal{B}(f)$ induces
isomorphisms between $H_{i}(BX,R)$ and $H_{i}(BY,R)$. ∎

We see two updates. The first one is essential, it requires Lemma 2.45 and
operates some equivalence propagating in a chain of length equal to the
cardinality of $Y$. The second follows automatically from the functoriality of
all used constructions.

##  3\. Apparatus for main results

###  3.1. Persistence objects and related constructions

We have two types of persistence objects with similar definitions. There is a
general notion of a persistence object such that these definitions fall into
special cases.

######  Definition 3.1.

Consider $I$ — the poset category of a fixed linearly ordered set. There is a
sequence category $Fun(I,\mathcal{C})$ of functors from $I$ to some category
$\mathcal{C}$. We call objects of this category persistence objects over
$\mathcal{C}$

######  Example 3.2.

  * •

Persistence complex is a persistence object over the category of chain
complexes;

  * •

Persistence $R$-module is a persistence object over the category of
$R$-modules; [7, Definition 2.15]

  * •

Persistence simplicial set is a persistence object over the category $sSet$;

  * •

Persistence poset is a persistence object over $Pos$;

  * •

Persistence topological space is a persistence object over $Top$.

######  Definition 3.3.

We call images of morphisms in $I$ the structure maps of a persistence object
over $\mathcal{C}$.

We use the notation $(X,\phi)$ for ‘‘Persistence object X with the family of
structure maps $\phi$ over fixed indexing category $I$’’. We use notation
$\phi_{ij}$ for a structure map between $X_{i}$ and $X_{j}$.

Consider $\mathcal{F}$ — a functor from $\mathcal{C}$ to $\mathcal{D}$. It
naturally extends to a functor between $Fun(I,\mathcal{C})$ and
$Fun(I,\mathcal{D})$. Let $P$ be a persistence poset. Apparently,
$\mathcal{B}(P)$ is a persistence topological space.

######  Definition 3.4.

A persistence topological space $X$ is called $\varepsilon$-acyclic over $R$
if for all indices
$H_{i}(X,R)\stackrel{{\scriptstyle\varepsilon}}{{\sim}}H_{i}(pt,R)$.

######  Definition 3.5.

A finite sequence of finite posets is called a persistence poset of finite
type.

######  Remark 3.6.

The definition of a persistence poset has a foundational issue — since there
are no maps to an empty set, structure maps are in general not defined. In
particular, this issue is unavoidable in the case of posets of finite type. We
prefer to resolve this issue by adjoining a contradiction symbol $\bot$ to a
class of functions. If there is no candidate for a function $\phi_{ij}$ in
$Pos$, we set $\phi_{ij}=\bot$ and ignore these functions in reasonings
similar to ‘‘something is defined by all images of $x$ under the structure
maps’’.

######  Proposition 3.7.

Let $X$ be a persistence poset of finite type. Then $\mathcal{B}(X)$ has
homology modules of finitely presented type.

Proof. The nerve of an empty poset is an empty simplicial set. Hence
$\mathcal{N}(X)$ is a finite sequence of simplicial sets. Since a finite poset
has only a finite number of chains, each component is finite, hence
$\mathcal{N}(X)$ has finite type. The homology of $\mathcal{B}(X)$ can be
computed as the homology of this simplicial set. Hence homology modules of
$\mathcal{B}(X)$ are of finitely presented type as quotients of finite rank
free modules by finite rank free modules. ∎

######  Definition 3.8.

Let $(X,\phi)$ and $(Y,\psi)$ be persistence posets and $f:X\to Y$ be a
morphism. Consider the series of mapping cylinders of posets $M(f_{i})$. We
can define the structure maps $\chi$ on this series as $\phi$ on $X$ and
$\psi$ on $Y$. They preserve orders $R_{f_{i}}$ by definition of structure
maps. The persistence poset $(M(f),\chi_{i})$ is a mapping cylinder of a map
of persistence posets. We also have canonical inclusions $i_{X}$ and $i_{Y}$.

We can also define a subobject in $Fun(I,Pos)$.

######  Definition 3.9.

Consider persistence poset $(X,\phi)$ and the collection of subsets
$Y_{i}\subset X_{i}$ for $i\in I$. If for any $i\in I$, $y\in Y_{i}$ and for
any $j>i$ we have $\phi_{ij}(y)\in Y_{j}$ then $(Y,\phi)$ is called a
persistence subposet of $(X,\phi)$.

We can also define an element of a persistence poset.

######  Definition 3.10.

A persistence subposet $x$ of $(X,\phi)$ with at most one element in each
component is called an element of $X$.

######  Remark 3.11.

Consider the persistent poset $(X,\phi)$ of finite type. For each element
$x\in X$, there exists a minimal index $i$ such that $x_{i}\neq\emptyset$.
Then $x$ is completely defined by its images under structure maps.

One can consider component-wise order on $X$ and work with sets like the
following.

######  Example 3.12.

Let $x$ be an element of the persistence subposet $(X,\phi)$. Then $X_{<x}$ is
the poset of elements component-wise less than $x$.

$X_{<x}$ contains component-wise only elements comparable with $x$. Since
structure maps are order-preserving, $X_{<x}$ is a persistence subposet.

Finally, we give a definition of persistence covering.

######  Definition 3.13.

Assume a poset $X$ splits into a union of subposets $X_{j}$. Then every
subposet $X_{j}$ has its own classifying space $BX_{j}$. If these spaces (or
minimal open sets containing them in $BX$) cover the whole $BX$, they are
called a persistence covering.

This definition gives an example of how structures in the category of
persistence posets can be transferred to other persistence categories. It is
possible to reformulate the definition as internal to the category of
persistence topological spaces but we prefer to keep a more constructive way.

###  3.2. Order extension principle for persistence posets

In his proof of the Quillen-McCord theorem Barmak relies on the order
extension principle. To be able to transfer Barmak’s proof to the persistent
case we have to stress a similar statement for persistence posets.

######  Definition 3.14.

An extension of persistence poset $X$ is a series of partially-ordered
extensions of $X_{i}$ such that the structure maps of $X$ are well-defined on
these extensions. If extensions of all components are linear, we call this
series a linear extension.

######  Proposition 3.15.

Transfer of order. Let $f$ be a morphism between posets $X$ and $Y$ and
$\overline{Y}$ be a linear extension of $Y$. Then $f$ induces partially
ordered extension $\hat{X}$ of $X$ such that $f$ is well-defined as map
$\hat{X}\to\overline{Y}$.

Proof. Consider two incomparable points $a,b\in X$ and map
$f:X\to\overline{Y}$ which is well-defined. One of the following holds.

