From 78110ace2ef9097e36d719bcfd4c32d870b5ab6a Mon Sep 17 00:00:00 2001
From: Gerd Heber <gheber@hdfgroup.org>
Date: Sat, 30 Dec 2017 18:04:17 -0600
Subject: [PATCH] Ongoing Org experiment.

---
 doc/org/00_Overview.org        | 135 +++++++++++++-------------
 doc/org/01_Chain_Complexes.org | 168 +++++++++++++++++++++++++++++++++
 doc/org/Handbook.org           |   6 +-
 doc/org/diabolo.pdf            | Bin 0 -> 5166 bytes
 4 files changed, 239 insertions(+), 70 deletions(-)
 create mode 100644 doc/org/01_Chain_Complexes.org
 create mode 100644 doc/org/diabolo.pdf

diff --git a/doc/org/00_Overview.org b/doc/org/00_Overview.org
index 10e16fb..fcb5fbd 100644
--- a/doc/org/00_Overview.org
+++ b/doc/org/00_Overview.org
@@ -1,3 +1,8 @@
+#+BEGIN_SRC lisp :session 00 :exports none :results silent
+(ql:quickload "kenzo")
+(use-package :cat)
+#+END_SRC
+
 
 * Overview
 
@@ -9,11 +14,11 @@ labelled =*=, =M3= and =N4=. The faces 0 and 2 of the 4-simplex =N4= are
 identified with (pasted to) the 3-simplex =M3=, the others being contracted on
 the base point. To create the simplicial set one types simply:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar m23)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf m23 (moore 2 3))
 #+END_SRC
 
@@ -23,7 +28,7 @@ the base point. To create the simplicial set one types simply:
 The system answers that a Kenzo object has been created, with number 1 and
 type =SIMPLICIAL SET=. This object may be referenced by the symbol =m23=.
 
-#+BEGIN_SRC lisp :session cat
+#+BEGIN_SRC lisp :session 00
 m23
 #+END_SRC
 
@@ -35,7 +40,7 @@ complex induced by the simplicial set description. Here we compute the $H_i$
 from 0 to 4 included. When in the answer the component part is void, it means
 that the corresponding homology group is null.
 
-#+BEGIN_SRC lisp :session cat :results output
+#+BEGIN_SRC lisp :session 00 :results output
 (homology m23 0 5)
 #+END_SRC
 
@@ -180,11 +185,11 @@ As =m23= is a simplicial set, it is possible to create the cartesian product
 =m23xm23= by the function =crts-prdc= (= CaRTeSian PRoDuCt). This is a new
 simplicial set.
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar m23xm23)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf m23xm23 (crts-prdc m23 m23))
 #+END_SRC
 
@@ -194,14 +199,14 @@ simplicial set.
 Being a simplicial set, =m23xm23= is also a chain complex object and we may for
 instance ask for the basis in dimension 6.
 
-#+BEGIN_SRC lisp :session cat :exports both :results verbatim
+#+BEGIN_SRC lisp :session 00 :exports both :results verbatim
 (basis m23xm23 6)
 #+END_SRC
 
 #+RESULTS:
 : (<CrPr 1-0 N4 3-2 N4> <CrPr 1-0 N4 4-2 N4> <CrPr 1-0 N4 4-3 N4> <CrPr 1-0 N4 4-3-2 M3> <CrPr 1-0 N4 5-2 N4> <CrPr 1-0 N4 5-3 N4> <CrPr 1-0 N4 5-3-2 M3> <CrPr 1-0 N4 5-4 N4> <CrPr 1-0 N4 5-4-2 M3> <CrPr 1-0 N4 5-4-3 M3> ...)
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (length (basis m23xm23 6))
 #+END_SRC
 
@@ -220,18 +225,18 @@ two simplicial sets do not make sense, and Kenzo understands to work on the
 underlying chain complex =m23=. This tensor product is a new chain complex and
 we see that the basis in dimension 6 has only one element:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar t2m23)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf t2m23 (tnsr-prdc m23 m23))
 #+END_SRC
 
 #+RESULTS:
 : [K3 Chain-Complex]
 
-#+BEGIN_SRC lisp :session cat :exports both :results verbatim
+#+BEGIN_SRC lisp :session 00 :exports both :results verbatim
 (basis t2m23 6)
 #+END_SRC
 
@@ -243,7 +248,7 @@ cartesian product space: as chain complexes, =m23xm23= and =t2m23= have the
 same homology groups, but the computations using the tensor product are
 considerably faster. The user can check Künneth's theorem.
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (homology m23xm23 0 8)
 #+END_SRC
 
