rem[theo]Remark

<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>18N10, 18N15,
18A05, 18A22, 18A40

# Lax comma categories: cartesian closedness, extensivity, topologicity, and
descent

Maria Manuel Clementino  Fernando Lucatelli Nunes and Rui Prezado  (1,2,3):
University of Coimbra, CMUC, Department of Mathematics, 3000-143 Coimbra,
Portugal
(2): Department of Information and Computing Sciences, Utrecht University, The
Netherlands

###### Abstract

We investigate the properties of lax comma categories over a base category
$X$, focusing on topologicity, extensivity, cartesian closedness, and descent.
We establish that the forgetful functor from ${\rm\sf Cat}//X$ to ${\rm\sf
Cat}$ is topological if and only if $X$ is large-complete. Moreover, we
provide conditions for ${\rm\sf Cat}//X$ to be complete, cocomplete, extensive
and cartesian closed. We analyze descent in ${\rm\sf Cat}//X$ and identify
necessary conditions for effective descent morphisms. Our findings contribute
to the literature on lax comma categories and provide a foundation for further
research in $2$-dimensional Janelidze’s Galois theory.

###### keywords:

lax comma categories, Grothendieck descent theory, Galois theory,
$2$-dimensional category theory, topological functor, effective descent
morphism, cartesian closed category, exponentiability

## Introduction

The ubiquitous notion of comma category has a natural $2$-dimensional lax
notion, called lax comma $2$-category (see, for instance, [15, Ch. I,5]). The
first two authors’ motivation for investigating this notion stems from the
fundamental role it plays in our approach to studying $2$-dimensional
counterparts of Janelidze’s Galois theory [11], particularly regarding its
interplay with lax orthogonal factorization systems [9].

While various remarkable insights exist in the literature (see, for instance,
[13, 16, 10, 34]), we identified a gap in the systematic exploration of
foundational properties crucial for advancing our research in the direction
suggested in [11].

The present paper builds upon our prior examination of lax comma ordered sets
[10], extending the scope to encompass lax comma categories ${\rm\sf Cat}//X$,
where ${\rm\sf Cat}$ denotes the category of small categories and $X$
represents a (possibly large) category. Our primary objective is to lay down
the groundwork for our ongoing work in Galois theory and descent theory
initiated in [11]. In pursuit of this aim, we concentrate on four fundamental
aspects: cartesian closedness, extensivity, topologicity, and descent.

In Section 2, we start by giving conditions under which the category ${\rm\sf
Cat}//X$ is cartesian closed, extensive and (co)complete, showing that the
properties of the base category $X$ play a crucial role in determining the
properties of ${\rm\sf Cat}//X$. In this direction, we also establish that, if
$X$ is small, the forgetful functor ${\rm\sf Cat}//X\rightarrow{\rm\sf Cat}$
is topological if and only if $X$ is complete (which is equivalent to say that
$X$ is a complete lattice).

The study and characterization of effective descent morphisms have a rich and
intricate history. The work of Reiterman-Tholen [38] in the realm of
topological spaces, reformulated by Clementino-Hofmann [5], exemplifies the
depth and complexity of this classification problem. Several notable
contributions have significantly advanced our understanding of effective
descent morphisms in categories of categorical structures. Notably, the study
of effective descent morphisms in the categories of internal categories [22],
the investigation into categories of enriched categories (see, for instance,
[7], [23, Theorem 9.11] and [35]), and the study of effective descent
morphisms in categories of generalized multicategories (see, for instance, [6,
36, 37]) have each imparted pivotal insights. In Section 3, we initiate our
investigation into effective descent morphisms in ${\rm\sf Cat}//X$. In this
direction, we show that the forgetful functor ${\rm\sf
Cat}//X\rightarrow{\rm\sf Cat}$ preserves effective descent morphisms. Despite
the seemingly basic nature of ${\rm\sf Cat}//X$, we unveil that the full
characterization of effective descent morphisms in this category remains an
open challenge. This work serves as the inception of our exploration into this
descent aspect of ${\rm\sf Cat}//X$.

Our investigation on basic categorical properties of ${\rm\sf Cat}//X$
presented herein sheds light on the general properties of lax comma categories
and lays the foundation for our future work in generalized aspects of lax
comma objects in $2$-categories, particularly in the context of
$2$-dimensional Janelidze-Galois theory. In Section 4, we give some further
comments and point to future work.

This paper serves as a formal exposition of our findings, contributing to the
basic literature on lax comma categories and lax comma $2$-categories in
general.

#### Acknowledgements

We thank the anonymous referee for their insightful and prompt feedback, which
included valuable remarks, suggestions, and additional references.

The authors acknowledge partial financial support by Centro de Matemática da
Universidade de Coimbra (CMUC), funded by the Portuguese Government through
FCT/MCTES, DOI 10.54499/UIDB/00324/2020.

This research was supported through the programme “Oberwolfach Leibniz
Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2022.

##  1 Preliminaries

We denote by ${\rm\sf Cat}$ the $2$-category of small categories, functors and
natural transformations. In this section, we recall the explicit definition of
lax comma categories ${\rm\sf Cat}//X$. For additional general fundamental
aspects, we recommend consulting [15, I,5] and [11].

###### Definition 1.1

Given a (possibly large) category $X$, we denote by ${\rm\sf Cat}//X$ the
category defined by the following.

  * –

The objects are pairs $(W,a)$ in which $W$ is a small category and
$a:W\rightarrow X$ is a functor.

  * –

A morphism in ${\rm\sf Cat}//X$ between objects $(W,a)$ and $(Y,b)$ is a pair

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in which $f:W\rightarrow X$ is a functor and $\gamma$ is a natural
transformation.

If $(f,\gamma):(W,a)\rightarrow(Y,b)$ and $(g,\chi):(Y,b)\rightarrow(Z,c)$ are
morphisms of ${\rm\sf Cat}//X$, the composition is defined by $\left(g\circ
f,\left(\chi\ast{\rm id}_{f}\right)\cdot\gamma\right)$, that is to say, the
composition of the morphisms $g$ and $f$ with the pasting

$\textstyle{W}$$\textstyle{Y}$$\scriptstyle{f}$$\textstyle{Y}$$\textstyle{Z}$$\scriptstyle{g}$$\textstyle{W}$$\textstyle{X}$$\scriptstyle{a}$$\textstyle{Z}$$\textstyle{X}$$\scriptstyle{c}$$\textstyle{Y}$$\textstyle{X}$$\textstyle{\scriptstyle
b}$ $\scriptstyle{\gamma}$ $\scriptstyle{\chi}$

of the natural transformations $\chi$ and $\gamma$. Finally, with the
definitions above, the identity on the object $(W,a)$ is the morphism $({\rm
id}_{W},{\rm id}_{a})$.

Herein, the category ${\rm\sf Cat}//X$ is called the lax comma category over
$X$.

A morphism $(f,\gamma)$ in ${\rm\sf Cat}//X$ is called strict if $\gamma$ is
the identity. It should be noted that the comma category ${\rm\sf Cat}/X$ is
the subcategory of ${\rm\sf Cat}//X$ consisting of the same objects but only
the strict morphisms.

While the category ${\rm\sf Cat}//X$ can be endowed with a natural two-
dimensional structure, our current investigation primarily centers on its
foundational one-categorical structure. We briefly touch upon the two-
dimensional aspects in Section 4. For the reader interested in two-dimensional
features, we refer to [11, 33]. Further investigation of two-dimensional
aspects is reserved for future research endeavors.

###  1.1 Lax comma categories as total spaces

We recall that lax comma categories are fibred over ${\rm\sf Cat}$. This
fibred structure over ${\rm\sf Cat}$ is pivotal as it enables us to derive
several essential properties and insights concerning limits, colimits,
extensivity, topologicity, among other properties. We refer to [14], [20,
A1.1.7 and B1.3.1], [31] and [30, Section 6] for basic aspects on fibrations
and Grothendieck constructions.

Given a (possibly large) category $X$, the forgetful functor defined in
(1.1.1) is a fibration.

$U:{\rm\sf Cat}//X\rightarrow{\rm\sf Cat},\qquad(W,a)\mapsto
W,\qquad(f,\gamma)\mapsto f.$ (1.1.1)

Denoting by ${\rm\sf Cat}\left[-,X\right]:{\rm\sf Cat}^{\rm
op}\rightarrow{\rm\sf CAT}$ the functor that takes every small category $W$ to
the category ${\rm\sf Cat}\left[W,X\right]$ of functors and natural
transformations, we have that

$\textstyle{\sum{{\rm\sf Cat}\left[{-,X}\right]}}$$\textstyle{{{\rm\sf
Cat}}//X}$$\scriptstyle{\cong}$$\textstyle{\sum{{\rm\sf
Cat}\left[{-,X}\right]}}$$\textstyle{{\rm\sf
Cat}}$$\scriptstyle{\mathsf{U}}$$\textstyle{{{\rm\sf
Cat}}//X}$$\textstyle{{\rm\sf Cat}}$$\scriptstyle{U}$

commutes, where we denote by $\displaystyle\sum{\rm\sf Cat}\left[-,X\right]$
the Grothendieck construction of the $2$-functor ${\rm\sf
Cat}\left[-,X\right]$, and $\mathsf{U}$ is the associated fibration (forgetful
functor).

###  1.2 Adjoints

Clearly, if $X$ has an initial object $\mathsf{0}$, then (1.1.1) has a left
adjoint; namely

$L:{\rm\sf Cat}\rightarrow{\rm\sf Cat}//X,\qquad
W\mapsto\underline{\mathsf{0}},\qquad(f:W\rightarrow
Y)\mapsto\left(f,\iota\right)$ (1.2.1)

where $\underline{\mathsf{0}}$ denotes the functor constantly equal to the
initial object, and
$\iota:\underline{\mathsf{0}}\rightarrow\underline{\mathsf{0}}\circ f$ is the
only natural transformation. Dually, (1.1.1) has a right adjoint if $X$ has a
terminal object.

##  2 Basic properties

In this section, we study cartesian closedness, extensivity and topologicity
of lax comma categories. As a foundational step in our study, we also provide
an explicit construction of pullbacks, equalizers and coequalizers in lax
comma categories.

A fundamental underpinning for our investigation of limits and colimits in lax
comma categories lies in the pivotal observation that the forgetful functor
$U:{\rm\sf Cat}//X\rightarrow{\rm\sf Cat}$ is a fibration. For a more nuanced
understanding of limits and colimits in the context of fibred categories, we
recommend consulting [14], [30, Section 6], and [31].

###  2.1 Cartesian closedness

We start by studying the cartesian structure of the category ${\rm\sf
Cat}//X$. We note that, whenever $X$ has products, for each small category $W$
and any functor $g:W\rightarrow Y$, ${\rm\sf Cat}\left[W,X\right]$ has
products, and the change-of-base functors ${\rm\sf Cat}\left[g,X\right]$
preserve products.