  * •

f(b) < f(a)

  * •

f(a) < f(b)

  * •

f(a) = f(b)

If strict inequality holds, we can impose a single relation on $a$ and $b$ —
we inherit relation from images.
If equality holds we do not add any new relation. ∎

######  Proposition 3.16.

Left propagation of linear extension. Assume indexing set $I$ of persistence
poset $(X,\phi)$ is converse well-founded and there exists maximal index $i$
such that $X_{j}=\emptyset$ for any $j>i$. Then $(X,\phi)$ has a linear
extension.

Proof. We can extend the order on the component $X_{i}$ to linear. Given this
order, we can transfer it to the left via all structure maps. We obtain an
extension of $(X,\phi)$ because all preimages of incomparable elements were
incomparable and we have equipped them with compatible orders. Now let’s
assume we obtained linear orders in components $X_{j}$ for $j>j_{0}$ by this
construction. Then $X_{j_{0}}$ can be linearly extended. The proposition
follows by transfinite induction and by simple induction if $X$ is finite as a
sequence. ∎

This statement can also be seen as a corollary of a more general proposition.
One can consider set $E(X,\phi)$ of extensions of persistence poset $(X,\phi)$
with partial order defined as follows: let $Y,Z$ be extensions of $X$, then
$Y\geqslant Z$ if and only if $Y$ is an extension of $Z$. Since underlying
sets of these extensions are always the same we can identify elements of
$E(X,\phi)$ with tuples of order relations on components of $X$.

######  Proposition 3.17.

Every linearly ordered subset of $E(X,\phi)$ has an upper bound in
$E(X,\phi)$.

Proof. Let $\\{R^{s}|\;s\in S\\}$ be a linearly ordered subset of $E(X,\phi)$
indexed by set $S$. Consider $R=\bigcup R^{s}$ where the union is taken
component-wise. Assume for some elements $a,b\in X_{i}$ for some $i$ some
$\phi_{ij}$ cannot be defined on them as an order-preserving map. Then there
exists $s\in S$ such that $a$ and $b$ are comparable in extension $R^{s}$. But
$R^{s}$ is an extension, hence $\phi_{ij}$ is defined on both $a$ and $b$. By
contradiction, the proposition follows. ∎

######  Proposition 3.18.

Persistent order extension principle. Every persistence poset $(X,\phi)$ has a
linear extension.

Proof. By proposition 3.17, conditions of Zorn’s lemma are fulfilled. By
Zorn’s lemma $E(X,\phi)$ has maximal element $M$. Assume this element is not a
linear extension of $(X,\phi)$.

Then in some $M_{i}$, there exists incomparable pair $(a,b)$. Consider
$\phi_{ij}(a)$ and $\phi_{ij}(b)$ for all $j>i$. Suppose that
$\phi_{ij_{1}}(a)>\phi_{ij_{1}}(b)$ and $\phi_{ij_{2}}(a)<\phi_{ij_{2}}(b)$
for some $j_{2}>j_{1}$. Then the map $\phi_{j_{1}j_{2}}$ cannot be defined.
Hence for any $j$, the relation between images of $a$ and $b$ has the same
sign if exists. If there exists such $j$ that relation between $\phi_{ij}(a)$
and $\phi_{ij}(b)$ exists we define relation between $a$ and $b$ accordingly.
Otherwise, we define it arbitrarily. We can propagate the extension to the
right by order preservation and to the left — all preimages of $a$ and $b$
were incomparable and can be equipped with compatible orders.

We have constructed a proper extension of $M$. By contradiction, $M$ must be a
linear extension of $(X,\phi)$. ∎

###  3.3. Approximation distances

We can infer some stability results on persistence modules, using notions
transferred from the category of graded modules. During this subsection, we
assume conditions of Theorem 1 or Theorem 2 to be satisfied.

######  Proposition 3.19.

Let $A$, $B$ be two persistence modules such that $d(A,0)\leqslant\varepsilon$
and $d(B,0)\leqslant\varepsilon$. Then $d(A\oplus B,0)\leqslant\varepsilon$.

Proof. Recall that an index shift maps to multiplication by $m$ in a category
of graded modules. We have $mA\oplus mB=m(A\oplus B)$ by definition of a
direct sum. The result follows by Proposition 2.23 via Theorems 1 and 2. ∎

######  Proposition 3.20.

Let $A$, $B$ be two persistence modules such that $d(A,0)\leqslant\varepsilon$
and $d(B,0)\leqslant\varepsilon$. Then $d(A\otimes B,0)\leqslant\varepsilon$.

Proof. The result follows from the bilinearity of the tensor product and
Theorems 1 and 2. ∎

######  Proposition 3.21.

Let $P=\ldots\to P_{n}\to P_{n-1}\to\ldots$ be a persistence complex such that
$P_{i}\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$ for all $i\in I$. Then
the homology modules of $P$ are $\varepsilon$-interleaved with $0$.

Proof. Assume $d_{i}$ is a differential in a complex. We know that
$0\to\operatorname{im}{d_{i+1}}\to\ker{d_{i}}\to H_{i}(P)\to 0$ is exact and
that $0\to\ker{d_{i}}\to P_{i}\to P_{i-1}\to 0$ is exact. The result follows
from the application of Lemma 1 twice. ∎

######  Proposition 3.22.

Category $Fun(I,$R$-Mod)$ has enough projectives.

######  Remark 3.23.

This proposition is known to be true without conditions on $I$. [11, Page 2]

Since we have enough projectives, we can compute derived functors. We need the
following proposition.

######  Proposition 3.24.

Let $R$ be a commutative ring, $A$ and $B$ — persistence $R$-modules such that
either $A$ or $B$ is $\varepsilon$-interleaved with $0$. Then
$\operatorname{Tor}_{i}^{R}(A,B)\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$.

Proof. Since $R$ is commutative,
$\operatorname{Tor}_{i}^{R}(A,B)=\operatorname{Tor}_{i}^{R}(B,A)$. Without
loss of generality assume $B\stackrel{{\scriptstyle\varepsilon}}{{\sim}}0$.
Let $P$ be the projective resolution of $A$. By Proposition 3.20, after taking
the tensor product we obtain a sequence of modules, $\varepsilon$-interleaved
with $0$. The proposition follows by Proposition 3.21. ∎

We can also derive the result about exact sequences.

######  Proposition 3.25.

Let $A\xrightarrow{f}B\xrightarrow{\phi}C\xrightarrow{g}D$ be an exact
sequence in the category of persistence modules. Then if
$d(A,0)\leqslant\varepsilon$ and $d(D,0)\leqslant\varepsilon$, then
$B\stackrel{{\scriptstyle 4\varepsilon}}{{\sim}}C$.

Proof. In s.e.s $0\to\ker{f}\hookrightarrow
A\xrightarrow{f}\operatorname{im}f\to 0$ the $\operatorname{im}f$ is
$\varepsilon$-trivial. By exactness, it is equal to $K=\ker\phi$. On the other
side from $0\to\ker{g}\xrightarrow{g}D\to D\to 0$ there follows that
$d(I=\operatorname{im}\phi,0)\leqslant\varepsilon$. Hence
$K\stackrel{{\scriptstyle 2\varepsilon}}{{\sim}}I$ by triangle inequality.