@@ -518,11 +523,11 @@ In =Kenzo=, a non-degenerate simplex of $\mathrm{K}(\mathbb{Z}, 1)$ in
 dimension $n$ will be simply a list of $n$ non-null integers, for instance:
 =(2 3 4 5)=. In dimension 0, the only simplex is =NIL= (the base point).
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar kz1)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf kz1 (k-z 1))
 #+END_SRC
 
@@ -532,7 +537,7 @@ dimension $n$ will be simply a list of $n$ non-null integers, for instance:
 But this object is also a /coalgebra/ and an /algebra/, and we may see the
 effect of the respective induced /coproduct/ and /product/: 
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (cprd kz1 4 '(2 3 4 5)))
 #+END_SRC
 
@@ -546,7 +551,7 @@ effect of the respective induced /coproduct/ and /product/:
 : <1 * <TnPr (2 3 4 5) NIL>>
 : ------------------------------------------------------------------------------
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (aprd kz1 6 (tnpr 2 '(1 2) 4 '(3 4 5 6))))
 #+END_SRC
 
@@ -576,11 +581,11 @@ morphisms. The degree of the combination is indicated by the information:
 In the same way, we may create the Abelian simplicial groups
 $\mathrm{K}(\mathbb{Z}/2\mathbb{Z}, n)$:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar k-z2-2)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf k-z2-2 (k-z2 2))
 #+END_SRC
 
@@ -650,7 +655,7 @@ Let us play now with the sphere $S^3$ and its loop spaces. $S^3$ and
 $\Omega^2 S^3$ are created by respective calls to the functions =sphere= and
 =loop-space=. Then we compute the $H_4$ and $H_5$ of $\Omega^2 S^3$:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar s3)
 #+END_SRC
 
@@ -661,18 +666,18 @@ $\Omega^2 S^3$ are created by respective calls to the functions =sphere= and
 #+RESULTS:
 : [K174 Simplicial-Set]
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar o2s3)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf o2s3 (loop-space s3 2))
 #+END_SRC
 
 #+RESULTS:
 : [K191 Simplicial-Group]
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (homology o2s3 4 6)
 #+END_SRC
 
@@ -854,11 +859,11 @@ Component Z/2Z
 
 Let us take now the first loop space $\Omega^1 S^3$
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar os3)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf os3 (loop-space s3))
 #+END_SRC
 
@@ -871,11 +876,11 @@ original sphere. In fact, the object created by the command
 $\texttt{(loop3 0 's3 1)}$ is the "word" $S3^1$ belonging to the Kan simplicial
 version $G(S^3)$ (a simplicial group) of the loop space $\Omega S^3$.
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar L1)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (setf L1 (loop3 0 's3 1)))
 #+END_SRC
 
@@ -886,11 +891,11 @@ Let us consider also the 2-degeneracy of the base point of the loop space. In
 the printed result, the user will recognize the degeneracy $\eta_1\eta_0$ of
 the null loop, base point of $\Omega^1 S^3$: 
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar null-simp)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (setf null-simp (absm 3 +null-loop+)))
 #+END_SRC
 
@@ -905,11 +910,11 @@ following call. It means that we "paste" to the space =os3= a 3-simplex named
 collapsed on the base point; in this way the attaching map
 $(S^2 \rightarrow \Omega S^3)$ has degree 2.
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar dos3)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf dos3 (disk-pasting os3 3 '<D3> (list L1 null-simp L1 null-simp)))
 #+END_SRC
 
@@ -918,7 +923,7 @@ $(S^2 \rightarrow \Omega S^3)$ has degree 2.
 
 Let us compute a few homology groups of the new space =dos3=:
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (homology dos3 2 4)
 #+END_SRC
 
@@ -993,18 +998,18 @@ Homology in dimension 3 :
 But more interesting, let us build the loop space of the object =dos3= and let
 us compute the homology in dimension 5:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar odos3)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf odos3 (loop-space dos3))
 #+END_SRC
 
 #+RESULTS:
 : [K471 Simplicial-Group]
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (homology odos3 5)
 #+END_SRC
 
@@ -1289,21 +1294,21 @@ Let us continue with the Kan theory. First, we check that $S^3$ is not of type
 Kan and that $\Omega S^3$ is indeed of type Kan and a non-Abelian simplicial
 group.
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (typep s3 'kan)
 #+END_SRC
 
 #+RESULTS:
 : NIL
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (typep os3 'kan)
 #+END_SRC
 
 #+RESULTS:
 : T
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (typep os3 'simplicial-group)
 #+END_SRC
 