Since $\sum{\rm\sf Cat}\left[-,X\right]\cong{\rm\sf Cat}//X$, by general
results on products of fibred categories, the observation above allows us to
conclude Propositions 2.1 and 2.2 (see, for instance, [14, 31] and [30,
Section 6] for further details).

###### Proposition 2.1

If $X$ has a terminal object, then so does ${\rm\sf Cat}//X$.

More precisely, the terminal object of ${\rm\sf Cat}//X$ is given by
$\left(\mathsf{1},\overline{\mathsf{1}}\right)$ where $\mathsf{1}$ is the
terminal category, and $\overline{\mathsf{1}}$ denotes the functor
$\overline{\mathsf{1}}:\mathsf{1}\rightarrow X$ whose image is the terminal
object in $X$.

###### Proposition 2.2

If $X$ has binary products then so does ${\rm\sf Cat}//X$.

More precisely, if $(W,a),(Y,b)$ are objects of ${\rm\sf Cat}//X$, then the
object $(W\times Y,a\underline{\times}b)$, where
$a\underline{\times}b(w,y)=a(w)\times b(y)$, is the product $(W,a)\times(Y,b)$
in ${\rm\sf Cat}//X$.

Furthermore, if $X$ has products, so does ${\rm\sf Cat}//X$. More explicitly,
if $\left(W_{i},a_{i}\right)_{i\in L}$ is a family of objects in ${\rm\sf
Cat}//X$,

$\prod_{i\in L}\left(W_{i},a_{i}\right)\cong\left(\prod_{i\in
L}W_{i},\underline{a}\right),$

where $\displaystyle\underline{a}\left(x_{i}\right)_{i\in L}=\prod_{i\in
L}a_{i}(x_{i})$.

Being a natural generalization of the setting of [10], the study of
exponentials fits the setting of [31]. However, we stick to an explicit
presentation of the exponentials. More precisely, to streamline the
computation of exponentials in ${\rm\sf Cat}//X$, we employ the concept of
ends, as outlined in [21, 3.10] and recalled below.

Assuming that $X$ is cartesian closed and $x,y$ are objects of $X$, we denote
by

$\mathrm{p}_{x}:x\times y\rightarrow x$

the projection, and by $x\Rightarrow y$ the exponential; that is to say,

$x\Rightarrow-:X\rightarrow X$

is the functor right adjoint to $x\times-:X\rightarrow X$.

It is worth noting that a complete category inherently possesses all (${\rm\sf
Set}$-enriched/ordinary) ends. Specifically, for any small category $W$ and
any functor

$T:W^{\text{op}}\times W\rightarrow X,$

the end $\displaystyle\int_{W}T=\int_{w\in W}T(w,w)$ is given by the equalizer
of the following diagram:

$\textstyle{\displaystyle\prod_{w\in\mathrm{obj}\left(W\right)}{T(w,w)}}$$\textstyle{\displaystyle\prod_{(w,y)\in\mathrm{obj}\left(W\times{W}\right)}{\prod_{h\in{W(w,y)}}{T(w,y)}}}$$\scriptstyle{t_{0}}$$\textstyle{\displaystyle\prod_{w\in\mathrm{obj}\left(W\right)}{T(w,w)}}$$\textstyle{\displaystyle\prod_{(w,y)\in\mathrm{obj}\left(W\times{W}\right)}{\prod_{h\in{W(w,y)}}{T(w,y)}}}$$\scriptstyle{t_{1}}$

where, for the component corresponding to $(w,y,h)\in\mathrm{obj}\left(W\times
W\right)\times W(w,y)$, we have that $t_{0}$ and $t_{1}$ are respectively
induced by $T(w,h)\cdot\mathrm{p}_{T(w,w)}$ and
$T(h,y)\cdot\mathrm{p}_{T(y,y)}$.

###### Theorem 2.3

Let $X$ be a cartesian closed category. We assume that $(W,a),(Y,b)$ are
objects in ${\rm\sf Cat}//X$ such that (2.1.1) exists in $X$ for any functor
$h:W\rightarrow Y$. In this setting, the exponential $(W,a)\Rightarrow(Y,b)$
in ${\rm\sf Cat}//X$ exists and is given by $\left({\rm\sf
Cat}\left[W,Y\right],b^{a}\right)$.

$b^{a}(h)\colon=\int_{w\in W}\left(a(w)\Rightarrow b\cdot h(w)\right)$ (2.1.1)

Consequently, if $X$ is a complete cartesian closed category, so is ${\rm\sf
Cat}//X$.

######  Proof 2.1.

Indeed, we have that:

$\displaystyle{\rm\sf Cat}//X\left((W,a)\times(Z,c),(Y,b)\right)$
$\displaystyle\cong$ $\displaystyle{\rm\sf Cat}//X\left((W\times
Z,a\underline{\times}c),(Y,b)\right)$ $\displaystyle\cong$
$\displaystyle\coprod_{f\in{\rm\sf Cat}\left(W\times Z,Y\right)}{\rm\sf
Cat}\left[W\times Z,X\right]\left(a\underline{\times}c,b\cdot f\right)$
$\displaystyle\cong$ $\displaystyle\coprod_{f\in{\rm\sf Cat}\left(W\times
Z,Y\right)}\int_{(w,z)\in W\times Z}X\left(a(w)\times c(z),b(f(w,z))\right)$
$\displaystyle\cong$ $\displaystyle\coprod_{f\in{\rm\sf Cat}\left(W\times
Z,Y\right)}\int_{z\in Z}\int_{w\in W}X\left(c(z),a(w)\Rightarrow
b(f(w,z))\right)$ $\displaystyle\cong$
$\displaystyle\coprod_{\dot{f}\in{\rm\sf Cat}\left(Z,{\rm\sf
Cat}\left[W,Y\right]\right)}\int_{z\in Z}X\left(c(z),\int_{w\in
W}\left(a(w)\Rightarrow b(\dot{f}(z)(w))\right)\right)$ $\displaystyle\cong$
$\displaystyle\coprod_{\dot{f}\in{\rm\sf Cat}\left(Z,{\rm\sf
Cat}\left[W,Y\right]\right)}\int_{z\in
Z}X\left(c(z),b^{a}\left(\dot{f}(z)\right)\right)$ $\displaystyle\cong$
$\displaystyle\coprod_{\dot{f}\in{\rm\sf Cat}\left(Z,{\rm\sf
Cat}\left[W,Y\right]\right)}{\rm\sf
Cat}\left[Z,X\right]\left(c(z),b^{a}\left(\dot{f}(z)\right)\right)$
$\displaystyle\cong$ $\displaystyle{\rm\sf
Cat}//X\left(\left(Z,c\right),\left({\rm\sf
Cat}\left[W,Y\right],b^{a}\right)\right)$

###  2.2 Pullbacks

We assume that (2.2.1) is a pullback in ${\rm\sf Cat}$.

$\textstyle{P}$$\textstyle{W}$$\scriptstyle{{j}}$$\textstyle{P}$$\textstyle{Z}$$\scriptstyle{{h}}$$\textstyle{W}$$\textstyle{Y}$$\scriptstyle{{f}}$$\textstyle{Z}$$\textstyle{Y}$$\scriptstyle{{g}}$
(2.2.1)

Since the change-of-base functors of the indexed category ${\rm\sf
Cat}\left[-,X\right]$ preserve limits (and, in particular, pullbacks), we have
that:

###### Theorem 2.4

If $X$ is a category with pullbacks, then ${\rm\sf Cat}//X$ has pullbacks.
Explicitly, (2.2.3) is a pullback in ${\rm\sf Cat}//X$ provided that (2.2.1)
is a pullback in ${\rm\sf Cat}$ and (2.2.2) is a pullback for any object $t$
in $P$.
$\textstyle{d(t)}$$\textstyle{a\circ{j}(t)}$$\scriptstyle{{(j,\varphi)}}$$\textstyle{d(t)}$$\textstyle{c\circ{j}(t)}$$\scriptstyle{{\left(h,\zeta\right)}}$$\textstyle{a\circ{j}(t)}$$\textstyle{b\circ\left(f\circ{j}\right)(t)=b\circ\left(g\circ{h}\right)(t)}$$\scriptstyle{{\left(f,\gamma\right)}}$$\textstyle{c\circ{j}(t)}$$\textstyle{b\circ\left(f\circ{j}\right)(t)=b\circ\left(g\circ{h}\right)(t)}$$\scriptstyle{{\left(g,\chi\right)}}$
(2.2.2)
$\textstyle{(P,d)}$$\textstyle{(W,a)}$$\scriptstyle{{(j,\varphi)}}$$\textstyle{(P,d)}$$\textstyle{(Z,c)}$$\scriptstyle{{\left(h,\zeta\right)}}$$\textstyle{(W,a)}$$\textstyle{(Y,b)}$$\scriptstyle{{\left(f,\gamma\right)}}$$\textstyle{(Z,c)}$$\textstyle{(Y,b)}$$\scriptstyle{{\left(g,\chi\right)}}$
(2.2.3)

###  2.3 Coproducts and Extensivity

Let $X$ be a category with an initial object $\mathsf{0}$. For any small
category $W$, the fibre ${\rm\sf Cat}\left[W,X\right]$ has an initial object
$\overline{\mathsf{0}}:W\rightarrow X$ given by the functor constantly equal
to $\mathsf{0}$. Moreover, clearly, the indexed category

${\rm\sf Cat}\left[-,X\right]:{\rm\sf Cat}^{\rm op}\rightarrow{\rm\sf CAT}$
(2.3.1)

is (infinitary) extensive in the sense of [30, Section 6.6]; namely, we mean
that ${\rm\sf Cat}\left[-,X\right]$ preserves products of ${\rm\sf Cat}^{\rm
op}$, and ${\rm\sf Cat}$ has coproducts. Therefore, by [30, Corollary 35], we
have:

######  Proposition 1.

If $X$ is a category with an initial object $\mathsf{0}$, then ${\rm\sf
Cat}//X$ has coproducts.

Explicitly, the category ${\rm\sf Cat}//X$ has an initial object defined by
$\left(\mathsf{0},\overline{\mathsf{0}}:\emptyset\rightarrow X\right)$.

Furthermore, if $(W_{i},a_{i})_{i\in L}$ is a family of objects in ${\rm\sf
Cat}//X$, then

$\coprod_{i\in L}(W_{i},a_{i})\cong\left(\coprod_{i\in
L}W_{i},\left[a_{i}\right]_{i\in L}\right)$

where $\displaystyle\left[a_{i}\right]_{i\in L}:\coprod_{i\in
L}W_{i}\rightarrow X$ is the induced functor.