We obtain an exact sequence $0\to K\to B\xrightarrow{\phi}C\to I\to 0$. This
sequence decomposes into sequences $0\to K\to B\to\operatorname{coIm}\phi\to
0$ and $0\to\operatorname{Im}\phi\to C\to I\to 0$. By lemma 2.24 we have that
$d(B,\operatorname{coIm}\phi)\leqslant 2\varepsilon$ and
$d(C,\operatorname{Im}\phi)\leqslant 2\varepsilon$. Coimage and image are
pointwise canonically isomorphic by the first isomorphism theorem for modules,
hence $d(B,C)\leqslant 4\varepsilon$. ∎

##  4\. Main results

######  Proposition 4.1.

Let $A$ and $B$ be two persistence topological spaces with at least one of
them being $\varepsilon$-acyclic over $R$. Then $A\star B$ is
$\varepsilon$-acyclic over $R$.

Proof. All $\operatorname{Tor}$-functors from Proposition 2.44 are
$\varepsilon$-interleaved with $0$ by Proposition 3.24. Hence by Proposition
3.19 the right hand side of expression of Proposition 2.44 is
$\varepsilon$-equivalent to $0$.

Note that on the left-hand side we operate a component-wise join. The
Proposition 2.44 is true component-wise and is true as a statement about
persistence modules by gluing over structure maps. These structure maps are
not assumed to be known, in particular, for persistent
$\operatorname{Tor}_{1}^{R}(...)$. ∎

######  Proposition 4.2.

Let $x$ be an element of $(X,\phi)$. Then coverings (2.1) of all components of
$X$ form the persistence covering $\mathcal{U}$ with covering sets $U_{1}$ —
preimage of $\operatorname{st}(\mathcal{N}(x))$ under nerve functor and
$U_{2}=X\setminus\\{x\\}$.

Proof. It suffices to check that $X\setminus\\{x\\}$ and preimage of
$st(\mathcal{N}(x))$ are persistence subposets. It is evident for
$X\setminus\\{x\\}$. Elements in the preimage of
$\operatorname{st}(\mathcal{N}(x_{i}))$ are exactly elements comparable to
$x_{i}$. Since structure maps preserve order, they do not move comparable
elements to incomparable ones. Hence the preimage also forms a subposet. ∎

######  Lemma 4.3.

Let $(X,\phi)$ be a persistence poset and for $x=(\ldots,x_{i},\ldots)\in X$
either $\mathcal{B}(X_{<x})$ or $\mathcal{B}(X_{>x})$ is
$\varepsilon$-acyclic. Then persistent homology of
$\mathcal{B}(X\setminus\\{x\\})$ and $\mathcal{B}(X)$ are
$4\varepsilon$-interleaved.

Proof.

By Proposition 4.1 $\left|\operatorname{lk}(\mathcal{N}(x))\right|$ is
$\varepsilon$-acyclic.

Given persistence covering we can define Mayer-Vietoris exact sequence on
persistence homology modules component-wise by gluing sequences for components
over structure maps. Proposition 3.25 yields the lemma. ∎

######  Remark 4.4.

If a map $f$ is a component-wise homotopy equivalence, it induces
$0$-interleaving of homology modules.

We are now ready to adapt known proof to Quillen-McCord theorem for
persistence posets.

######  Theorem 6.

The persistent homological Quillen-McCord theorem
Assume $X,Y$ are persistence posets of finite type indexed by very good monoid
$I$, $f:X\to Y$ is an order-preserving map. Let $m$ be the number of elements
of $Y$ and $R$ be a PID. Assume that the persistence topological space
$\mathcal{B}(f^{-1}(Y_{\leqslant y}))$ is $\varepsilon$-acyclic over $R$ for
any $y=(\ldots,y_{i},\ldots)\in Y$. Then $BX$ and $BY$ are
$4m\varepsilon$-interleaved over $R$.

Proof. Let $\overline{X}$ be a linear extension of $X$. Recall that an element
of a persistence poset has an initial element $x_{i}\in X_{i}$. We can
enumerate all elements of $X$ by lexicographic order of pairs $(i,r)$ with $i$
— index of a component in which element is born and $r$ — the number of its
initial element in order on $\overline{X_{i}}$.

Let $Y^{r}\subset M(f)$ be the union of $Y$ and the first $r$ elements of $X$.
It is a persistence subposet of $M(f)$.

Consider $Y^{r}_{>x_{r}}=Y_{\geqslant f(x_{r})}$. The persistence space
$\mathcal{B}(Y_{\geqslant f(x_{r})})$ is a component-wise cone with apex
$\mathcal{B}(f(x_{r}))$. It is component-wise contractible, therefore
$\mathcal{B}(Y^{r-1})\hookrightarrow\mathcal{B}(Y^{r})$ is a component-wise
homotopy equivalence by Lemma 2.42. By iteration, the map
$\mathcal{B}(i_{Y}):\mathcal{B}(Y^{0})=\mathcal{B}(Y)\hookrightarrow\mathcal{B}(M(f))=\mathcal{B}(Y^{n})$
is a component-wise homotopy equivalence between $BY$ and $M(f)$. Note that
persistence structure is not used here.

Now consider a linear extension of $Y$ with the enumeration of elements and
$X^{r}\subset M(f)$ constructed analogously. We have
$X^{r-1}_{<y_{r}}=f^{-1}(Y_{\leqslant y_{r}})$. The classifying space of the
latter is $\varepsilon$-acyclic over $\mathbb{F}$ by the assumption of the
theorem. Hence homology modules of $\mathcal{B}(X^{r})$ and
$\mathcal{B}(X^{r-1})$ are $4\varepsilon$-interleaved. By transitivity of
$\varepsilon$-equivalence homology of $BX$ and $M(f)$ are
$4m\varepsilon$-interleaved.

We have that $H_{i}(BX)\stackrel{{\scriptstyle
4m\varepsilon}}{{\sim}}H_{i}(M(f))$ and $H_{i}(M(f))\stackrel{{\scriptstyle
0}}{{\sim}}H_{i}(BY)$ for all $i$. Hence $H_{i}(BX)\stackrel{{\scriptstyle
4m\varepsilon}}{{\sim}}H_{i}(BY)$. ∎

We expect the stronger statement with no conditions on $R$ to be true with
another error multiple and that it can be proved using the technique of this
paper by considering error propagation in the Kunneth spectral sequence.

Acknowledgements. The author is thankful to his supervisor Anton Ayzenberg for
multiple reviews of the article, invaluable support, and the link to the
article [5] — the starting point of the work. The author is also thankful to
Ivan Limonchenko for his review.