@@ -1320,22 +1325,22 @@ group.
 Let us create the word $L2=(S3)^2$, i.e an object belonging to $\Omega S^3$ and
 let us apply the product of the underlying algebra upon $L2 \otimes L2$:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar L2)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (setf L2 (loop3 0 's3 2)))
 #+END_SRC
 
 #+RESULTS:
 : <<Loop[S3\2]>>
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar square)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (setf square (aprd os3 4 (tnpr 2 L2 2 L2))))
 #+END_SRC
 
@@ -1354,11 +1359,11 @@ We see that the result is a linear combination of words composed from
 degeneracies of =L2=. The following instruction selects the generator part of
 the second element of the previous combination.
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar L4)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (setf L4 (gnrt (second (cmbn-list square)))))
 #+END_SRC
 
@@ -1370,11 +1375,11 @@ Let us use the lisp function =mapcar= (one among the various iteration
 functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object
 =L4=, this list is a "Kan horn".
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar horn)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (setf horn (mapcar #'(lambda (i) (face os3 i 4 l4)) '(1 2 3 4))))
 #+END_SRC
 
@@ -1386,11 +1391,11 @@ The function =kfll= tries to find a filling of this "Kan horn", and we see that
 the face 2 of the resulting simplex (which is very different from =L4=) is the
 same as the face 2 of =L4=.
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar kan-simplex)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (setf kan-simplex (kfll os3 0 4 horn)))
 #+END_SRC
 
@@ -1398,7 +1403,7 @@ same as the face 2 of =L4=.
 : 
 : <AbSm - <<Loop[3-1 S3\2][2-1 S3\-2][2-0 S3\2][1-0 S3\-2][2-1 S3\2][3-1 S3\-2][1-0 S3\2][3-1 S3\2][3-0 S3\-2][1-0 S3\-2][3-0 S3\2][2-0 S3\-2][1-0 S3\2][3-0 S3\-2][2-0 S3\2][3-0 S3\2]>>> 
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (face os3 2 4 kan-simplex))
 #+END_SRC
 
@@ -1406,7 +1411,7 @@ same as the face 2 of =L4=.
 : 
 : <AbSm - <<Loop[0 S3\2][2 S3\2]>>> 
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (princ (second horn))
 #+END_SRC
 
@@ -1422,25 +1427,25 @@ ${\cal G}$. In our case, as $\Omega S^3$ in non-Abelian, the result is not a
 simplicial group but only a simplicial set. We verify that the $H_4$, which in
 principle should be isomorphic to $H_4 S^3 = 0$, is really null.
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar cls-os3)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf cls-os3 (classifying-space os3))
 #+END_SRC
 
 #+RESULTS:
 : [K938 Simplicial-Set]
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (typep cls-os3 'simplicial-group)
 #+END_SRC
 
 #+RESULTS:
 : NIL
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (homology cls-os3 4)
 #+END_SRC
 
@@ -1486,11 +1491,11 @@ We take again $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}, 3)$ whose $H_3$ is
 $\mathbb{Z}/2\mathbb{Z}$. First the fundamental cohomology class is
 constructed:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar ch3)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf ch3 (chml-clss m23 3))
 #+END_SRC
 
@@ -1500,11 +1505,11 @@ constructed:
 Then the function =z2-whitehead= is called to build a fibration over the
 simplicial set =m23= canonically associated to the cohomology class =ch3=.
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar f3)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf f3 (z2-whitehead m23 ch3))
 #+END_SRC
 
@@ -1513,11 +1518,11 @@ simplicial set =m23= canonically associated to the cohomology class =ch3=.
 
 Then the total space of the fibration is built:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar x4)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf x4 (fibration-total f3))
 #+END_SRC
 
@@ -1527,7 +1532,7 @@ Then the total space of the fibration is built:
 The $H_4$ of this total space is the $\pi_4$ of
 $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
 
-#+BEGIN_SRC lisp :session cat :exports both :results output
+#+BEGIN_SRC lisp :session 00 :exports both :results output
 (homology x4 3 5)
 #+END_SRC
 
@@ -1650,29 +1655,29 @@ Component Z/2Z
 We may now iterate the process, to compute the $\pi_5$ of
 $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar ch4)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf ch4 (chml-clss x4 4))
 #+END_SRC
 
 #+RESULTS:
 : [K1318 Cohomology-Class on K1303 of degree 4]
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar f4)
 #+END_SRC
 
-#+BEGIN_SRC lisp :session cat :exports both
+#+BEGIN_SRC lisp :session 00 :exports both
 (setf f4 (z2-whitehead x4 ch4))
 #+END_SRC
 
 #+RESULTS:
 : [K1333 Fibration K1237 -> K1319]
 
