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@ -8,15 +8,15 @@
"\n",
"## Introduction\n",
"\n",
"A *chain complex* $(C_p, d_p)$ is a collection of free $\\mathbb{Z}$-modules $(C_p)$, one for each\n",
"In the Kenzo program, a *chain complex* $(C_p, d_p)$ is a collection of free $\\mathbb{Z}$-modules $(C_p)$, one for each\n",
"$p \\in \\mathbb{Z}$, together with a homomorphism $d_p : C_p \\rightarrow C_{p-1}$, such\n",
"that, for all $p$, $d_{p-1} \\circ d_p = 0$. (See, for example, __P.J. Giblin__ in *Graph, Surface and Homology*, Chapman and Hall Math. series, 1981.)\n",
"\n",
"__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\n",
"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\n",
"\n",
"$$f_p : C_p \\rightarrow C'_{p+k}.$$\n",
"\n",
"This is expressed by the following diagram, generally __not assumend commutative__.\n",
"This is expressed by the following diagram, generally not assumend commutative .\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
@ -28,8 +28,8 @@
"Three types of morphisms are most generally considered.\n",
"\n",
"1. $k=0$. If the commutativity relation $d'_p\\circ f_p=f_{p-1}\\circ d_p$ holds for every $p$, then the morphism $f$ is an ordinary chain complex morphism or *chain map*.\n",
"2. $k=-1$. If $(C_p,d_p)=(C'_p,d'_p)$ and $f_p=d_p$, then $f$ is the *differential* of the chain complex $(C_p,d_p)$ and, in fact, this differential is implemented in the ${\\tt Kenzo}$ program as a morphism of degree $-1$.\n",
"3. $k=+1$. In this case, $f$ is usually a *homotopy operator*, that is, some relation $$d'_{p+1}\\circ f_p+f_{p-1}\\circ d_p=g_p-g'_p$$ is satisfied for two (ordinary) chain complex morphisms $g$ and $g'$.\n",
"2. $k=-1$. If $(C_p,d_p)=(C'_p,d'_p)$ and $f_p=d_p$, then $f$ is the *differential* of the chain complex $(C_p,d_p)$ and, in fact, this differential is implemented in the ${\\tt Kenzo}$ program as a morphism of degree $-1$. Also, if $d_p \\neq f_p$ and $(d_{p-1}+f_{p-1})(d_p+f_p) = 0$ for every $p$, then $f_\\ast$ is a perturbation of the differential $d_\\ast$.\n",
"3. $k=+1$. In this case, $f$ is usually a *homotopy operator*, that is, some relation $$d'_{p+1}\\circ f_p+f_{p-1}\\circ d_p=g_p-g'_p$$ is satisfied for two (ordinary) chain complex morphisms $g_\\ast$ and $g'_\\ast$, and $f_\\ast$ is a homotopy between $g_\\ast$ and $g'_\\ast$.\n",
"\n",
"For technical reasons, these three types of morphisms have been implemented in the `Kenzo` program\n",
"in a unique type.\n",
@ -56,15 +56,17 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"In this simplicial complex, called here *diabolo*, there are 3 associated chain groups.\n",
"In this simplicial complex, called here *diabolo*, there are 3 non-null chain groups.\n",
"\n",
"* $C_0$, the free $\\mathbb{Z}$-module on the set of vertices $\\{s_0,s_1,s_2,s_3,s_4,s_5\\}$\n",
"* $C_1$, the free $\\mathbb{Z}$-module on the set of edges $\\{s_{01},s_{02},s_{12},s_{23},s_{34},s_{35}, s_{45}\\}$\n",
"* $C_2$, the free $\\mathbb{Z}$-module on the set of triangles (here a singleton) $\\{s_{345}\\}$\n",
"\n",
"The elements of either of those groups $C_p$ are linear integer combinations of the corresponding basis (set of $\\sigma_i$'s), i.e., elements of the form\n",
"$$\\sum\\lambda_i\\sigma_i, \\quad \\lambda_i\\in\\mathbb{Z}.$$\n",
"An element $\\sigma_i$ of any basis is also called a *generator* and in our specific case this generator will be represented by a Lisp symbol. For instance, $s_{45}$ will be translated into `s45`. However, the user should be aware that in more realistic usage of the software, generators may be of any type. A product such as $\\lambda_i\\sigma_i$ is called a *term* and a sum of terms, a *combination*.\n",