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# Multiple electron acceleration instances during a series of solar
microflares observed simultaneously at X-rays and microwaves

Marina Battaglia University of Applied Sciences and Arts Northwestern
Switzerland, 5210 Windisch, Switzerland  Rohit Sharma University of Applied
Sciences and Arts Northwestern Switzerland, 5210 Windisch, Switzerland
Yingjie Luo (骆英杰) Center for Solar-Terrestrial Research, New Jersey Institute
of Technology, 323 M L King Jr Blvd, Newark, NJ 07102-1982, USA  Bin Chen (陈彬)
Center for Solar-Terrestrial Research, New Jersey Institute of Technology, 323
M L King Jr Blvd, Newark, NJ 07102-1982, USA  Sijie Yu (余思捷) Center for Solar-
Terrestrial Research, New Jersey Institute of Technology, 323 M L King Jr
Blvd, Newark, NJ 07102-1982, USA  Säm Krucker University of Applied Sciences
and Arts Northwestern Switzerland, 5210 Windisch, Switzerland  Space Sciences
Laboratory, University of California, 7 Gauss Way, 94720 Berkeley, USA

###### Abstract

Even small solar flares can display a surprising level of complexity regarding
their morphology and temporal evolution. Many of their properties, such as
energy release and electron acceleration can be studied using highly
complementary observations at X-ray and radio wavelengths. We present X-ray
observations from the Reuven Ramaty High Energy Solar Spectroscopic Imager
(RHESSI) and radio observations from the Karl G. Jansky Very Large Array (VLA)
of a series of GOES A3.4 to B1.6 class flares observed on 2013 April 23. The
flares, as seen in X-ray and extreme ultraviolet (EUV), originated from
multiple locations within active region NOAA 11726. A veritable zoo of
different radio emissions between 1 GHz and 2 GHz was observed co-temporally
with the X-ray flares. In addition to broad-band continuum emission, broad-
band short-lived bursts and narrow-band spikes, indicative of accelerated
electrons, were observed. However, these sources were located up to 150 arcsec
away from the flaring X-ray sources but only some of these emissions could be
explained as signatures of electrons that were accelerated near the main flare
site. For other sources, no obvious magnetic connection to the main flare site
could be found. These emissions likely originate from secondary acceleration
sites triggered by the flare, but may be due to reconnection and acceleration
completely unrelated to the co-temporally observed flare. Thanks to the
extremely high sensitivity of the VLA, not achieved with current X-ray
instrumentation, it is shown that particle acceleration happens frequently and
at multiple locations within a flaring active region.

Sun: flares — Sun: X-rays, gamma rays — Sun: corona

##  1 Introduction

Solar flares are magnetic energy release events in the solar atmosphere that
efficiently accelerate particles to relativistic speeds. The most direct
signatures of flare accelerated electrons are found at X-ray and radio
wavelengths. Observations at these wavelengths are highly complementary (e.g.
White et al., 2011) and allow us to diagnose the locations and mechanisms of
particle acceleration in solar flares and to study how particles are
transported close to the Sun and away from the Sun.

Figure 1: Panel A: VLA median cross-power dynamic spectrum between 1 GHz and 2
GHz. The orange and red lines are the RHESSI 4$-$8 keV and 10$-$18 keV
lightcurves, respectively (arbitrary scaling). Flares were identified as
individual peaks in the 10$-$18 keV lightcurve and labelled 1 to 7. The GOES
lightcurve at 1$-$8 Å is given in dark grey (axis on the righthand side).
Panel B: X-ray lightcurves as in Panel A. Additionally, the frequency
integrated radio lightcurve is given as blue line. Panel C: Inset between
20:45:25 UT and 20:45:45 UT, showing narrow-band spikes in the decay phase of
F6.

Gyrosynchrotron emission at frequencies above $\sim$ 2 GHz and X-ray emission
at a few tens of keV originating from high energetic, nonthermal electrons are
often found to be tightly correlated (e.g. Melnikov et al., 2002; Krucker et
al., 2010). In a statistical study on the timings of X-ray and radio emission
at 17 GHz and 34 GHz, Krucker et al. (2020) found that this may in fact hold
for the majority of larger flares. In single-event analysis, such common
origin of the emissions leads to a more complete scenario of electron
acceleration than observations at one wavelength alone (e.g. Gary et al.,
2018). Chen et al. (2021) performed an in-depth study of a source above the
top of magnetic loops (above-the-loop-top source). Such sources have been
interpreted as the (primary) acceleration site (e.g. Masuda et al., 1994;
Krucker et al., 2010; Chen & Petrosian, 2012; Krucker & Battaglia, 2014;
Battaglia et al., 2019). Combining spatially resolved X-ray and radio spectra,
Chen et al. (2021) inferred the energetic electron spectrum at the
acceleration site over a much wider energy range than would be possible from
observations at either wavelength alone.
However, very often, radio emission is not observed from the same location as
X-rays. This is typically the case for different types of coherent emission
observed below $\sim 2$ GHz. In many instances, such emission is due to beams
of accelerated electrons whose X-ray signatures are too faint to be detected
with current instrumentation (e.g. Saint-Hilaire et al., 2009; Glesener &
Fleishman, 2018; Chen et al., 2018), that are propagating away from the Sun on
open field lines. In many of these cases it is found that the electrons were
accelerated at the same site as electrons that become trapped in closed
magnetic loops. However, a number of authors found such emissions to originate
from a different, secondary acceleration site (e.g. Battaglia & Benz, 2009;
Benz et al., 2011; Carley et al., 2016; Chen et al., 2015, 2019). Large solar
flares usually have a long duration (up to several hours) and often display a
complex morphology with multiple loop systems, hence it is not surprising that
multiple acceleration sites are found. However, even smaller flares, so-called
microflares (Hannah et al., 2011) can display a remarkable complexity as
demonstrated by e.g. Sharma et al. (2020), Glesener et al. (2020), Duncan et
al. (2021), and Vievering et al. (2021). Understanding this complexity and
different features observed at both, radio and X-ray wavelengths, is the key
to understanding where and how electron acceleration takes place and how
electrons are transported.

Here, we present a series of microflares, ranging in GOES class from B8 to
B9.3 (A3.4 to B1.6 background-subtracted), that were observed simultaneously
at X-rays by the Reuven Ramaty High Energy Solar Spectroscopy Imager (RHESSI,
Lin et al., 2002), at radio frequencies between 1 GHz and 2 GHz by the Karl G.
Jansky Very Large Array (VLA, Perley et al., 2011), and at extreme ultraviolet
(EUV) wavelengths by the Atmospheric Imaging Assembly (AIA, Lemen et al.,
2012) on board the Solar Dynamics Observatory (SDO, Pesnell et al., 2012). The
flares happened on 2013 April 23 over a duration of fifty minutes. They all
originated from active region NOAA 11726 (N13W49) but from different locations
within it. During the observations, a veritable zoo of different radio
emissions was observed, most, but not all, temporally associated with X-ray
flares. However, there was no spatial association between the radio emissions
and the X-ray flare locations, suggesting multiple acceleration instances and
sites before, during and after the main flares, and energetic particles that
gained access to large or open field lines far away from the main acceleration
site. This work is intended as an overview of all events, representing the
”bigger picture”. Individual events will be analysed in-depth in a follow-up
publication.