Since ${\rm\sf Cat}$ is (infinitary) extensive, we can also conclude that:

###### Theorem 2.5

If $X$ is a category with an initial object $\mathsf{0}$, then ${\rm\sf
Cat}//X$ is (infinitary) extensive.

######  Proof 2.2.

The result follows from an infinitary version of the argument given in [30,
Theorem 41].

###  2.4 Coequalizers

To study the coequalizers of ${\rm\sf Cat}//X$, we start by observing that
(1.1.1) is a bifibration provided that $X$ is cocomplete. More conveniently
put in our context, this means that:

${\rm\sf Cat}\left[f,X\right]:{\rm\sf Cat}\left[Y,X\right]\rightarrow{\rm\sf
Cat}\left[W,X\right]$ (2.4.1)

has a left adjoint for any functor $f:W\rightarrow Y$. More precisely,
assuming that $X$ is cocomplete, ${\rm\sf Cat}\left[f,X\right]$ has a left
adjoint given by the pointwise left Kan extension

$\mathrm{lan}_{f}:{\rm\sf Cat}\left[W,X\right]\rightarrow{\rm\sf
Cat}\left[Y,X\right].$ (2.4.2)

We refer the reader to [12, 21, 23] for pointwise Kan extensions. With this
observation in mind, we get:

###### Theorem 2.6

If $X$ is cocomplete, so is ${\rm\sf Cat}//X$.

Explicitly, we have that (2.4.6) is a coequalizer in ${\rm\sf Cat}//X$
provided that:

  * •

(2.4.5) is the coequalizer in ${\rm\sf Cat}$;

  * •

(2.4.3) is the coequalizer of the parallel morphisms (2.4.4) in ${\rm\sf
Cat}\left[C,X\right]$, where each unlabeled arrow in (2.4.4) is induced by the
appropriate counit, and $\varphi^{t}$ is the mate of $\varphi$ w.r.t the
adjunction $\mathrm{lan}_{j}\dashv{\rm\sf Cat}\left[j,X\right]$.

$\textstyle{\mathrm{lan}_{jf}a=\mathrm{lan}_{jg}a}$$\textstyle{\mathrm{lan}_{j}b}$$\textstyle{\mathrm{lan}_{j}b}$$\textstyle{d}$$\scriptstyle{{{\varphi}^{t}}}$$\textstyle{\mathrm{lan}_{jf}a=\mathrm{lan}_{jg}a}$$\textstyle{\mathrm{lan}_{j}b}$
(2.4.3)

$\textstyle{\mathrm{lan}_{jg}a}$$\textstyle{\mathrm{lan}_{jg}(b\circ{g})}$$\scriptstyle{{\mathrm{lan}_{jg}\gamma}}$$\textstyle{\mathrm{lan}_{jg}(b\circ{g})}$$\textstyle{\mathrm{lan}_{j}b}$$\textstyle{\mathrm{lan}_{jf}a}$$\textstyle{\mathrm{lan}_{jf}(b\circ{f})}$$\scriptstyle{{\mathrm{lan}_{jf}\chi}}$$\textstyle{\mathrm{lan}_{jf}(b\circ{f})}$$\textstyle{\mathrm{lan}_{j}b}$
(2.4.4)

$\textstyle{W}$$\textstyle{Y}$$\scriptstyle{{g}}$$\textstyle{Y}$$\textstyle{C}$$\scriptstyle{{j}}$$\textstyle{W}$$\textstyle{Y}$$\scriptstyle{{f}}$
(2.4.5)

$\textstyle{(W,a)}$$\textstyle{(Y,b)}$$\scriptstyle{{(g,\chi)}}$$\textstyle{(Y,b)}$$\textstyle{(C,d)}$$\scriptstyle{{(j,\varphi)}}$$\textstyle{(W,a)}$$\textstyle{(Y,b)}$$\scriptstyle{{(f,\gamma)}}$
(2.4.6)

######  Proof 2.3.

The coproducts are given by Proposition 1.

As for the coequalizers, the direct verification is straightforward, and the
result follows from general results on the total categories of bifibrations
(see [14, 31]).

###  2.5 Topologicity

In order to examine the topologicity of the functor $U:{\rm\sf
Cat}//X\rightarrow{\rm\sf Cat}$, we fully rely on the characterization of [17,
Theorem 5.9.1]. In other words, $U:{\rm\sf Cat}//X\rightarrow{\rm\sf Cat}$ is
topological if, and only if, $U$ is a bifibration whose fibres are large-
complete.

###### Theorem 2.7

$U:{\rm\sf Cat}//X\rightarrow{\rm\sf Cat}$ is topological if, and only if, $X$
is large-complete.

######  Proof 2.4.

If $X$ is large-complete, then so is every fibre ${\rm\sf Cat}(A,X)$ of $U$.
Reciprocally, if $U$ is topological, then the fiber ${\rm\sf Cat}(1,X)\cong X$
is large-complete.

###### Corollary 2.8

Let $X$ be a small category. $U:{\rm\sf Cat}//X\rightarrow{\rm\sf Cat}$ is
topological if, and only if, $X$ is a complete lattice.

##  3 Effective descent morphisms

The study of effective descent morphisms, exhibiting a pivotal role in
Grothendieck descent theory, has far-reaching implications across various
subjects (see, for instance, [2, 3, 4]). Beyond their instrumental role,
effective descent morphisms stand out as a fascinating subject on their own
right, representing a distinctive subclass of stable regular epimorphisms. By
definition, they are morphisms that show how bundles/morphisms over their
codomain can be characterized as bundles over their domain endowed with an
algebraic structure. We refer the reader to [18, 19, 23, 26] for further
aspects of effective descent morphisms.

We recall that, explicitly, a morphism $q:u\rightarrow v$ in a category $X$
with pullbacks along $q$ is of effective descent if the change-of-base functor

$q^{\ast}:X/v\rightarrow X/u$ (3.0.1)

is monadic. Moreover, we are usually also interested in other cases, even if
$q^{\ast}$ is not monadic. More precisely, if the Eilenberg-Moore comparison
functor is fully faithful (respectively, faithful), $q$ is said to be of
descent (respectively, almost descent). In this section, we embark on the
study of effective descent morphisms in the lax comma category ${\rm\sf
Cat}//X$.

###  3.1 Fundamentals

Before delving into our study, we shortly recall the basic technique to study
(effective) descent morphisms in a category. We refer the reader to [23,
Section 1] for a further overview.

If $q$ is a morphism of a category $X$ with pullbacks, one can study if $q$ is
effective descent by investigating the monadicity of $q^{\ast}$ via Beck’s
monadicity theorem (see, for instance, [32, Chapter VI], [24, Corollary 1.2],
[27], and [26]). This proves to be quite fruitful in some special cases: for
instance, one can conclude that, in locally cartesian closed categories,
effective descent morphisms are the same as the stable regular epimorphisms.

Beyond Beck’s monadicity theorem, most general results we have are about
reflection of properties by functors. More precisely, pullback-preserving
fully faithful functors do not generally reflect all effective descent
morphisms, but provide us with the following classical result (see, for
instance, [19, 2.6] and [23, Theorem 1.3]).

###### Theorem 3.1

Let $V:X\rightarrow N$ be a fully faithful pullback-preserving functor between
categories with pullbacks. We assume that $V(q:e\rightarrow b)$ is an
effective descent morphism. The morphism $q$ is of effective descent in $X$ if
and only if it satisfies the following property: whenever (3.1.1) is a
pullback in $N$, there is an object $x$ in $X$ such that $V(x)\cong n$.

$\textstyle{V(y)}$$\textstyle{n}$$\textstyle{V(y)}$$\textstyle{V(e)}$$\textstyle{n}$$\textstyle{V(b)}$$\textstyle{V(e)}$$\textstyle{V(b)}$$\scriptstyle{{V(q)}}$
(3.1.1)

The result above is one of the reasons why the main technique to study
(effective) descent morphisms in a category $X$ is about fully embedding $X$
into a category $N$ whose descent behaviour we know about. We refer the reader
to [23, Section 1] for further observations about functors reflecting descent
properties of morphisms.

###  3.2 Preservation of effective descent morphisms

Although the main classical results in the study of effective descent
morphisms, such as Theorem 3.1, focus on understanding whether they are
reflected by fully faithful, pullback preserving functors, our main result,
Theorem 3.3, is about preservation of effective descent morphisms.

Throughout this subsection, we assume that $X$ has a strict initial object
$\mathsf{0}$. We start by considering the left adjoint

$L:{\rm\sf Cat}\rightarrow{\rm\sf Cat}//X$ (3.2.1)

to the forgetful functor $U:{\rm\sf Cat}//X\rightarrow{\rm\sf Cat}$, defined
in 1.2. That is to say:

######  Lemma 2.

$L$ is fully faithful and preserves limits of non-empty diagrams. In
particular, it preserves pullbacks and non-empty products.

$L$ preserves the terminal object if, and only if, $X$ is equivalent to the
terminal category.

######  Proof 3.1.

We have $UL(C)=C$ for any small category $C$, hence $L$ is fully faithful.

Since the initial object in $X$ is strict by hypothesis, we get that the limit
of any non-empty diagram involving the initial object is isomorphic to the
initial object. Therefore, it follows by the constructions of pullbacks and
non-empty products in ${\rm\sf Cat}//X$ (see Theorem 2.2.1 and Proposition
2.2) that $L$ preserves pullbacks and non-empty products.

Moreover, since $L\left(\mathsf{1}\right)=\left(\mathsf{1},\mathsf{0}\right)$,
we conclude by Proposition 2.1 that $L\left(\mathsf{1}\right)$ is the terminal
object if and only if $\mathsf{0}\cong\mathsf{1}$ in $X$.

Since $X$ has a strict initial object by hypothesis, then having zero object
is equivalent to having $X\simeq 1$.

###### Theorem 3.2

The functor $L$ reflects effective descent morphisms.

######  Proof 3.2.

Given any functor $p:E\rightarrow B$ in ${\rm\sf Cat}$, we have that, if
(3.2.2) is a pullback in ${\rm\sf Cat}//X$, then $d=\underline{\mathsf{0}}$
since $\mathsf{0}$ is strict and $\gamma:d\Rightarrow\underline{\mathsf{0}}$.

$\textstyle{L(Z)=(Z,\underline{\mathsf{0}})}$$\textstyle{(W,d)}$$\textstyle{L(Z)=(Z,\underline{\mathsf{0}})}$$\textstyle{L(E)=(E,\underline{\mathsf{0}})}$$\textstyle{(W,d)}$$\textstyle{L(B)=(B,\underline{\mathsf{0}})}$$\scriptstyle{{(f,{\gamma})}}$$\textstyle{L(E)=(E,\underline{\mathsf{0}})}$$\textstyle{L(B)=(B,\underline{\mathsf{0}})}$$\scriptstyle{{L(p)}}$
(3.2.2)

This proves that, by Theorem 3.1, $p$ is of effective descent provided that
$L(p)$ is of effective descent.