"The elements of either of those groups $C_p$ are linear integer combinations of the corresponding basis $\\beta_p$, i.e. elements of the form:\n",
"$$\n",
"\\sum_{b \\in \\beta_p} \\lambda_b b.\n",
"$$\n",
"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*.\n",
"\n",
"### Representation of a combination\n",
"\n",
@ -76,25 +78,25 @@
"1. The degree of the combination corresponding to the index $p\\in\\mathbb{Z}$ of the group $C_p$ to which this combination belongs.\n",
"2. The list of the internal representation of the terms, namely the list of pairs $(\\lambda_i\\cdot\\sigma_i)$.\n",
"\n",
"This choice of representation implaies that only homogeneous combinations will be considered. A type `CMBN` and a printing method are provided with this internal representation. The external form of a combination is shown in the examples.\n",
"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.\n",
"\n",
"### Ordering the generators\n",
"\n",
"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 (ex: the lexicographical ordering n the symbols). For programming convenience, an enumerated type `CMPR` has been defined:\n",
"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 (ex: the lexicographical ordering on the symbols). For programming convenience, an enumerated type `CMPR` has been defined:\n",
"```lisp\n",
"(deftype cmpr() '(member :less :equal :greater))\n",
"```\n",
"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:\n",
"\n",
"`(f-cmpr n1 n2)` *[Function]* \n",
"> Return `:less`, `:equal`, `:greater`, according to the result of the canonical comparison of both integers `n1` and `n2`.\n",
"`(f-cmpr` *n1 n2*`)` *[Function]* \n",
"> Return `:less`, `:equal`, `:greater`, according to the result of the canonical comparison of both integers *n1* and *n2*.\n",
"\n",
"`(s-cmpr symbol1 symbol2)` *[Function]* \n",
"> Return `:less`, `:equal`, `:greater`, according to the result of the lisp comparison functions on the strings `(symbol-name symbol1)` and `(symbol-name symbol2)`\n",
"`(s-cmpr` *symbol1 symbol2*`)` *[Function]* \n",
"> Return `:less`, `:equal`, `:greater`, according to the result of the lisp comparison functions on the strings `(symbol-name` *symbol1*`)` and `(symbol-name` *symbol2*`)`\n",
"\n",
"`(l-cmpr list1 list2)` *[Function]* \n",
"> Return `:less`, `:equal`, `:greater`, according to the lexicographical ordering of the two lists `list1`\n",
"and `list2` representing legal generators in this implementation.\n",
"`(l-cmpr ` *list1 list2*`)` *[Function]* \n",
"> Return `:less`, `:equal`, `:greater`, according to the lexicographical ordering of the two lists *list1*\n",
"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.\n",
"\n",
"#### Examples"
]
@ -155,50 +157,47 @@
"\n",
"The software provides a set of functions, methods, and macros to create or modify combinations.\n",
"\n",
"`(term-cmbn dgr cf gnr)` *[Macro]* \n",
"`(term-cmbn` *dgr cf gnr*`)` *[Macro]* \n",
"> Construct the combination of degree `dgr` with unique term `cf * gnr`.\n",
"\n",
"`(cmbn dgr cf1 gnr1 cf2 gnr2 ... cfn gnrn)` *[Function]* \n",
"> Construct a combination of degree `dgr`, sum of the terms $cf_i * gnr_i$. The sequence of pairs $\\lbrace cf_i\\ gnr_i\\rbrace$ has an undefinite length and may be void. In this case, the combination is a null combination of degree `dgr`. \n",
"`(cmbn` *dgr cf1 gnr1 cf2 gnr2 ... cfn gnrn*`)` *[Function]* \n",