In Section 2 an overview of the active region, the X-ray flares and radio
emissions is given, followed by a detailed description of imaging and spectral
analysis in both X-rays and radio wavelengths (Section 3). In Section 4, a
detailed investigation of the temporal and spatial association between radio
emissions and X-ray emission in the different flares is made. The findings are
discussed and interpreted in Section 5. Conclusions are drawn in Section 6

Flare | X-ray | X-ray | Dominant Radio feature | Radio image
---|---|---|---|---
No. | peak-time (UT) | integration time (UT) | in dynamic spectra | integration time
1 | 20:17:42 | 20:15:00 - 20:19:30 | no VLA data | $-$
2 | 20:26:10 | 20:27:10 - 20:29:10 | none | 20:26:41.5-20:26:42.5
3 | 20:34:25 | 20:33:30 - 20:35:36 | broad-band continuum | 20:34:30-20:34:31
4 | 20:36:47 | 20:35:40 - 20:38:36 | broad-band continuum | 20:37:31-20:37:32
5 | 20:40:51 | 20:40:00 - 20:42:36 | short-lived burst | 20:40:54 - 20:40:55
6 | 20:44:31 | 20:43:50 - 20:46:02 | none | 20:44:30-20:44:31
6b | 20:44:31 |  | short-lived burst during decay phase | 20:45:21-20:45:22
6s | 20:44:31 |  | narrow-band spikes during decay phase | 20:45:25.90 - 20:45:25.95
7 | 20:56:04 | 20:55:20 - 20:57:00 | none | 20:56:05.5 - 20:56:06.5
Table 1: X-ray peak-times in 10$-$18 keV lightcurve, X-ray image intergration
time, radio feature identified in dynamic spectrum and radio image integration
times.

##  2 Event overview

The flares were observed on April 23 2013 between 20:10 UT and 21:00 UT.
Figure 1 shows the median radio cross-power dynamic spectrum between 1 and 2
GHz along with the GOES 1$-$8 Å lightcurve and RHESSI X-ray lightcurves at two
energy bands (4$-$8 keV and 10$-18$ keV). Seven individual flares were
identified based on X-ray emission in the RHESSI 10$-$18 keV lightcurves, the
highest photon energies that RHESSI detected from these flares. The individual
flares were numbered 1 to 7, as indicated in the Figure. We will refer to them
as F1 through F7. The corresponding radio sources will be denoted F1R through
F7R. For all flares, except F3 and F4, the intensity at 10 $-$ 18 keV returned
to background level between each flare. Since X-ray images of F4 showed an
additional source, not visible during F3, these two peaks were treated as
individual flares rather than one flare with two peaks. The 10-18 keV emission
of all flares except F1 is quite impulsive, suggesting potential nonthermal
emission while the lightcurve of F1 is more gradual in both energy ranges,
hinting at purely thermal emission. A definite distinction is only possible
through spectroscopy (see Section 3.1).

During the whole duration of the observations, various types of radio
emissions were visible in the dynamic spectrum: broad-band continuum emission
above 1.5 GHz, most prominently during F3 to F7, broad-band short-lived bursts
during F5 and F6, and narrow-band spikes in the decay phase of F6. The latter
two will be referred to as F6bR and F6sR. Even though these emissions were
temporally associated with the X-ray flares, there was no clear time
correlation. Table 1 gives as summary of the observed features, including the
X-ray peak-times, the time integrations of the RHESSI and VLA images (see
Section 3) and a short description of the observed features in the radio
dynamic spectrum.

##  3 Data analysis

X-ray images and spectra were generated at the high-energy X-ray peak-time of
each of the seven flares. Radio images and spectra were generated at the same
times as the X-ray data products. Additionally, images and spectra of the
burst and the spikes in the decay phase of F6 were generated, as described in
detail below.

###  3.1 X-ray imaging and spectroscopy

RHESSI images were generated using the CLEAN algorithm (Högbom, 1974; Hurford
et al., 2002) for two energy bands: $4-8$ keV (thermal emission) and $10-18$
keV (potentially nonthermal emission). For the image reconstruction, detectors
5, 6, 7, and 8 were chosen to limit noise contributed by detectors with finer
grids either because there was no flux modulation or because of reduced
detector sensitivity during that time of the mission. The image integration
time was chosen to include the whole high-energy peak but minimum 60 s to
achieve good count statistics, especially in the higher energy band. The image
integration times are listed in Table 1. X-ray spectra were generated using
the same integration times as for the images. For spectral fitting, detector 6
was used as, at that point in the RHESSI mission, this detector had the
highest sensitivity with relatively low background. All flares were fitted
with a single thermal component at the lowest energies (compare Table 2).
However, spectral fitting of the higher energies proved difficult. After
careful background subtraction, only a few counts were observed at energies
between 10$-$18 keV, most prominently in F4 and F7. Since detector pileup
could be excluded as origin of this emission, we interpreted it as weak
signatures of accelerated, nonthermal electrons and took the following
approach to obtain an upper limit on the nonthermal electron spectrum. The
single thermal component fitted previously was fixed and an additional
nonthermal thick-target component was fitted up to $\sim$18 keV, depending on
the flare. Then both, the thermal and nonthermal component were fitted
simultaneously. Even though a nonthermal component could be fitted in this
manner, the parameters were not well constrained with uncertainties that were
larger than the fit parameters in many cases. An example spectrum is given in
Figure 2. The nonthermal spectrum shown in the figure corresponds to a total
electron flux of $F_{e}=1.0\times 10^{34}$s-1, electron spectral index
$\delta=4.9$ and low-energy cutoff $E_{low}=6.5$ and has to be seen as an
upper limit. An additional difficulty was posed by F1 and F4 for which two
X-ray sources could be imaged simultaneously. Hence, these flares actually
constituted of two separate events happening simultaneously and the full Sun
spectra are superpositions of both. For larger flares, individual spectra of
flaring sources can be isolated through imaging spectroscopy (e.g. Emslie et
al., 2003; Battaglia & Benz, 2006; Simões & Kontar, 2013; Krucker & Battaglia,
2014). However, the sources here were too faint to be imaged at more than two
energy bands. Note that a fitting model consisting of two purely thermal
components with different temperatures would be another possibility. Such a
model resulted in even larger uncertainties of the fit parameters and was
discarded.