-#+BEGIN_SRC lisp :session cat :exports none
+#+BEGIN_SRC lisp :session 00 :exports none
 (defvar x5)
 #+END_SRC
 
diff --git a/doc/org/01_Chain_Complexes.org b/doc/org/01_Chain_Complexes.org
new file mode 100644
index 0000000..ccec767
--- /dev/null
+++ b/doc/org/01_Chain_Complexes.org
@@ -0,0 +1,168 @@
+#+BEGIN_SRC lisp :session 01 :exports none :results silent
+(ql:quickload "kenzo")
+(use-package :cat)
+#+END_SRC
+
+* Chain Complexes
+
+** Introduction
+
+In the Kenzo program, a /chain complex/[fn:1]$(C_p, d_p)$ is a collection of free
+\(\mathbb{Z}\)-modules $(C_p)$, one for each \(p\in\mathbb{Z}\), together with
+a homomorphism $d_p : C_p \rightarrow C_{p-1}$, such that, for all
+$p$, $d_{p-1} \circ d_p = 0$.
+
+*In the Kenzo program*, a /morphism/ $f=(f_p)$, of degree $k$, from a chain
+complex $(C_p, d_p)$ to another $(C'_p, d'_p)$ is a collection of homomorphisms
+
+$$f_p : C_p \rightarrow C'_{p+k}.$$
+
+This is expressed by the following diagram, generally not assumed commutative.
+
+#+BEGIN_EXPORT latex
+\[
+\begin{aligned}
+\cdots & \leftarrow & C_{p-1} & \quad\xleftarrow{d_p} & C_p & \quad\xleftarrow{d_{p+1}} & C_{p+1}  & \quad\leftarrow & \cdots \\
+       &    & f_{p-1} \Bigg\downarrow &    & f_p \Bigg\downarrow &    & f_{p+1} \Bigg\downarrow \\
+\cdots & \leftarrow & C'_{p+k-1} & \quad\xleftarrow{d'_{p+k}} & C'_{p+k} &
+       \quad\xleftarrow{d'_{p+k+1}} & C'_{p+k+1} & \quad\leftarrow & \cdots
+\end{aligned}
+\]
+
+#+END_EXPORT
+
+Three types of   morphisms are most generally considered.
+
+1. $k=0$. If the commutativity relation $d'_p\circ f_p=f_{p-1}\circ d_p$ holds
+   for every $p$, then the morphism $f$ is an ordinary chain complex morphism
+   or /chain map/.
+2. $k=-1$. If $(C_p,d_p)=(C'_p,d'_p)$ and $f_p=d_p$, then $f$ is the
+   /differential/ of the chain complex $(C_p,d_p)$ and, in fact, this
+   differential is implemented in the =Kenzo= program as a morphism of
+   degree $-1$. Also, if $d_p \neq f_p$ and $(d_{p-1}+f_{p-1})(d_p+f_p) = 0$
+   for every $p$, then $f_\ast$ is a perturbation of the differential $d_\ast$.
+3. $k=+1$. In this case, $f$ is usually a /homotopy operator/, that is, some
+   relation $$d'_{p+1}\circ f_p+f_{p-1}\circ d_p=g_p-g'_p$$ is satisfied for
+   two (ordinary) chain complex morphisms $g_\ast$ and $g'_\ast$, and $f_\ast$
+   is a homotopy between $g_\ast$ and $g'_\ast$.
+
+For technical reasons, these three types of morphisms have been implemented in
+the =Kenzo= program in a unique type.
+
+[fn:1] *P.J. Giblin* in /Graph, Surface and Homology/, Chapman and Hall Math.
+series, 1981.
+
+** Generators, terms, and combinations
+
+To become familiar with the Lisp functions implementing the chain complexes,
+the best is to begin with an example. /Simplicial complexes/ are good
+candidates for this purpose, and we shall take as a typical example the
+following simplicial complex.
+
+[[./diabolo.pdf]]
+
+In this simplicial complex, called here /diabolo/, there are 3 non-null chain
+groups.
+
+- $C_0$, the free \(\mathbb{Z}\)-module on the set of vertices
+  $\{s_0,s_1,s_2,s_3,s_4,s_5\}$
+- $C_1$, the free \(\mathbb{Z}\)-module on the set of edges
+  $\{s_{01},s_{02},s_{12},s_{23},s_{34},s_{35}, s_{45}\}$
+- $C_2$, the free \(\mathbb{Z}\)-module on the set of triangles (here a
+  singleton) $\{s_{345}\}$
+
+The elements of either of those groups $C_p$ are linear integer combinations of
+the corresponding basis $\beta_p$, i.e. elements of the form:
+$$
+\sum_{b \in \beta_p} \lambda_b b.
+$$
+For example $2 s_{02} - 3 s_{34} \in C_1$. An element $b$ of the basis
+$\beta_p$ of $C_p$ is called a /generator/ of $C_p$ and in our specific case