Our first result on the preservation of effective descent morphisms follows
from the well-known result that effective descent morphisms are stable under
pullback, that is, if $p$ is of effective descent, then so is $q^{\ast}(p)$ in
any category $X$.

######  Lemma 3.

$LU(f,\gamma)$ is of effective descent if $(f,\gamma)$ is of effective
descent.

######  Proof 3.3.

Consider $({\rm
id}_{Y},\iota_{b}):\left(Y,\underline{\mathsf{0}}\right)\rightarrow\left(Y,b\right)$
where $\iota_{b}$ is the unique $2$-cell
$\iota_{b}:\underline{\mathsf{0}}\Rightarrow b$.

We have that $LU(f,\gamma)=\left({\rm
id}_{Y},\iota_{b}\right)^{\ast}(f,\gamma)$ and, hence, whenever $(f,\gamma)$
is of effective descent so is the pullback $LU(f,\gamma)$.

As a consequence, we get:

###### Theorem 3.3

The functor $U:{\rm\sf Cat}//X\rightarrow{\rm\sf Cat}$ preserves effective
descent morphisms provided that $X$ has pullbacks and a strict initial object.

######  Proof 3.4.

If $\left(f,\gamma\right)$ is of effective descent in ${\rm\sf Cat}//X$, then
so is $LU(f,\gamma)$ by Lemma 3. Finally, we can conclude that $U(f,\gamma)=f$
is an effective descent morphism by Theorem 3.2.

##  4 Further comments

We have pointed out that it is natural to consider ${\rm\sf Cat}//X$ as a
$2$-category. We refer the reader to [11, 33] and future work for more
details. We recall the $2$-dimensional structure of ${\rm\sf Cat}//X$ below.

###### Definition 4.1

Given a small category $X$, we consider ${\rm\sf Cat}//X$ as a $2$-category
with the $2$-cells given by the following.

  * –

A $2$-cell between morphisms $(f,\gamma)$ and $(f^{\prime},\gamma^{\prime})$
is given by a $2$-cell $\zeta:f\Rightarrow f^{\prime}$ such that the equation

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holds.

The $2$-category ${\rm\sf Cat}//X$ is called the lax comma $2$-category of
${\rm\sf Cat}$ over $X$, while we call the underlying category the lax comma
category ${\rm\sf Cat}//X$.

###  4.1 Monadicity and lax monadicity

It is well known that the comma category ${\rm\sf Cat}/X$ is isomorphic to the
category of coalgebras and (strict) morphisms of the ($2$-)comonad $T_{X}$
whose underlying ($2$-)endofunctor is given by $W\mapsto W\times X$. In
particular, this means that ${\rm\sf Cat}/X$ is cocomplete and the forgetful
functor ${\rm\sf Cat}/X\rightarrow{\rm\sf Cat}$ is $2$-comonadic.

${\rm\sf Cat}//X$ fits into this picture as the $2$-category of strict
$T_{X}$-coalgebras and lax $T_{X}$-morphisms (see, for instance, [25] for
basic definitions). This observation shows that it would be interesting to
understand better the general interest in studying descent in categories of
(strict) algebras and lax morphisms between them.

###  4.2 Free fibrations

With the definition above, we have that ${\rm\sf Cat}//X$ can be naturally
embedded in ${\rm\sf CAT}\left[X^{\rm op},{\rm\sf Cat}\right]$, as ${\rm\sf
CAT}//X$ is equivalent to the $2$-category of free fibrations over $X$. This
embedding preserves pullbacks and, hence, it can be used to further get
results on effective descent morphisms in ${\rm\sf Cat}//X$. However, a
strategy to get a characterization would probably rely on a generalization of
the techniques presented in [8].

###  4.3 Preservation of effective descent morphisms

For basic definitions involved in the comment below, we refer the reader to
[23, Section 1]. As pointed out in Section 3, on one hand, in the study of
descent, one usually relies on results about the reflection of properties. For
instance, besides Theorem 3.1, we recall that:

  * •

$G$ reflects almost descent morphisms if it is a pullback-preserving faithful
functor;

  * •

$G$ reflects descent morphisms if it is a pullback-preserving fully faithful
functor.

On the other hand, there aren’t many tools in the literature about the
preservation of effective descent morphisms. In this sense, Theorem 3.3 gives
rise to the question of whether one can find a general setting or framework
for which the preservation of effective descent morphisms holds. We leave this
investigation to future work.

###  4.4 Effective descent morphisms w.r.t. other indexed categories

We left the problem of fully characterizing effective descent morphisms
(w.r.t. the basic fibration) in ${\rm\sf Cat}//X$ open.

We present, herein, another natural problem on effective descent morphisms
arising from our considerations; namely, for each $2$-category $\mathbb{A}$
and object $W$ of $\mathbb{A}$, we have the indexed category

$\mathbb{A}\left(-,W\right):\mathbb{A}_{0}^{\rm op}\rightarrow{\rm\sf CAT},$

where $\mathbb{A}_{0}$ is the underlying category, and
$\mathbb{A}\left(X,W\right)$ is the category of morphisms $X\rightarrow W$ in
$\mathbb{A}$.

In particular, considering ${\rm\sf Cat}//X$ as a $2$-dimensional category, we
have the indexed category

${\rm\sf Cat}//X\left(-,\left(W,a\right)\right):\left({\rm\sf
Cat}//X\right)^{\rm op}\rightarrow{\rm\sf CAT}$

for each $\left(W,a\right)$ in ${\rm\sf Cat}//X$. More relevantly, we can
extrapolate and consider the indexed category

$\mathfrak{F}={\rm\sf Cat}//X\left(-,\left(X,{\rm
id}_{X}\right)\right):\left({\rm\sf Cat}//X\right)^{\rm op}\rightarrow{\rm\sf
CAT}.$

Akin [39, 28], it is natural to consider the problem of characterizing
effective $\mathfrak{F}$-descent morphisms. Following the insights of [28], we
understand that such a study will rely on understanding lax epimorphisms in
${\rm\sf Cat}//X$ (see, for instance, [1, 29, 28]). We leave this study with
further $2$-dimensional considerations for future work.

## References

  * [1] J. Adámek, R. El Bashir, M. Sobral, and J. Velebil.  On functors which are lax epimorphisms.  Theory Appl. Categ., 8:509–521, 2001.
  * [2] F. Borceux and G. Janelidze.  Galois theories, volume 72 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2001.
  * [3] R. Brown and G. Janelidze.  Van Kampen theorems for categories of covering morphisms in lextensive categories.  J. Pure Appl. Algebra, 119(3):255–263, 1997.
  * [4] R. Brown and G. Janelidze.  Galois theory and a new homotopy double groupoid of a map of spaces.  Appl. Categ. Structures, 12(1):63–80, 2004.
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incollectioninproceedings

# Conformal Parametrisation of Loxodromes
by Triples of Circles

Vladimir V. Kisil and James Reid  School of Mathematics
University of Leeds
Leeds LS2 9JT
UK<EMAIL_ADDRESS>http://www.maths.leeds.ac.uk/~kisilv/ School of
Mathematics
The University of Manchester
Manchester M13 9PL
UK<EMAIL_ADDRESS>Dedicated to Prof. Wolfgang Sprößig on the
occasion of his 70${}^{\text{th}}$ birthday

###### Abstract.

We provide a parametrisation of a loxodrome by three specially arranged
cycles. The parametrisation is covariant under fractional linear
transformations of the complex plane and naturally encodes conformal
properties of loxodromes. Selected geometrical examples illustrate the usage
of parametrisation. Our work extends the set of objects in Lie sphere
geometry—circle, lines and points—to the natural maximal conformally-invariant
family, which also includes loxodromes.

###### Key words and phrases:

loxodrome, fractional linear transformations, logarithmic spiral, cycle, Lie
geometry, Möbius map, Fillmore–Springer–Cnops construction

###### 2010 Mathematics Subject Classification:

Primary 51B10; Secondary 51B25, 30C20, 30C35.

##  1\. Introduction

It is easy to come across shapes of logarithmic spirals, as on Fig. 1(a),
looking either on a sunflower, a snail shell or a remote galaxy. It is not
surprising since the fundamental differential equation $\dot{y}=\lambda y$,
$\lambda\in\mathbb{C}{}$ serves as a first approximation to many natural
processes.

(a) (b)

Figure 1. A logarithnic spiral (a) and its image under a fractional linear
transformation—loxodrome (b).

The main symmetries of complex analysis are build on the fractional linear
transformation (FLT):

(1) $\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}:\
z\mapsto\frac{\alpha z+\beta}{\gamma z+\delta},\quad\text{ where
}\alpha,\beta,\gamma,\delta\in\mathbb{C}{}\text{ and
}\det\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}\neq 0.$

Thus, images of logarithmic spirals under FLT, called loxodromes, as on Fig.
1(b) shall not be rare. Indeed, they appear in many occasions from the
stereographic projection of a rhumb line in navigation to a preferred model of
a Carleson arc in the theory singular integral operators
[BoettcherKarlovich01a, BishopBoettcherKarlovichSpitkovsky99a]. Furthermore,
loxodromes are orbits of one-parameter continuous groups of FLT of loxodromic
type [Beardon95]*§ 4.3 [Simon11a]*§ 9.2 [Vasilevski08a]*§ 9.2.

This setup motivates a search for effective tools to deal with FLT-invariant
properties of loxodromes. They were studied from a differential geometry point
of view in several recent paper [Bolt07a, Porter93a, Porter98a, Porter98b,
Porter07a]. In this work we develop a “global” description which matches the
Lie sphere geometry framework, see Rem. 3.

The outline of the paper is as follows. After preliminaries on FLT and
invariant geometry of cycles (Section 2) we review the basics of logarithmic
spirals and loxodromes (Section 3). A new parametrisation of loxodromes is
introduced in Section 4 and several examples illustrate its usage in Section
5. Section 6 frames our work within a wider approach [Kisil15a, Kisil14b],
which extends Lie sphere geometry. A brief list of open questions concludes
the paper.

##  2\. Preliminaries: Fractional Linear Transformations and Cycles

In this section we provide some necessary background in Lie geometry of
circles, fractional-linear transformations and Fillmore–Springer–Cnops
construction (FSCc). Regretfully, the latter remains largely unknown in the
context of complex numbers despite of its numerous advantages. We will have
some further discussion of this in Rem. 3 below.