Flare | Emission measure | Temperature
---|---|---
No. | $(10^{46}\rm{cm^{-3}})$ | (MK)
1 | $4.1\pm 0.3$ | $10.7\pm 0.1$
2 | $4.5\pm 0.5$ | $10.0\pm 0.2$
3 | $3.5\pm 0.3$ | $11.4\pm 0.1$
4 | $1.3\pm 0.9$ | $14.3\pm 1.7$
5 | $1.3\pm 0.9$ | $14.3\pm 1.7$
6 | $1.7\pm 0.1$ | $12.6\pm 0.2$
7 | $1.2\pm 1.8$ | $11.5\pm 0.7$
Table 2: Temperature and emission measure from X-ray spectral fitting

###  3.2 Radio imaging and spectroscopy

The VLA observations (under the observing program VLA/13A-384) were carried
out using two subarrays. One subarray observed in the 1–2 GHz ($\lambda=15$–30
cm) L band with 13 antennas, and another in the 2–4 GHz ($\lambda=7.5$–15 cm)
S band with 13 antennas. In the present study, we focus on the 1–2 GHz L band
data where the radio bursts are present. The L band data had a spectral
resolution of 2 MHz and temporal resolution of 50 ms in both the right-hand-
and left-hand-circular polarization (RCP and LCP). The spectral range of the
observations was divided uniformly into 8 spectral windows. Each spectral
window had 64 2-MHz-wide frequency channels. The L-band subarray was in
D-configuration with a total of 13 antennas used for the observations. The
longest baseline was 1031 m, corresponding to an angular resolution of
$70^{\prime\prime}\times 40^{\prime\prime}$ at $1.5$ GHz at the time of the
observations.

CLEAN images in both RCP and LCP were made during the X-ray peak-time of each
flare and, additionally, for the burst and the spikes in the decay phase of
F6. A time-integration of one second and frequency integration of 20 MHz were
used for all images except the spikes. The latter were only detected at the
lowest frequencies, extremely short-lived and much brighter than the other
sources. Hence, they were imaged at full time resolution of 50 ms and full
frequency resolution of 2 MHz. A circular beam with FWHM of 70 arcsec at 1
GHz, linearly decreasing to 40 arcsec at 2 GHz, was used for image
convolution. The radio emission observed in the dynamic spectrum was dominated
by background emission of the active region that acted as a strong background
source. Hence, background subtraction was performed in the visibility domain
to isolate the flaring sources. As background time, the time interval between
20:08:00 and 20:08:01 UT was chosen (20:45:24.5-20:45:25.0 UT for F6sR).
Figure 2 shows images of the background source and the burst in LCP during F6
at selected frequencies for comparison.

Figure 2: Panels A and B: Example of RHESSI count spectrum and corresponding
image (F4). Panel A shows the RHESSI count spectrum where the range $4-8$ keV
is colored blue and the range $10-18$ keV is colored pink. The grey line gives
the background level. The image in panel B is an AIA Fe XVIII image with
contours from the RHESSI images at $4-8$ keV and $10-18$ keV overlaid in the
respective colors. In this case, the spectrum and images suggest some flaring
emission, of the same order as the background level, up to 15 keV. Panel C
gives the photon spectrum fitted with a thermal component (blue) plus a
nonthermal thick target component (green). The red curves represents the
combined model. Panel D: Comparison of the radio source locations at selected
frequencies (colors) between active region thermal emission and the radio
burst during F6. Contours are at 90% and 99% levels of maximum emission in
each image. Panel E: Radio brightness temperature spectra, color coded
according to legend, where 6bR denotes the radio burst during F6. Colored dots
indicate the frequencies for which a source position was determined. Panel F:
Radio brightness temperature spectra of the short-lived spikes (F6sR) during
F6.

The centroid location of the burst sources was determined as a function of
frequency by fitting a 2D Gaussian at a flux level of $>90\%$ of the maximum
emission in each image. Radio spectra were computed from the brightness
temperature maps by selecting the maximum brightness temperature of the source
in images for which a source location could be fitted. At some frequencies the
images were dominated by noise and no location could be fitted. In that case,
no data point is shown in Figure 2. The uncertainties of the brightness
temperature spectra are given as the root-mean-square noise of the images,
excluding the flaring source.

Imaging and extraction of spectra was done for both polarizations. However,
only the spikes during F6 were highly polarized ($<-$0.8), where the degree of
circular polarization was defined as (RCP-LCP)/(RCP+LCP). F6b had a
polarization degree between $-$0.7 and 0.2, depending on frequency. All other
sources were polarized to less than 30%. For further analysis and
interpretation, we used the LCP data.

###  3.3 EUV and magnetic field imaging

AIA images and magnetic field maps from the Helioseismic and Magnetic Imager
(HMI, Scherrer et al., 2012) on SDO were processed using the standard data
processing pipeline for these instruments. Using the method developed by Del
Zanna (2013), we also constructed AIA maps of the Fe XVIII line, which has a
peak formation temperature of 8 MK (Del Zanna et al., 2015), typical for
smaller flares.

##  4 Results

First, we investigate the X-ray location of each flare relative to the other
flares and to flaring emission seen at EUV wavelengths and an HMI magnetogram,
followed by presentation of the radio source locations relative to each-other
and to the X-ray sources and EUV emission features.

###  4.1 X-ray source locations

Figure 3: Top row: Locations of X-ray sources at 4$-$8 keV relative to EUV
emission in the Fe XVIII line (left) and an HMI magnetogram (right). On the
AIA FFe XVIII-image the X-ray sources are indicated by the 50%, 70%, and 90%
contours of RHESSI CLEAN. Numbers next to the sources give the flare number
according to the legend on the right. On the right, the location of maximum
emission in RHESSI images is indicated by crosses. Here, the location of
weaker X-ray sources during F1 and F4 are also indicated near coordinates
[650,300] and [680,260], respectively. Different symbols are used for better
visibility with arbitrary symbol size. The starting times of the RHESSI images
are given to the right in the respective colors along with the flare number.
Bottom row: AIA 171 Å image and HMI image overlaid with the frequency-averaged
location of the radio sources associated with each flare. The size of the
cross gives the standard deviation of the frequency-dependent source
locations. For better visibility, the location and standard deviation of the
spike locations is indicated with an ellipse. The locations of F6bR and F6sR
are marked, respectively.