"> Construct a combination of degree `dgr`, sum of the terms $cf_i * gnr_i$. The sequence of pairs $\\lbrace cf_i\\ gnr_i\\rbrace$ has an undefinite length and may be void. In this case, the combination is a null combination of degree `dgr`. It is the responsability of the user to properly order the terms of the combination, otherwise later calculations with this combination could work erroneously. This point can be checked by the `do-control` function. **[I suggest to insert here two appropriate examples.]**\n",
"\n",
"`(cmbn-p object)` *[Function]* \n",
"`(cmbn-p` *object*`)` *[Function]* \n",
"> Test is `object` is a legal combination.\n",
"\n",
"`(cmbn-degr cmbn)` *[Macro]* \n",
"`(cmbn-degr` *cmbn*`)` *[Macro]* \n",
"> Get the degree (an integer) of the combination `cmbn`.\n",
"\n",
"`(cmbn-list cmbn)` *[Macro]* \n",
"> Get the list of the terms of the combination `cmbn`. Beware: a term is not a `Kenzo` object. One may select the coefficient (an integer) or the generator - a `Kenzo` object - respectively by the macros `cffc` and `gnrt`.\n",
"`(cmbn-list` *cmbn*`)` *[Macro]* \n",
"> Get the list of the terms of the combination `cmbn`. Beware: a term is not a structure object, it is a simple `cons` `(cffc . gnrt)`, a pair coefficient - generator. Given a term, one may select the coefficient (an integer) or the generator - a `Kenzo` object - respectively by the macros `cffc` and `gnrt`.\n",
"\n",
"`(zero-cmbn dgr)` *[Function]* \n",
"`(zero-cmbn` *dgr*`)` *[Function]* \n",
"> Create an instance of a null combination in the degree `dgr`.\n",
"\n",
"`(zero-pure-dffr cmbn)` *[Function]* \n",
"> Create a null combination of degree $degr(cmbn) - 1$.\n",
"\n",
"`(cmbn-zero-p cmbn)` *[Macro]* \n",
"`(cmbn-zero-p` *cmbn*`)` *[Macro]* \n",
"> Test if `cmbn` is a null combination in any degree.\n",
"\n",
"`(cmbn-non-zero-p cmbn)` *[Macro]* \n",
"`(cmbn-non-zero-p` *cmbn*`)` *[Macro]* \n",
"> Test if `cmbn` is a non-null combination in any degree.\n",
"\n",
"`(cmbn-opps cmbn)` *[Function]*\n",
"`(cmbn-opps` *cmbn*`)` *[Function]*\n",
"> Create a combination opposite to `cmbn`.\n",
"\n",
"`(n-cmbn n cmbn)` *[Function]* \n",
"`(n-cmbn` *n cmbn*`)` *[Function]* \n",
"> Create a combination multiple of `cmbn` by the factor `n`.\n",
"\n",
"`(2cmbn-add cmpr cmbn1 cmbn2)` *[Function]*\n",
"> Create a combination, sum of both combinations `cmbn1` and `cmbn2`. The first argument, `cmpr`, must be a function or macro relevant to compare the generators of the involved combinations, in order to return an ordered combination.\n",
"`(2cmbn-add cmpr` *cmbn1 cmbn2*`)` *[Function]*\n",
"> Create a combination, sum of both combinations `cmbn1` and `cmbn2`. The first argument, `cmpr`, must be the comparison function available in the slot `cmpr` of the underlying chain complex, required to properly merge the terms of both combinations.\n",
"\n",
"`(ncmbn-add cmpr cmbn1 cmbn2 ... cmbnk)` *[Function]* \n",
"`(ncmbn-add` *cmpr cmbn1 cmbn2 ... cmbnk*`)` *[Function]* \n",
"> Create a combination, sum of the indefinite number of combinations $cmbn_i$. As to the first argument, `cmpr`, see the function `2cmbn-add`.\n",
"\n",
"`(2n-2cmbn cmpr n1 cmbn1 n2 cmbn2)` *[Function]* \n",
"`(2n-2cmbn` *cmpr n1 cmbn1 n2 cmbn2*`)` *[Function]* \n",
"> Build the combination `n1 * cmbn1 + n2 * cmbn2`. Both integers `n1` and `n2` must be\n",
"non null. As to the first argument, `cmpr`, see the function `2cmbn-add`.\n",
"\n",
"`(2cmbn-sbtr cmpr cmbn1 cmbn2)` *[Function]* \n",
"`(2cmbn-sbtr` *cmpr cmbn1 cmbn2*`)` *[Function]* \n",
"> Create a combination, difference of `cmbn1` and `cmbn2`. As to the first argument, `cmpr`, see the function `2cmbn-add`.\n",
"\n",
"#### Examples"
@ -249,7 +248,7 @@
},
"outputs": [],
"source": [
"(def term3 (third *)) ;; not a Kenzo object!"
"(def term3 (third *)) ;; a simple cons!"