The right way [Simon11a]*§ 9.2 to think about FLT (1) is through the
_projective complex line_ $P\mathbb{C}{}$. It is the family of cosets in
$\mathbb{C}^{2}{}$ with respect to the equivalence relation
$\begin{pmatrix}w_{1}\\\ w_{2}\end{pmatrix}\sim\begin{pmatrix}\alpha w_{1}\\\
\alpha w_{2}\end{pmatrix}$ for all nonzero $\alpha\in\mathbb{C}{}$.
Conveniently $\mathbb{C}{}$ is identified with a part of $P\mathbb{C}{}$ by
assigning the coset of $\begin{pmatrix}z\\\ 1\end{pmatrix}$ to
$z\in\mathbb{C}{}$. Loosely speaking
$P\mathbb{C}{}=\mathbb{C}{}\cup\\{\infty\\}$, where $\infty$ is the coset of
$\begin{pmatrix}1\\\ 0\end{pmatrix}$. The pair $[w_{1}:w_{2}]$ with $w_{2}\neq
0$ gives _homogeneous coordinates_ for $z=w_{1}/w_{2}\in\mathbb{C}{}$. Then,
the linear map $\mathbb{C}^{2}{}\rightarrow\mathbb{C}^{2}{}$

(2) $M:\ \begin{pmatrix}w_{1}\\\
w_{2}\end{pmatrix}\mapsto\begin{pmatrix}w_{1}^{\prime}\\\
w_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}\alpha w_{1}+\beta w_{2}\\\ \gamma
w_{1}+\delta w_{2}\end{pmatrix},\text{ where }M=\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}\in\mathrm{GL}_{2}(\mathbb{C}{})$

factors from $\mathbb{C}^{2}{}$ to $P\mathbb{C}{}$ and coincides with (1) on
$\mathbb{C}{}\subset P\mathbb{C}{}$.

Generic equations of cycle in real and complex coordinates $z=x+\mathrm{i}y$
are:

(3) $k(x^{2}+y^{2})-2lx-2ny+m=0\qquad\text{or}\qquad
kz\bar{z}-\bar{L}z-{L}\bar{z}+m=0\,,$

where $(k,l,n,m)\in\mathbb{R}^{4}{}$ and $L=l+\mathrm{i}n$. This includes
lines (if $k=0$), points as zero-radius circles (if $l^{2}+n^{2}-mk=0$) and
proper circles otherwise. Homogeneity of (3) suggests that $(k,l,m,n)$ shall
be considered as homogeneous coordinates $[k:l:m:n]$ of a point in three-
dimensional projective space $P\mathbb{R}^{3}{}$.

The homogeneous form of cycle’s equation (3) for $z=[w_{1}:w_{2}]$ can be
written111Of course, this is not the only possible presentation. However, this
form is particularly suitable to demonstrate FLT-invariance (8) of the cycle
product below. using matrices as follows:

(4)
$kw_{1}\bar{w}_{1}-\bar{L}w_{1}\bar{w}_{2}-{L}\bar{w}_{1}w_{2}+mw_{2}\bar{w}_{2}=\begin{pmatrix}-\bar{w}_{2}&\bar{w}_{1}\end{pmatrix}\begin{pmatrix}\bar{L}&-m\\\
k&-{L}\end{pmatrix}\begin{pmatrix}{w}_{1}\\\ w_{2}\end{pmatrix}=0.$

From now on we identify a cycle $C$ given by (3) with its $2\times 2$ matrix
$\begin{pmatrix}\bar{L}&-m\\\ k&-{L}\end{pmatrix}$, this is called the
_Fillmore–Springer–Cnops construction_ (FSCc) . Again, $C$ shall be treated up
to the equivalence relation $C\sim tC$ for all real $t\neq 0$. Then, the
linear action (2) corresponds to some action on $2\times 2$ cycle matrices by
the intertwining identity:

(5)
$\begin{pmatrix}-\bar{w}^{\prime}_{2}&\bar{w}^{\prime}_{1}\end{pmatrix}\begin{pmatrix}\bar{L}^{\prime}&-m^{\prime}\\\
k^{\prime}&-{L}^{\prime}\end{pmatrix}\begin{pmatrix}{w}^{\prime}_{1}\\\
w^{\prime}_{2}\end{pmatrix}=\begin{pmatrix}-\bar{w}_{2}&\bar{w}_{1}\end{pmatrix}\begin{pmatrix}\bar{L}&-m\\\
k&-{L}\end{pmatrix}\begin{pmatrix}{w}_{1}\\\ w_{2}\end{pmatrix}.$

Explicitly, for $M\in\mathrm{GL}_{2}(\mathbb{C}{})$ those actions are:

(6) $\begin{pmatrix}{w}^{\prime}_{1}\\\
w^{\prime}_{2}\end{pmatrix}=M\begin{pmatrix}{w}_{1}\\\
w_{2}\end{pmatrix},\quad\text{and}\quad\begin{pmatrix}\bar{L}^{\prime}&-m^{\prime}\\\
k^{\prime}&-{L}^{\prime}\end{pmatrix}=\bar{M}\begin{pmatrix}\bar{L}&-m\\\
k&-{L}\end{pmatrix}M^{-1}\,,$

where $\bar{M}$ is the component-wise complex conjugation of $M$. Note, that
FLT $M$ (1) corresponds to a linear transformation $C\mapsto
M(C):=\bar{M}CM^{-1}$ of cycle matrices in (6). A quick calculation shows that
$M(C)$ indeed has real off-diagonal elements as required for a FSCc matrix.

This paper essentially depends on the following

######  Proposition 1.

Define a _cycle product_ of two cycles $C$ and $C^{\prime}$ by:

(7) $\left\langle
C,C^{\prime}\right\rangle:=\mathop{\mathrm{tr}}(C\bar{C}^{\prime})=L\bar{L}^{\prime}+\bar{L}L^{\prime}-mk^{\prime}-km^{\prime}.$

Then, the cycle product is FLT-invariant:

(8) $\left\langle M(C),M(C^{\prime})\right\rangle=\left\langle
C,C^{\prime}\right\rangle\qquad\text{for any
}M\in\mathrm{SL}_{2}(\mathbb{C}{})\,.$

###### Proof.

Indeed:

$\displaystyle\left\langle M(C),M(C^{\prime})\right\rangle$
$\displaystyle=\mathop{\mathrm{tr}}(M(C)\overline{M(C^{\prime})})$
$\displaystyle=\mathop{\mathrm{tr}}(\bar{M}CM^{-1}M\bar{C}^{\prime}\bar{M}^{-1})$
$\displaystyle=\mathop{\mathrm{tr}}(\bar{M}C\bar{C}^{\prime}\bar{M}^{-1})$
$\displaystyle=\mathop{\mathrm{tr}}(C\bar{C}^{\prime})$
$\displaystyle=\left\langle C,C^{\prime}\right\rangle,$

using the invariance of trace. ∎

Note that the cycle product (7) is _not_ positive definite, it produces a
Lorentz-type metric in $\mathbb{R}^{4}{}$. Here are some relevant examples of
geometric properties expressed through the cycle product:

######  Example 2.

  1. (1)

If $k=1$ (and $C$ is a proper circle), then $\left\langle
C,C\right\rangle\\!/2$ is equal to the square of radius of $C$. In particular
$\left\langle C,C\right\rangle=0$ indicates a zero-radius circle representing
a point.

  2. (2)

If $\left\langle C_{1},C_{2}\right\rangle=0$ for non-zero radius cycles
$C_{1}$ and $C_{2}$, then they intersects at the right angle.

  3. (3)

If $\left\langle C_{1},C_{2}\right\rangle=0$ and $C_{2}$ is zero-radius
circle, then $C_{1}$ passes the point represented by $C_{2}$.

In general, a combination of (6) and (8) yields that a consideration of FLT in
$\mathbb{C}{}$ can be replaced by linear algebra in the space of cycles
$\mathbb{R}^{4}{}$ (or rather $P\mathbb{R}^{3}{}$) with an indefinite metric,
see [GohbergLancasterRodman05a] for the latter.

Figure 2. Orthogonal elliptic (green-dashed) and hyperbolic (red-solid)
pencils of cycles. Left drawing shows the standard case and the right—generic,
which is the image of the standard pencils under FLT.

A spectacular (and needed later) illustration of this approach is orthogonal
pencils of cycles. Consider a collection of all cycles passing two different
points in $\mathbb{C}{}$, it is called an _elliptic pencil_. A beautiful and
non-elementary fact of the Euclidean geometry is that cycles orthogonal to
every cycle in the elliptic pencil fill the entire plane and are disjoint, the
family is called a _hyperbolic pencil_. The statement is obvious in the
standard arrangement when the elliptic pencil is formed by straight
lines—cycles passing the origin and infinity. Then, the hyperbolic pencil
consists of the concentric circles, see Fig. 2. For the sake of completeness,
a _parabolic pencil_ (not used in this paper) formed by all circles touching a
given line at a given point, [Kisil12a]*Ex. 6.10 contains further extensions
and illustrations. See [Vasilevski08a]*§ 11.8 for an example of cycle pencils’
appearance in operator theory.

This picture trivialises a bit in the language of cycles. A pencil of cycles
(of any type!) is a linear span $tC_{1}+(1-t)C_{2}$ of two arbitrary different
cycles $C_{1}$ and $C_{2}$ from the pencil. Again, this is easier to check for
the standard pencils. A pencil is elliptic, parabolic or hyperbolic depending
on which inequality holds [Kisil12a]*Ex. 5.28.ii:

(9) $\left\langle C_{1},C_{2}\right\rangle^{2}\lesseqqgtr\left\langle
C_{1},C_{1}\right\rangle\left\langle C_{2},C_{2}\right\rangle.$

Then, the orthogonality of cycles on the plane is exactly their orthogonality
as vectors with respect to the indefinite cycle product (7). For cycles in the
standard pencils this is immediately seen from the explicit expression of the
product $\left\langle
C,C^{\prime}\right\rangle=L\bar{L}^{\prime}+\bar{L}L^{\prime}-mk^{\prime}-km^{\prime}$
in cycle components. Finally, linearization (6) of FLT in the cycle space
shows that a pencil (i.e. a linear span) is transformed to a pencil and FLT-
invariance (8) of the cycle product guarantees that the orthogonality of two
pencils is preserved.

######  Remark 3.

A sketchy historic overview (we apologise for any important omission!) starts
from the concept of Lie sphere geometry, see [Benz07a]*Ch. 3 for a detailed
presentation. It unifies circles, lines and points, which all are called
cycles in this context (analytically it is already in (3)). The main invariant
property of Lie sphere geometry is _tangential contact_. The first radical
advance came from the observation that cycles (through their parameters in
(3)) naturally form a linear or projective space, see [Pedoe95a]
[Schwerdtfeger79a]*Ch. 1. The second crucial step is the recognition that the
cycle space carries out the FLT-invariant indefinite metric [Benz07a]*Ch. 3
[Kirillov06]*§ F.4. At the same time some presentations of cycles by $2\times
2$ matrices were used [Simon11a]*§ 9.2 [Schwerdtfeger79a]*Ch. 1 [Kirillov06]*§
F.4. Their main feature is that FLT in $\mathbb{C}{}$ corresponds to a some
sort of linear transform by matrix conjugation in the cycle space. However,
the metric in the cycle space was not expressed in terms of those matrices.