Figure 3 summarizes the locations of the X-ray flares relative to features in
AIA 171 Å and Fe XVIII images and an HMI magnetogram. The AIA images were
composed by averaging images at 20:27:01 UT, 20:45:01 UT, and 20:54:59 UT. The
AIA 171 Å wavelength band is most sensitive to cooler plasma around 1 MK while
the Fe XVIII images show plasma at 8 MK. The locations of the X-ray sources at
4$-8$ keV are given as contour levels relative to the maximum in each image on
the AIA image and as symbols on the HMI image. The magnetogram shows a complex
magnetic topology with large positive and negative polarity patches at either
side of the active region, interspersed with smaller patches of either
polarity. EUV images show multiple loops at different temperatures. A large
loop system, best visible in Fe XVIII images, seems to connect the large
positive and negative magnetic field patches. Smaller loops are visible
between the large loops. X-ray and radio images show sources from multiple
locations within the active region over the fifty minutes of observations. F1
and F2 originated from the base of a larger loop system, visible at 171 Å and
where associated with an EUV jet. There was a second, fainter source imaged
during F1 that was associated with a different loop system seen in the Fe
XVIII images. F3 to F6 originated from a small loop system, clearly visible in
the Fe XVIII images but not at the same location as F1 and F2. An additional,
fainter source was observed during F4 that had no such obvious EUV
counterpart. F7 originated from yet another location, close to the footpoint
of an EUV loop.

###  4.2 Radio source locations

Figure 3 gives the frequency-averaged centroid locations of the radio sources
during each X-ray flare. The error bars give the $1\sigma$ standard deviation,
representing the frequency-dependent scatter of the sources. All radio sources
were located Eastward of the hot EUV loops and the locations with strongest
magnetic field. None of them were co-spatial with any of the X-ray sources nor
with the EUV loops.

For the brightest radio features (F5R, F6bR, and F6sR) we also investigate the
locations as a function of frequency. These are shown in Figure 4, overlaid on
a composite AIA image and individually over AIA 171 Å images. The color table
in the EUV images was adapted such that faint loops become more visible. For
F5R and F6bR we restricted us to centroid locations for which the maximum flux
of the source was larger than 0.3 MK. As mentioned before, none of these
sources coincide with any of the X-ray sources. F5R shows a systematic
displacement of the centroid location as a function of frequency perpendicular
to a large, faint loop visible at 171 Å. F6bR and F6sR originate from about
the same location, close to a fan-like structure. The scatter of the
frequency-dependent centroids in F6bR is considerable and there is no clear
trend, other than that lower frequencies seem to originate preferentially
further north than higher frequencies. Within F6sR, the scatter is smaller
than in F6bR, but again no systematic displacement of the locations as
function of frequency can be observed.

Figure 4: Radio source locations as a function of frequency for F5R, F6R and
F6sR. The top image is a composite of a time-averaged AIA 171 Å (red) image
and an Fe XVIII image (blue). Below are close-ups of the centroid locations of
three features as a function of frequency overlaid on a AIA171 Å image closest
in time to the burst time. The locations as function of frequency are color-
coded.

###  4.3 Spectra

####  4.3.1 X-ray spectra

Table 2 summarizes the results from the thermal spectral fitting as described
in Section 3.1. As an example of a fitted spectrum, Figure 2 shows the
background-subtracted count rate spectrum of F4. The background-subtracted
count rate above 10 keV was around the background level, yet two sources could
be imaged in this energy range. The temperatures of the flares ranged between
10.0 MK in F2 to 14.3 MK in F4 and F5. The emission measure was lowest in F7
at $1.2\times 10^{46}$ cm-3 and highest in F2 at $4.2\times 10^{46}$ cm-3.
Even though a nonthermal component could be fitted in F4 and F7, the
uncertainties were too large (same order or larger as the fit parameters) for
the result to be interpreted with confidence.

####  4.3.2 Radiospectra

The radio spectra shown in Figure 2 show the maximum brightness temperature as
a function of frequency in LCP polarization. Except for F5R, F6sR, and F6bR
the brightness temperature was low (less than $\sim$ 0.5 MK) or, depending on
the frequency, at the noise level of the image. Emission during F5R was
detected at almost all frequencies and peaked at $\sim$ 1.4 GHz. F6bR had the
highest brightness temperature among the broad-band features with a maximum
around 2.1 MK at 1 GHz, declining to 0.3 MK at 2 GHz. The maximum brightness
temperature of F6sR (spikes) was 5 MK. Since such bursts may originate from
fairly compact source regions, it is likely that they are not spatially
resolved. Indeed, the FWHM of the fitted 2D Gaussian was around 20 arcsec at
1.1 GHz while the restoring beam size was 60 arcsec. Hence these values should
be considered as lower limit.

##  5 Discussion

During a fifty minute time window, seven X-ray flares between GOES A3.4 and
B1.6 (background subtracted) were observed co-temporally with a variety of
radio emissions, such as broad-band bursts, narrow-band, short-lived spikes
and broad-band continuum emission at frequencies between 1 to 2 GHz. The
broad-band continuum emission can be attributed to background thermal emission
of the active region, while the short-lived bursts and spikes are indicative
of accelerated electrons. The X-ray flares displayed quite typical
characteristics regarding size, spatial origin, and energies, as will be
discussed in some more detail in Section 5.1. The radio sources were displaced
from the X-ray sources by more than 100 arcseconds despite close temporal
association with the X-ray emission. In Section 5.2 we will discuss possible
explanations for this displacement.

###  5.1 X-ray microflare locations and energies

The observed microflares were quite ordinary in the sense that they were short
and compact. It is noteworthy that, during F1 and F4, two sources from
different locations within the active region were imaged. Since the RHESSI
imaging technique favors the brightest source present, this means both of
these sources were of comparable brighthness and the X-ray lightcurves and
spectra consist of a superposition of emission from two separate flares that
were, however, too faint to be analysed individually through imaging
spectroscopy.
X-ray spectra indicate emission up to $\sim$ 14 keV. For all flares it was
possible to fit a thermal component. From the fitted temperature $T$ and
emission measure $EM$ one can calculate the thermal energy as

$E_{th}=3k_{B}T\sqrt{EM\times V}$ (1)

where $k_{B}$ is the Boltzmann constant and $V$ the flaring volume. Assuming a
filling factor of unity and a flaring volume of $V=A^{3/2}$ where $A$ is the
area of a circle of radius 10 arcsec, one finds thermal energies of the order
of $2.4\times 10^{28}$ erg to $4\times 10^{28}$ erg. Emission above $\sim$10
keV was at the background level and any spectral fitting model was not well
constrained, as described in Section 3.1. Using upper limit spectral
parameters (total electron flux $F_{e}=1.0\times 10^{34}$s-1, electron
spectral index $\delta=4.9$ and low-energy cutoff $E_{low}=6.5~{}$keV) the
total nonthermal power is calculated as

$P=\frac{\delta-1}{\delta-2}F_{e}E_{low}=1.4\times 10^{26}\mathrm{erg\
s^{-1}.}$ (2)

Both, thermal and nonthermal energies are consistent with earlier findings
using RHESSI observations for similarly sized flares (Stoiser et al., 2007;
Hannah et al., 2008; Inglis & Christe, 2014).