+this generator will be represented by a Lisp symbol. For instance, $s_{45}$
+will be translated into =s45=. However, the user should be aware that in more
+realistic usage of the software, generators may be of any type. A product such
+as $\lambda_i\sigma_i$ is called a /term/ and a sum of terms, a /combination/.
+
+
+*** Representation of a combination
+
+A combination is represented internally in the system by a structure having the
+following general form:
+
+$$\#\texttt{S(CMBN}\quad\texttt{DEGREE P}\quad\texttt{LIST (}\texttt{(}
+\lambda_1\cdot\sigma_1\texttt{)}\dots\texttt{(}\lambda_k\cdot\sigma_k
+\texttt{)))}$$
+
+and containing
+
+1. The degree of the combination corresponding to the index \(p\in\mathbb{Z}\)
+   of the group $C_p$ to which this combination belongs.
+2. The list of the internal representation of the terms, namely the list of
+   pairs $(\lambda_i\cdot\sigma_i)$.
+
+This choice of representation implies that only homogeneous combinations will
+be considered. The type =CMBN= and a printing method are provided with this
+internal representation. The external form of a combination is shown in the
+examples.
+
+*** Ordering the generators
+
+In order to speed up the execution of algorithms involving combinations, the
+list of pairs $(\lambda_i.\sigma_i)$ is ordered by an adequate ordering
+function (e.g., the lexicographical ordering on the symbols). For programming
+convenience, an enumerated type =CMPR= has been defined:
+
+#+BEGIN_EXAMPLE
+(deftype cmpr() '(member :less :equal :greater))
+#+END_EXAMPLE
+
+A number of macros, functions and methods have been defined on usual sets
+(symbols, numbers, lists, etc.) taking their value in the set
+=[:less, :equal, :greater]=. Of course, the user may define its own function
+for a particular case. There exists functions to compare various couples of
+usual items:
+
+=f-cmpr= /n1 n2/ | /[Function]/
++ Return =:less=, =:equal=, =:greater=, according to the result of the
+  canonical comparison of both integers /n1/ and /n2/.
+
+=s-cmpr= /symbol1 symbol2/ | /[Function]/ 
++ Return  =:less=, =:equal=, =:greater=, according to the result of the Lisp
+  comparison functions on the strings =(symbol-name= /symbol1/ =)= and
+  =(symbol-name= /symbol2/ =)=
+
+=l-cmpr= /list1 list2/ | /[Function]/ 
++ Return =:less=, =:equal=, =:greater=, according to the lexicographical
+  ordering of the two lists /list1/ and /list2/, making sense only if /list1/
+  and /list2/ are lists made of integers, symbols and/or lists; in the last
+  case, the qualifier /list/ is to be understood recursively.
+
+**** Examples
+
+#+BEGIN_SRC lisp :session 01 :exports both
+(f-cmpr 123 789)
+#+END_SRC
+
+#+RESULTS:
+: :LESS
+
+#+BEGIN_SRC lisp :session 01 :exports both
+(s-cmpr 'circulation 'circular)
+#+END_SRC
+
+#+RESULTS:
+: :GREATER
+
+#+BEGIN_SRC lisp :session 02 :exports both
+(s-cmpr 'qwerty 'qwerty)
+#+END_SRC
+
+#+RESULTS:
+: :EQUAL
+
+#+BEGIN_SRC lisp :session 02 :exports both
+(l-cmpr '(1 a b) '(1 a))
+#+END_SRC
+
+#+RESULTS:
+: :GREATER
diff --git a/doc/org/Handbook.org b/doc/org/Handbook.org
index 9420d1c..dab414e 100644
--- a/doc/org/Handbook.org
+++ b/doc/org/Handbook.org
@@ -2,10 +2,6 @@
 #+AUTHOR: Julio Rubio Garcia, Francis Sergeraert, Yvon Siret
 #+DATE: April 26, 1999
 
-#+BEGIN_SRC lisp :session cat :exports none :results silent
-(ql:quickload "kenzo")
-(use-package :cat)
-#+END_SRC
-
 #+INCLUDE: "Foreword.org"
 #+INCLUDE: "00_Overview.org"
+#+INCLUDE: "01_Chain_Complexes.org"
diff --git a/doc/org/diabolo.pdf b/doc/org/diabolo.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..cc5cc76b6fb39260ceff98609c050d39ec1ba09e
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