All three ingredients—matrix presentation with linear structure and the
invariant product–came happily together as Fillmore–Springer–Cnops
construction (FSCc) in the context of Clifford algebras [Cnops02a]*Ch. 4
[FillmoreSpringer90a]. Regretfully, FSCc have not yet propagated back to the
most fundamental case of complex numbers, cf. [Simon11a]*§ 9.2 or somewhat
cumbersome techniques used in [Benz07a]*Ch. 3. Interestingly, even the
founding fathers were not always strict followers of their own techniques, see
[FillmoreSpringer00a].

A combination of all three components of Lie cycle geometry within FSCc
facilitates further development. It was discovered that for the smaller group
$\mathrm{SL}_{2}(\mathbb{R}{})$ there exist more types—elliptic, parabolic and
hyperbolic–of invariant metrics in the cycle space [Kisil06a] [Kisil05a]
[Kisil12a]*Ch. 5. Based on the earlier work [Kirillov06], the key concept of
Lie sphere geometry— _tangency of two cycles_ $C_{1}$ and $C_{2}$— was
expressed through the cycle product (7) as [Kisil12a]*Ex. 5.26.ii:

$\left\langle C_{1}+C_{2},C_{1}+C_{2}\right\rangle=0\,\quad\text{for
}C_{1},C_{2}\text{ normalised such that }\left\langle
C_{1},C_{1}\right\rangle=\left\langle C_{2},C_{2}\right\rangle=1.$

(Furthermore, $C_{1}+C_{2}$ is the zero-radius cycle representing the point of
contact.) FSCc is useful in consideration of the Poincaré extension of Möbius
maps [Kisil15a] and continued fractions [Kisil14a]. In theoretical physics
FSCc nicely describes conformal compactifications of various space-time models
[HerranzSantander02b] [Kisil06b] [Kisil12a]*§ 8.1. Last but not least, FSCc is
behind the Computer Algebra System (CAS) operating in Lie sphere geometry
[Kisil05b] [Kisil14b]. FSCc equally well covers not only the field of complex
numbers but rings of dual and double numbers as well [Kisil12a]. New usage of
FSCc will be given in the following sections in applications to loxodromes.

##  3\. Fractional Linear Transformations and Loxodromes

In aiming for a covariant description of loxodromes we start from the
following definition.

######  Definition 4.

A _standard logarithmic spiral_ (SLS) with parameter $\lambda\in\mathbb{C}{}$
is the orbit of the point $1$ under the (disconnected) one-parameter subgroup
of FLT of diagonal matrices

(10) $D_{\lambda}(t)=\begin{pmatrix}\pm\mathrm{e}^{\lambda t/2}&0\\\
0&\mathrm{e}^{-\lambda t/2}\end{pmatrix},\qquad t\in\mathbb{R}{}.$

######  Remark 5.

Our SLS is a _union_ of two branches, each of them is a logarithmic spiral in
the common sense. The three-cycle parametrisation of loxodromes presented
below will becomes less elegant if those two branches need to be separated.
Yet, we draw just one “positive” branch on Fig. 3 to make it more transparent.

SLS is the solution of the differential equation $z^{\prime}=\lambda z$ with
the initial value $z(0)=\pm 1$ and has the parametric equation
$z(t)=\pm\mathrm{e}^{\lambda t}$. Furthermore, we obtain the same orbit for
$\lambda_{1}$ and $\lambda_{2}\in\mathbb{C}{}$ if $\lambda_{1}=a\lambda_{2}$
for real $a\neq 0$ through a re-parametrisation of the time $t\mapsto at$.
Thus, SLS is identified by the point $[\Re(\lambda):\Im(\lambda)]$ of the real
projective line $P\mathbb{R}{}$. Thereafter the following classification is
useful:

######  Definition 6.

SLS is

  * •

_positive_ , if $\Re(\lambda)\cdot\Im(\lambda)>0$;

  * •

_degenerate_ , if $\Re(\lambda)\cdot\Im(\lambda)=0$;

  * •

_negative_ , if $\Re(\lambda)\cdot\Im(\lambda)<0$.

Informally: a positive SLS unwinds counterclockwise, a negative—clockwise.
Degenerate SLS is the unit circle if $\Im(\lambda)\neq 0$ and the punctured
real axis $\mathbb{R}{}\setminus\\{0\\}$ if $\Re(\lambda)\neq 0$. If
$\Re(\lambda)=\Im(\lambda)=0$ then SLS is the single point $1$.

######  Definition 7.

A _logarithmic spiral_ is the image of a SLS under a complex affine
transformation $z\mapsto\alpha z+\beta$, with $\alpha$,
$\beta\in\mathbb{C}{}$. A _loxodrome_ is an image of a SLS under a generic FLT
(1).

Obviously, a complex affine transformation is FLT with the upper triangular
matrix $\begin{pmatrix}\alpha&\beta\\\ 0&1\end{pmatrix}$. Thus, logarithmic
spirals form an affine-invariant (but not FLT-invariant) subset of loxodromes.
Thereafter, loxodromes (and their degenerate forms—circles, straight lines and
points) extend the notion of cycles from the Lie sphere geometry, cf. Rem. 3.

By the nature of Defn. 7, the parameter $\lambda$ and the corresponding
classification from Defn. 6 remain meaningful for logarithmic spirals and
loxodromes. FLTs eliminate distinctions between circles and straight lines,
but for degenerate loxodromes ($\Re(\lambda)\cdot\Im(\lambda)=0$) we still can
note the difference between two cases of $\Re(\lambda)\neq 0$ and
$\Im(\lambda)\neq 0$: orbits of former are whole circles (straight lines)
while latter orbits are only arcs of circles (segments of lines).

The immediate consequence of Defn. 7 is

######  Proposition 8.

The collection of all loxodromes is a FLT-invariant family. Degenerate
loxodromes—(arcs of) circles and (segments) of straight lines—form a FLT-
invariant subset of loxodromes.

As mentioned above, SLS is completely characterised by the point
$[\Re(\lambda):\Im(\lambda)]$ of the real projective line $P\mathbb{R}{}$
extended by the additional point $[0:0]$222Pedantic consideration of the
trivial case $\Re(\lambda)=\Im(\lambda)=0$ will be often omitted in the
following discussion.. In the standard way, $[\Re(\lambda):\Im(\lambda)]$ is
associated with the real value
$\tilde{\lambda}:=2\pi\Re(\lambda)/\Im(\lambda)$ extended by $\infty$ for
$\Im(\lambda)=0$ and symbol $\frac{0}{0}$ for the
$\Re(\lambda)=\Im(\lambda)=0$ cases. Geometrically,
$a=\exp(\tilde{\lambda})\in\mathbb{R}_{+}{}$ represents the next point after
$1$, where the given SLS branch meets the real positive half-axis after one
full counterclockwise turn. Obviously, $a>1$ and $a<1$ for positive and
negative SLS, respectively. For a degenerate SLS:

  1. (1)

with $\Im(\lambda)\neq 0$ we obtain $\tilde{\lambda}=0$ and $a=1$;

  2. (2)

with $\Re(\lambda)\neq 0$ we consistently _define_ $a=\infty$.

In essence, a loxodrome $\Lambda$ is defined by the pair
$(\tilde{\lambda},M)$, where $M$ is the FLT mapping $\Lambda$ to SLS with the
parameter $\tilde{\lambda}$. While $\tilde{\lambda}$ is completely determined
by $\Lambda$, a map $M$ is not.

######  Proposition 9.

  1. (1)

The subgroup of FLT which maps SLS with the parameter $\tilde{\lambda}$ to
itself consists of products $D_{\tilde{\lambda}}(t)R^{\varepsilon}$,
$\varepsilon=0,1$ of transformations $D_{\tilde{\lambda}}(t)=D_{\lambda}(t)$,
$\lambda=\tilde{\lambda}+2\pi\mathrm{i}$ (10) and branch-swapping reflections:

(11) $R=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}:z\mapsto-z^{-1}\,.$

  2. (2)

Pairs $(\tilde{\lambda},M)$ and $(\tilde{\lambda}^{\prime},M^{\prime})$ define
the same loxodrome if and only if

    1. (a)

$\tilde{\lambda}=\tilde{\lambda}^{\prime}$;

    2. (b)

$M=D_{\tilde{\lambda}}(t)R^{\varepsilon}M^{\prime}$ for $\varepsilon=0,1$ and
$t\in\mathbb{R}{}$.

######  Remark 10.

Often loxodromes appear as orbits of one-parameter continuous subgroup of
loxodromic FLT, which are characterised by a non-real trace [Beardon95]*§ 4.3
[Simon11a]*§ 9.2 [Vasilevski08a]*§ 9.2. In the above notations such a subgroup
is $MD_{\tilde{\lambda}}(t)M^{-1}$, thus the common presentation is not much
different from the above $(\tilde{\lambda},M)$-parametrisation.

##  4\. Three-cycle Parametrisation of Loxodromes

Although pairs $(\tilde{\lambda},M)$ provide a parametrisation of loxodromes,
the following alternative is more operational. It is inspired by the
orthogonal pairs of elliptic and hyperbolic pencils described in Section 2.

######  Definition 11.

A _three-cycle parametrisation_ $\\{C_{1},C_{2},C_{3}\\}$ of a non-degenerate
SLS $\tilde{\lambda}$ satisfies the following conditions:

  1. (1)

$C_{1}$ is the straight line passing the origin;

  2. (2)

$C_{2}$ and $C_{3}$ are two circles with their centres at the origin;

  3. (3)

$\Lambda$ passes the intersection points $C_{1}\cap C_{2}$ and $C_{1}\cap
C_{3}$; and

  4. (4)

A branch of $\Lambda$ makes one full counterclockwise turn between
intersection points $C_{1}\cap C_{2}$ and $C_{1}\cap C_{3}$ belonging to a ray
in $C_{1}$ from the origin.

We say that three-cycle parametrisation is _standard_ if $C_{1}$ is the real
axis and $C_{2}$ is the unit circle, then
$C_{3}=\\{z:\left|z\right|=\exp(\tilde{\lambda})\\}$. A three-cycle
parametrisation can be consistently extended to a degenerate SLS $\Lambda$ as
follows:

  * $\tilde{\lambda}=0$:

any straight line $C_{1}$ passing the origin and the unit circles
$C_{2}=C_{3}=\Lambda$;

  * $\tilde{\lambda}=\infty$:

the real axis as $C_{1}=\Lambda$, the unit circle as $C_{2}$ and
$C_{3}=(0,0,0,1)$ being the zero-radius circle at infinity.