###  5.2 Radio sources as secondary acceleration sites?

The most noteworthy observation is the significant displacement between the
location of the radio burst sources and the X-ray sources. In a simple
scenario in which electrons are accelerated during a flare and the signatures
of the same accelerated electron population are observed at both, X-ray and
radio wavelengths, one would expect the locations of the radio sources to be
close to the flaring site as seen in X-rays where the radio emission can be
either gyrosynchrotron emission from electrons trapped in flaring loops or
coherent emission from electron beams escaping along open field lines visible
as e.g. type III bursts. However, only some of the observations presented here
can be explained with this simplest scenario.

Observationally one can distinguish three cases:
1) Both, X-ray and radio sources, originate from the same magnetic loop
structure (F7).
2) Radio emission does not originate from the same magnetic structure as X-ray
emission, but both structures are magnetically connected (F4, F5).
3) Radio emission originates from a different magnetic structure seemingly not
connected to the original X-ray flare site (F6).

To investigate the magnetic connectivity of the active region and confirm the
cases outlined above, we performed non-linear force free magnetic field
extrapolations (NLFFF) using the gx_simulator package (part of the IDL
SolarSoftware distribution; Nita et al. 2015), shown in Figure 5.

Figure 5: Non-linear force free magnetic field extrapolations seen from above
(Panel A) and from the side (Panel B). The approximate locations of the X-ray
flares are indicated with white ellipses and white labels. The locations of
the brightest corresponding radio sources are indicated with purple shapes and
labels.

The extrapolations show four major structures: small, closed magnetic loops at
the western side of the active region (labelled W in Figure 5); large, open
and closed field lines in the North (N); a closed loop structure centered
between the two mentioned before (C), and another, separate structure to the
East (E). Except for the last one, the magnetic loops found by the NLFFF are
also visible at EUV wavelengths. Due to projection effects it is difficult to
locate the sources on the magnetic field model exactly and the available data
do not provide enough constraints to locate them based on density or height
arguments, hence, the likely locations are indicated with elliptic shapes in
the Figure, where white ellipses outline the most likely origin of the X-ray
emission and purple shapes indicate the likely origin of the corresponding
radio emission deduced by qualitative comparison of the EUV features and radio
and X-ray source locations. In the following we provide some physical
scenarios and explanations for the cases listed above.

####  5.2.1 Scenario 1: Same electron population within same magnetic loop
(F7)

During F7, radio emission was observed co-temporally with X-ray emission and,
as can be seen in Figures 4 and 5, the sources at the two different
wavelengths originated from the same magnetic loop structure. These
observations are compatible with a scenario in which electrons are accelerated
during a flare, become trapped in a closed magnetic loop and emit radiation at
X-ray and radio wavelengths.

####  5.2.2 Scenario 2: Same electron accelerated electron population with
significant transport (F4, F5)

The radio emission in both of these events was observed co-temporally with the
high-energy X-ray emission with the strong burst in F5R happening 3 seconds
after the X-ray peak. The frequency dependent locations of F5R show no clear
trend, indicating that the electrons do not propagate away from the Sun along
open field lines as, in that case, one would expect a clear frequency
dependence of the source location (e.g., Chen et al., 2013, 2018). For both of
these cases it is likely that the emission is due to electrons that were
accelerated near location W and were either directly injected into one of the
field lines of loop system C or gained access to it through cross-field
diffusion.

####  5.2.3 Scenario 3: Secondary electron acceleration (F6b, F6s)

Both, F6bR and F6sR happened between 50 seconds and 54 seconds after the X-ray
peak time, and originated from field lines that seemingly are not connected to
the X-ray flare site. Short-lived spikes, similar to F6sR, at MHz to GHz
frequencies have commonly been interpreted as signatures of flare accelerated
electrons due to their high temporal association with hard X-rays (e.g. Benz,
1986; Guedel et al., 1991; Aschwanden & Guedel, 1992). However, imaging
studies of spikes showed that they can be displaced as much as 400 arcsec from
the main flare site (Benz et al., 2002; Khan & Aurass, 2006; Battaglia & Benz,
2009). Such findings suggested that spikes are signatures of secondary
acceleration, that typically happens higher in the Corona but is causally
linked to the main energy release. However, the connection to the main energy
release is not always obvious. Recently, Luo et al. (2021) observed stochastic
spike burst associated with an M-class flare for which the connection with the
flare was not obvious as is the case for F6bR and F6sR in the observations
presented here.

Since the NLFFF extrapolated field may deviate from the actual magnetic
connectivity in the Corona, particularly during flaring times, one explanation
is that we do not capture the fieldlines that connect regions W and E and
electrons were accelerated during the flare in region W and transported to
region E, similar to the second scenario. However, a more intriguing
explanation is that electrons were accelerated in-situ at a secondary
acceleration site, triggered by the original flare. Such sympathetic flares,
but also coronal mass ejections, filament eruptions and solar energetic
electron events have been known for quite some time (e.g. Pearce & Harrison,
1990; Wang et al., 2018, 2021), and have been found to even happen between
active regions. The exact mechanism that causes them is still under debate. It
has been suggested that the regions involved could be connected by large scale
coronal loops (e.g. Wang et al., 2001; Jiang et al., 2008) that are perturbed
but the original flare which results in reconnection events far away from the
original flare site.

##  6 Conclusions

The combined radio and X-ray observations shown here demonstrate that even
smaller flares can display a surprising level of complexity. Despite close
temporal association between radio and X-ray emission, the two types of
emissions originated from entirely different regions. While for some events it
can be argued, based on magnetic connectivity, that electrons were accelerated
during the primary energy release, in the other events, the radio emissions
are likely due to secondary energy release and particle acceleration, either
completely unrelated to the primary energy release or triggered by it in a way
yet to be understood. Thanks to the high sensitivity of the VLA it is shown
that nonthermal processes happen frequently and at many different locations of
an active region. Observations from future X-ray instruments that use focusing
optics, resulting in a much higher sensitivity and dynamic range as current
instruments, combined with VLA observations are needed to fully understand
these processes.

This work was supported by the Swiss National Science Foundation (grant no.
200021_175832). It makes use of public archival VLA data from the observing
program VLA/13A-384 carried out by the National Radio Astronomy Observatory
(NRAO). The authors acknowledge Tim Bastian, Dale Gary, and Stephen White for
their help in carrying out the observing program and Gregory Fleishman for
help with the GX-simulator package. Y.L. and B.C. are supported by NSF grant
AGS-1654382 to the New Jersey Institute of Technology. The NRAO is a facility
of the National Science Foundation (NSF) operated under cooperative agreement
by Associated Universities, Inc. Some of the figures within this paper were
produced using IDL colour-blind-friendly colour tables (see Wright, 2017).

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