]
},
{
@ -366,8 +365,8 @@
"\n",
"This class has 8 slots:\n",
"\n",
"1. `cmpr`, a comparison function or method for generators with a range in the set `[:less, :equal, :greater]`. It is very important to note that the generators to be compared are assumed to be of the *same degree*, i.e. they must belong to the same group $C_p$. As an exception with the general policy of the software, this degree is not explicitly precised. The implementor has chosen to avoid additional tests because in real problems the program spends a lot of time comparing generators.\n",
"2. `basis`, a lisp function giving the distinguished **ordered** basis of the free $\\mathbb{Z}$-modules ($C_p$). (Recall that in the software, only free chain complexes are considered.) When some components of the chain complex are not finitely generated, we say that the chain complex is *locally effective*. In this case the value of this slot must be the keyword `locally-effective`.\n",
"1. `cmpr`, a comparison function or method for generators with a range in the set `[:less, :equal, :greater]`. It is very important to note that the generators to be compared are assumed to be of the *same degree*, i.e. they must belong to the same group $C_p$. As an exception with the general policy of the program, this degree is not explicitly precised. The implementor has chosen to avoid additional tests because in real problems the program spends a lot of time comparing generators.\n",
"2. `basis`, a lisp function giving the distinguished **ordered** basis of the free $\\mathbb{Z}$-modules ($C_p$). (Recall that in the program, only free chain complexes are considered.) When some components of the chain complex are not finitely generated, we say that the chain complex is *locally effective*. In this case the value of this slot must be the keyword `locally-effective`.\n",
"3. `bsgn`, a lisp object of any type representing a distinguished generator in dimension 0, the base generator. \n",
"4. `dffr`, the differential morphism, instance of the class `MORPHISM`, defined hereafter. The pure lisp function corresponding to the differential homomorphism for each $p$ ($d_p : C_p \\rightarrow C_{p-1}$) is defined in the instance morphism object and not directly in the instance chain complex object.\n",
"5. `idnm`, an integer, number plate for this object. This is generated by the system in a sequential way, each time a new `Kenzo` object is created.\n",
@ -382,9 +381,8 @@
"To facilitate the construction of instances of the class `CHAIN-COMPLEX` and to free the user to call\n",
"the standard constructor `make-instance`, the software provides the function\n",
"\n",
"```lisp\n",
"build-chcm :cmpr cmpr :basis basis :bsgn bsgn :intr-dffr intr-dffr :strt strt :orgn orgn\n",
"```\n",
"`build-chcm`\n",
"> `:cmpr` *cmpr* `:basis` *basis* `:bsgn` *bsgn* `:intr-dffr` *intr-dffr* `:strt` *strt* `:orgn` *orgn*\n",
"\n",
"defined with keyword parameters. The returned value is an instance of the class `CHAIN-COMPLEX`. In particular, this function frees the user to build himself the instance of the class `MORPHISM` corresponding to the differential homomorphism. The keyword arguments of `build-chcm` are:\n",
"\n",
@ -393,7 +391,8 @@
"- `bsgn`, a generator, the base point of the underlying set.\n",
"- `intr-dffr`, a **lisp function** defining the differential homomorphism for each $p$ ($d_p : C_p \\rightarrow C_{p-1}$).\n",
"- `strt`, one of the two values: `:gnrt` or `:cmbn`, defining the mapping *strategy* of the differential homomorphism, either by generator or by combination. The default is `:gnrt`. The real connection between the arguments `intr-dffr` and `strt`will be detailed hereafter through typical examples. The general idea is the following: if the strategy is `:gnrt`, the `:intr-dffr` argument function uses two arguments, namely a degree and a generator of this degree and computes the boundary combination of this generator. If the strategy is `:cmbn`, the `:intr-dffr` argument function uses a combination as argument and computes the boundary combination of this argument. We recall that a combination contains its own degree.\n",
"- `orgn`, a list containing a relevant and carefully chosen comment about the `origin` of the chain complex. If, during a Lisp session, the user wishes to modify any slot of an existing chain complex, by calling again `build-chcm`, he must change also the comment, otherwise the new version of the object will not be created. This remark is valid for any kind of instantiation of `Kenzo` objects. For avoiding such a constraint, one may use the function `cat-init`, before the redefinition.\n",
"- `orgn`, a list containing a relevant and carefully chosen comment about the `origin` of the chain complex. If, during a Lisp session, the user wishes to modify any slot of an existing chain complex, by calling again `build-chcm`, he must change also the comment, otherwise the new version of the object will not be created. This remark is valid for any kind of instantiation of `Kenzo` objects. The function `cat-init` reinitializes the environment, that is, the previous relations `orgn` $\\rightarrow$ Kenzo objects are deleted.\n",
"\n",
"\n",
"After creation of an instance of chain complex, the function `build-chcm` pushes this\n",
"object in the list of already created chain complexes `*chcm-list*`.\n",
@ -560,16 +559,16 @@
"`(cat-init)` *[Function]*\n",