Since cycles are elements of the projective space, the following _normalised
cycle product_ :

(12) $\left[C_{1},C_{2}\right]:=\frac{\left\langle
C_{1},C_{2}\right\rangle}{\sqrt{\left\langle
C_{1},C_{1}\right\rangle\left\langle C_{2},C_{2}\right\rangle}}$

is more meaningful than the cycle product (7) itself. Note that,
$\left[C_{1},C_{2}\right]$ is defined only if neither $C_{1}$ nor $C_{2}$ is a
zero-radius cycle (i.e. a point). Also, the normalised cycle product is
$\mathrm{GL}_{2}(\mathbb{C}{})$-invariant in comparison to
$\mathrm{SL}_{2}(\mathbb{C}{})$-invariance in (8).

A reader will instantly recognise the familiar pattern of the cosine of angle
between two vectors appeared in (12). Simple calculations show that this
geometric interpretation is very explicit in two special cases of our
interest.

######  Lemma 12.

  1. (1)

Let $C_{1}$ and $C_{2}$ be two straight lines passing the origin with slopes
$\tan\phi_{1}$ and $\tan\phi_{2}$ respectively. Then
$C_{2}=D_{x+\mathrm{i}y}(1)C_{1}$ for transformation (10) with any
$x\in\mathbb{R}{}$ and $y=\phi_{2}-\phi_{1}$ satisfying the relations:

(13) $\left[C_{1},C_{2}\right]=\cos y\,.$

  2. (2)

Let $C_{1}$ and $C_{2}$ be two circles centred at the origin and radii $r_{1}$
and $r_{2}$ respectively. Then $C_{2}=D_{x+\mathrm{i}y}(1)C_{1}$ for
transformation (10) with any $y\in\mathbb{R}{}$ and
$x=\log(r_{2})-\log(r_{1})$ satisfying the relations:

(14) $\left[C_{1},C_{2}\right]=\cosh x\,.$

Note the explicit elliptic-hyperbolic analogy between (13) and (14). By the
way, both expressions produce real $x$ and $y$ due to inequality (9) for the
respective types of pencils. Now we can deduce the following properties of
three-cycle parametrisation.

######  Proposition 13.

For a given SLS $\Lambda$ with a parameter $\lambda$:

  1. (1)

Any transformation (10) maps a three-cycle parametrisation of $\Lambda$ to
another three-cycle parametrisation of $\Lambda$.

  2. (2)

For any two three-cycle parametrisations $\\{C_{1},C_{2},C_{3}\\}$ and
$\\{C_{1}^{\prime},C_{2}^{\prime},C_{3}^{\prime}\\}$ there exists
$t_{0}\in\mathbb{R}{}$ such that $C_{j}^{\prime}=D_{\lambda}(t_{0})C_{j}$ for
$D_{\lambda}(t_{0})$ (10) and $j=1,2,3$.

  3. (3)

The parameter $\tilde{\lambda}=2\pi\Re(\lambda)/\Im(\lambda)$ of SLS can be
recovered form its three-cycle parametrisation by the relation:

(15)
$\tilde{\lambda}=\operatorname{arccosh}\left[C_{2},C_{3}\right]\quad\text{and}\quad\lambda\sim\tilde{\lambda}+2\pi\mathrm{i}\,.$

###### Proof.

The first statement is obvious. For the second we take
$D_{\lambda}(t_{0}):\Lambda\rightarrow\Lambda$ which maps $C_{1}\cap C_{2}$ to
$C_{1}^{\prime}\cap C_{2}^{\prime}$, this transformation maps $C_{j}\mapsto
C_{j}^{\prime}$ for $j=1,2,3$. Finally, the last statement follows from (14).
∎

Note that expression (15) is FLT-invariant. Since any loxodrome is an image of
SLS under FLT we obtain a three-cycle parametrisation of loxodromes as
follows.

######  Proposition 14.

  1. (1)

Any three-cycle parametrisation $\\{C_{1},C_{2},C_{3}\\}$ of SLS has the
following FLT-invariant properties:

    1. (a)

$C_{1}$ is orthogonal to $C_{2}$ and $C_{3}$;

    2. (b)

$C_{2}$ and $C_{3}$ either disjoint or coincide333Recall that if
$C_{2}=C_{3}$, then SLS is degenerate and coincide with $C_{2}=C_{3}$..

  2. (2)

For any FLT $M$ and a three-cycle parametrisation
$\\{C_{1}^{\prime},C_{2}^{\prime},C_{3}^{\prime}\\}$ of SLS, three cycles
$C_{j}=M(C_{j}^{\prime})$, $j=1,2,3$ satisfy the above conditions (1a) and
(1b).

  3. (3)

For any triple of cycles $\\{C_{1},C_{2},C_{3}\\}$ satisfying the above
conditions (1a) and (1b) there exist a FLT $M$ such cycles
$\\{M(C_{1}),M(C_{2}),M(C_{3})\\}$ provide a three-cycle parametrisation of
SLS with the parameter $\tilde{\lambda}$ (15). FLT $M$ is uniquely defined by
the additional condition that $\\{M(C_{1}),M(C_{2}),M(C_{3})\\}$ is a
_standard_ parametrisation of SLS.

###### Proof.

The first statement is obvious, the second follows because properties (1a) and
(1b) are FLT-invariant.

For (3) in the degenerate case $C_{2}=C_{3}$: any $M$ which sends
$C_{2}=C_{3}$ to the unit circle will do the job. If $C_{2}\neq C_{3}$ we
explicitly describe below the procedure, which produces FLT $M$ mapping the
loxodrome to SLS. ∎

######  Procedure 15.

Two disjoint cycles $C_{2}$ and $C_{3}$ span a hyperbolic pencil $H$ as
described in Section 2. Then $C_{1}$ belongs to the elliptic $E$ pencil
orthogonal to $H$. Let $C_{0}$ and $C_{\infty}$ be the two zero-radius cycles
(points) from the hyperbolic pencil $H$. Every cycle in $E$, including
$C_{1}$, passes $C_{0}$ and $C_{\infty}$, we label those two in such a way
that

  * •

for a positive $\tilde{\lambda}$ cycle $C_{3}$ is between $C_{2}$ and
$C_{\infty}$; and

  * •

for a negative $\tilde{\lambda}$ cycle $C_{3}$ is between $C_{2}$ and $C_{0}$.

Here “between” for cycles means “between” for their intersection points with
$C_{1}$. Finally, let $C_{u}$ be any of two intersection points $C_{1}\cap
C_{2}$. Then, there exists the unique FLT $M$ such that $M:C_{0}\mapsto 0$,
$M:C_{u}\mapsto 1$ and $M:C_{\infty}\mapsto\infty$. We will call $M$ the
_standard FLT associated_ to the three-cycle parametrisation
$\\{C_{1},C_{2},C_{3}\\}$ of the loxodrome.

######  Remark 16.

To complement the construction of the standard FLT $M$ associated to the
three-cycle parametrisation $\\{C_{1},C_{2},C_{3}\\}$ from Procedure 15, we
can describe the inverse operation. For the loxodrome, which is the image of
SLS with the parameter $\lambda$ under FLT $M$, we define the _standard three-
cycle parametrisation_ $\\{M(\mathbb{R}{}),M(C_{u}),M(C_{\lambda})\\}$ as the
image of the standard parametrisation of the SLS under $M$. Here
$\mathbb{R}{}$ is the real axis, $C_{u}=\\{z:\left|z\right|=1\\}$ is the unit
circle and $C_{\lambda}=\\{z:\left|z\right|=\exp(\tilde{\lambda})\\}$.

Figure 3. Logarithmic spirals (left) and loxodrome (right) with associated
pencils of cycles. This is a combination of Figs. 1 and 2.

In essence, the previous proposition says that a three-cycle and $(\lambda,M)$
parametrisations are equivalent and delivers an explicit procedure producing
one from another. However, three-cycle parametrisation is more geometric,
since it links a loxodrome to a pair of orthogonal pencils, see Fig. 3.
Furthermore, cycles $C_{1}$, $C_{2}$, $C_{3}$ (unlike parameters $\lambda$ and
$M$) can be directly drawn on the plane to represent a loxodrome, which may be
even omitted.

##  5\. Applications of Three-Cycle Parametrisation

Now we present some examples of the usage of three-cycle parametrisation of
loxodromes. First, we want to resolve non-uniqueness in such parametrisations.
Recall, that a triple $\\{C_{1},C_{2},C_{3}\\}$ is non-degenerate if
$C_{2}\neq C_{3}$ and $C_{3}$ is not zero-radius.

######  Proposition 17.

Two non-degenerate triples $\\{C_{1},C_{2},C_{3}\\}$ and
$\\{C_{1}^{\prime},C_{2}^{\prime},C_{3}^{\prime}\\}$ parameterise the same
loxodrome if and only if all the following conditions are satisfied:

  1. (1)

Pairs $\\{C_{2},C_{3}\\}$ and $\\{C_{2}^{\prime},C_{3}^{\prime}\\}$ span the
same hyperbolic pencil. That is cycles $C_{2}^{\prime}$ and $C_{3}^{\prime}$
are linear combinations of $C_{2}$ and $C_{3}$ and vise versa.

  2. (2)

Pairs $\\{C_{2},C_{3}\\}$ and $\\{C_{2}^{\prime},C_{3}^{\prime}\\}$ define the
same parameter $\tilde{\lambda}$:

(16) $\left[C_{2},C_{3}\right]=\left[C_{2}^{\prime},C_{3}^{\prime}\right].$

  3. (3)

The elliptic-hyperbolic identity holds:

(17)
$\frac{\operatorname{arccosh}\left[C_{j},C_{j}^{\prime}\right]}{\operatorname{arccosh}\left[C_{2},C_{3}\right]}\equiv\frac{1}{2\pi}\arccos\left[C_{1},C_{1}^{\prime}\right]\pmod{1}\,,$

where $j$ is either $2$ or $3$.

###### Proof.

Necessity of (1) is obvious, since hyperbolic pencils spanned by
$\\{C_{2},C_{3}\\}$ and $\\{C_{2}^{\prime},C_{3}^{\prime}\\}$ shall be both
the image of concentric circles centred at origin under FLT $M$ defining the
loxodrome. Necessity of (2) is also obvious since $\tilde{\lambda}$ is
uniquely defined by the loxodrome. Necessity of (3) follows from the analysis
of the following demonstration of sufficiency.

For sufficiency, let $M$ be FLT constructed through Procedure 15 from
$\\{C_{1},C_{2},C_{3}\\}$. Then (1) implies that $M(C_{2}^{\prime})$ and
$M(C_{3}^{\prime})$ are also circles centred at origin. Then Lem. 12 implies
that the transformation $D_{x+\mathrm{i}y}(1)$, where
$x=\operatorname{arccosh}\left[C_{2},C_{2}^{\prime}\right]$ and
$y=\arccos\left[C_{1},C_{1}^{\prime}\right]$ maps $C_{1}^{\prime}$ and
$C_{2}^{\prime}$ to $C_{1}$ and $C_{2}$ respectively. Furthermore, from
identity (16) follows that the same $D_{x+\mathrm{i}y}(1)$ maps
$C_{3}^{\prime}$ to $C_{3}$. Finally, condition (17) means that
$x+\mathrm{i}(y+2\pi n)=a(\tilde{\lambda}+2\pi\mathrm{i})$ for
$a=x/\tilde{\lambda}$ and some $n\in\mathbb{Z}{}$. In other words
$D_{x+\mathrm{i}y}(1)=D_{\tilde{\lambda}}(a)$, thus $D_{x+\mathrm{i}y}(1)$
maps SLS with the parameter $\tilde{\lambda}$ to itself. Since
$\\{M(C_{1}),M(C_{2}),M(C_{3})\\}$ and
$\\{M(C_{1}^{\prime}),M(C_{2}^{\prime}),M(C_{3}^{\prime})\\}$ are two three-
cycle parametrisations of the same SLS, $\\{C_{1},C_{2},C_{3}\\}$ and
$\\{C_{1}^{\prime},C_{2}^{\prime},C_{3}^{\prime}\\}$ are two three-cycle
parametrisations of the same loxodrome. ∎

See Fig. 4 for an animated family of equivalent three-cycle parametrisations
of the same loxodrome (also posted at [Kisil16a]). Relation (17), which
correlates elliptic and hyperbolic rotations for loxodrome, regularly appears
in this context. The next topic provides another illustration of this.

######  Procedure 18.

To verify whether a loxodrome parametrised by three cycles
$\\{C_{1},C_{2},C_{3}\\}$ passes a point parametrised by a zero-radius cycle
$C_{0}$ we perform the following steps:

  1. (1)

Define the cycle

(18) $C_{h}=tC_{2}+(1-t)C_{3}\,,\qquad\text{where }t=-\frac{\left\langle
C_{0},C_{3}\right\rangle}{\left\langle C_{0},C_{2}-C_{3}\right\rangle}\,,$

which belongs to the hyperbolic pencil spanned by $\\{C_{2},C_{3}\\}$ and is
orthogonal to $C_{0}$, that is, passes the respective point.

  2. (2)

Find cycle $C_{e}$ from the elliptic pencil orthogonal to $\\{C_{2},C_{3}\\}$
which passes $C_{0}$. $C_{e}$ is the solution of the system of three linear
(with respect to parameters of $C_{e}$) equations, cf. Ex 2:

$\displaystyle\left\langle C_{e},C_{0}\right\rangle$ $\displaystyle=0\,,$
$\displaystyle\left\langle C_{e},C_{2}\right\rangle$ $\displaystyle=0\,,$
$\displaystyle\left\langle C_{e},C_{3}\right\rangle$ $\displaystyle=0\,.$

  3. (3)

Verify the elliptic-hyperbolic relation:

(19)
$\frac{\operatorname{arccosh}\left[C_{h},C_{2}\right]}{\operatorname{arccosh}\left[C_{2},C_{3}\right]}\equiv\frac{1}{2\pi}\arccos\left[C_{e},C_{1}\right]\pmod{1}\,.$

###### Proof.

Let $M$ be the standard FLT associated to $\\{C_{1},C_{2},C_{3}\\}$ from
Procedure 15. The point $C_{0}$ belongs to the loxodrome if the transformation
$D_{\tilde{\lambda}}(t)$ for some $t$ moves $M(C_{0})$ to the intersection
$M(C_{1})\cap M(C_{2})$. But $D_{x+\mathrm{i}y}(1)$ with
$x=\operatorname{arccosh}\left[C_{h},C_{2}\right]$ and
$y=\arccos\left[C_{e},C_{1}\right]$ maps $M(C_{h})\rightarrow M(C_{2})$ and
$M(C_{e})\rightarrow M(C_{1})$, thus it also maps $M(C_{0})\subset
M(C_{h})\cap M(C_{e})$ to $M(C_{1})\cap M(C_{2})$. Condition (19) guaranties
that $D_{x+\mathrm{i}y}(1)=D_{\tilde{\lambda}}(x/\tilde{\lambda})$, as in the
previous Prop. ∎

Our final example considers two loxodromes which may have completely different
associated pencils.

######  Procedure 19.

Let two loxodromes are parametrised by $\\{C_{1},C_{2},C_{3}\\}$ and
$\\{C_{1}^{\prime},C_{2}^{\prime},C_{3}^{\prime}\\}$. Assume they intersect at
some point parametrised by a zero-radius cycle $C_{0}$ (this can be checked by
Procedure 18, if needed). To find the angle of intersection we perform the
following steps:

  1. (1)

Use (18) to find cycles $C_{h}$ and $C_{h}^{\prime}$ belonging to hyperbolic
pencils, spanned by $\\{C_{2},C_{3}\\}$ and
$\\{C_{2}^{\prime},C_{3}^{\prime}\\}$ respectively, and both passing $C_{0}$.

  2. (2)

The intersection angle is

(20)
$\arccos\left[C_{h},C_{h}^{\prime}\right]-\arctan\left(\frac{\tilde{\lambda}}{2\pi}\right)+\arctan\left(\frac{\tilde{\lambda}^{\prime}}{2\pi}\right)\,,$

where $\tilde{\lambda}$ and $\tilde{\lambda}^{\prime}$ are determined by (15).

###### Proof.

A loxodrome intersects any cycle from its hyperbolic pencil with the fixed
angle $\arctan(\tilde{\lambda}/(2\pi))$. This is used to amend the
intersection angle $\arccos\left[C_{h},C_{h}^{\prime}\right]$ of cycles from
the respective hyperbolic pencils. ∎

######  Corollary 20.

Let a loxodrome parametrised by $\\{C_{1},C_{2},C_{3}\\}$ passes a point
parametrised by a zero-radius cycle $C_{0}$ as in Procedure 18. A non-zero
radius cycle $C$ is tangent to the loxodrome at $C_{0}$ if and only if two
conditions holds:

(21) $\begin{split}\left\langle C,C_{0}\right\rangle&=0\,,\\\
\arccos\left[C,C_{h}\right]&=\arctan\left(\frac{\tilde{\lambda}}{2\pi}\right)\,,\end{split}$

where $C_{h}$ is given by (18) and is $\tilde{\lambda}$ is determined by (15).

###### Proof.

The first condition simply verifies that $C$ passes $C_{0}$, cf. Ex 2. Cycle
$C$, as a degenerated loxodrome, is parametrised by $\\{C_{e},C,C\\}$, where
$C_{e}$ is any cycle orthogonal to $C$ and $C_{e}$ is not relevant in the
following. The hyperbolic pencil spanned by two copies of $C$ consists of $C$
only. Thus we put $C_{h}^{\prime}=C$, $\tilde{\lambda}^{\prime}=0$ in (20) and
equate it to $0$ to obtain the second identity in (21). ∎

##  6\. Discussion and Open Questions

It was mentioned at the end of Section 4 that a three-cycle parametrisation of
loxodromes is more geometrical than their presentation by a pair
$(\lambda,M)$. Furthermore, three-cycle parametrisation reveals the natural
analogy between elliptic and hyperbolic features of loxodromes, see (17) as an
illustration. Examples in Section 5 show that various geometrical questions
are explicitly answered in term of three-cycle parametrisation. Thus, our work
extends the set of objects in Lie sphere geometry—circle, lines and points—to
the natural maximal conformally-invariant family, which also includes
loxodromes. In practical terms, three-cycle parametrisation allows to extend
the library figure for Möbuis invariant geometry [Kisil14b] to operate with
loxodromes as well.

It is even more important, that the presented technique is another
implementation of a general framework [Kisil15a, Kisil14b], which provides a
significant advance in Lie sphere geometry. The Poincaré extension of FLT from
the real line to the upper half-plane was performed by a pair of orthogonal
cycles in [Kisil15a]. A similar extension of FLT from the complex plane to the
upper half-space can be done by a _triple of pairwise orthogonal cycles_.
Thus, triples satisfying FLT-invariant properties (1a) and (1b) of Prop. 14
present another non-equivalent class of cycle ensembles in the sense of
[Kisil15a]. In general, Lie sphere geometry _can be enriched by consideration
of cycle ensembles_ interrelated by a list of FLT-invariant properties
[Kisil15a]. Such ensembles become new objects in the extended Lie spheres
geometry and can be represented by points in a _cycle ensemble space_.

There are several natural directions to extend this work further, here are
just few of them:

  1. (1)

Link our “global” parametrisation of loxodromes with differential geometry
approach from [Bolt07a, Porter93a, Porter07a]. Our last Cor. 20 can be a first
step in this direction.

  2. (2)

Consider all FLT-invariant non-equivalent classes of three-cycle ensembles on
$\mathbb{C}{}$: pairwise orthogonal triples (representing points in the upper
half-space [Kisil15a]), triples satisfying properties (1a) and (1b) of Prop.
14 (representing loxodromes), etc.

  3. (3)

Extend this consideration for quaternions or Clifford algebras
[GuerlebeckHabetaSproessig08, MoraisGeorgievSprossig14a]. The previous works
[Porter98a, Porter98b] and availability of FSCc in this setup [Cnops02a]*Ch. 4
[FillmoreSpringer90a] make it rather promising.

  4. (4)

Consider Möbius transformations in rings of dual and double numbers [Kisil12a,
Kisil05a, Kisil15a, Kisil07a, Kisil09e, Kisil11a, Mustafa17a, BarrettBolt10a].
There are enough indications that the story will not be quite the same as for
complex numbers.

  5. (5)

Explore further connections of loxodromes with

     * •

Carleson curves and microlocal properties of singular integral operators
[BishopBoettcherKarlovichSpitkovsky99a, BoettcherKarlovich01a,
BarrettBolt08a]; or

     * •

applications in operator theory [Simon11a, Vasilevski08a].

Some combinations of those topics shall be fruitful as well.

## Acknowledgments

The second-named author was supported by the Summer UG Bursary Scheme (2017)
at the University of Leeds. An earlier work [Hurst13a] on this topic was
supported by the same scheme in 2013.

Graphics for this paper was prepared using Asymptote software [Asymptote]. We
supply an Asymptote library cycle2D for geometry of planar cycles under GPL
licence [GNUGPL] with the copy of this paper at arXiv.org.

[controls=true,width=.9poster=first]50_loxodromes1200

Figure 4. Animated graphics of equivalent three-cycle parametrisations of a
loxodrome. The green cycle is $C_{1}$, two red circles are $C_{2}$ and
$C_{3}$.

## References