"> Clear among others, the list `*chcm-list*`, list of user created chain complexes and reset the global counter to 1. The existing objects and in particular here the chain complexes, are not destroyed but they will not enter any more in account during the search process for duplicated objects. This remark is general for the other types of objects saved in specific lists. \n",
"\n",
"`(chcm n)` *[Function]*\n",
"`(chcm` *n*`)` *[Function]*\n",
"> Return from the list `*chcm-list*` the chain complex instance whose the Kenzo identification is `n`; if it does not exist, return `NIL`.\n",
"\n",
"`(cmpr object item1 item2)` *[Macro]*\n",
"`(cmpr` *object item1 item2*`)` *[Macro]*\n",
"> Apply the comparison function associated to the chain complex `object` to the two generators `item1` and `item2`.\n",
"\n",
"`(basis object n :dgnr)` *[Macro]*\n",
"> With only one argument (`object`), get the function attached to the slot `basis` of the chain complex object. With two arguments, get the distinguished basis of the group of degree `n` in the chain complex `object`. If the chain complex is locally effective, this function returns an error because, in some degrees, the corresponding set of generators is probably infinite. With a third argument, the keyword `:dgnr`, get also the *degenerate* elements of the basis in degree `n`.\n",
"`(basis` *object n*`)` *[Macro]*\n",
"> With only one argument (`object`), get the function attached to the slot `basis` of the chain complex object. With two arguments, get the distinguished basis of the group of degree `n` in the chain complex `object`. If the chain complex is locally effective, this function returns an error because, in some degrees, the corresponding set of generators is probably infinite.\n",
"\n",
"`(dffr chcm &rest)` *[Macro]* \n",
"`(dffr` *chcm* `&rest)` *[Macro]* \n",
"> Versatile macro to apply the differential morphism of the chain complex `chcm` either to a combination or a generator with a degree, respectively `(dffr chcm cmbn)` or `(dffr chcm degr gnrt)`. The macro `?`, described later, may be used for the same purpose.\n",
"\n",
"`(z-chcm)` *[Function]* \n",
@ -745,20 +744,15 @@
"1. The `:cmpr` keyword argument is a function returning `:equal` on any pair on generators (because there is a unique generator!).\n",
"2. The `:basis` keyword argument is a lisp function returning the null basis (the constant `+empty-list+ = ()`) for $p \\not= 0$ and the list `(:Z-gnrt)`, for $p=0$.\n",
"3. The base generator is of course `Z-gnrt`.\n",
"4. The `:intr-dffr` keyword argument is a lisp function defining the differential which, to any combination of degre $p$ of the chain complex, returns a *null combination* of degree $p-1$. This simple lisp function is also provided in `Kenzo` and is called `zero-intr-dffr`.\n",
"4. The `:intr-dffr` keyword argument is a lisp function defining the differential which, to any combination of degre $p$ of the chain complex, returns a *null combination* of degree $p-1$. This simple lisp function is also provided in `Kenzo` and is called `#'zero-intr-dffr`.\n",
"5. The `:strt` keyword argument is the combination strategy (`:cmbn`).\n",
"6. The `:orgn` keyword argument is the comment list `(zcc-constant)`.\n",
"\n",
"In the software `Kenzo`, the chain complex instance `ZCC` may be built, when needed, by the lisp statement: `(z-chcm)`. This statement may be used freely each time one needs this chain complex, since the system recognizes if it has already been created."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the program `Kenzo`, the chain complex instance `ZCC` may be built, when needed, by the lisp statement: `(z-chcm)`. This statement may be used freely each time one needs this chain complex, since the system recognizes if it has already been created.\n",
"\n",
"#### The chain complex `circle`\n",
"\n",
"On the model of the previous chain complex, one may define a function `circle` for building a chain complex having the same homology as the circle."
"On the model of the previous chain complex, one may define a function `circle` for building the chain complex associated to the standard model of the circle as a simplicial set."
]
},
{
@ -821,7 +815,7 @@
"\n",
"1. `sorc`, an object of the class `CHAIN-COMPLEX`, namely the *source* chain complex of this morphism.\n",
"2. `trgt`, an object of the class `CHAIN-COMPLEX`, namely the *target* chain complex of this morphism.\n",
"3. `degr`, an integer, the degree of the morphism. A morphism is supposed to associate to any element of degree $k$ of the source chain complex, an element of degree $k + degr$ of the target chain complex. For instance, the differential homomorphism is of degree -1.\n",
"3. `degr`, an integer, the degree of the morphism. A morphism is supposed to associate to any element of degree $k$ of the source chain complex, an element of degree $k + degr$ of the target chain complex. For instance, a differential homomorphism is of degree -1.\n",
"4. `intr`, a **pure lisp function** implementing the mathematical algorithm of the morphism and taking in account the strategy (`strt`).\n",
"5. `strt`, one of the two symbols `:gnrt`, `:cmbn`. What has already been said about the strategy of the differential morphism is generalized to any morphism: according to the value of the argument, `:gnrt` or `:cmbn`, the lisp function attached to the keyword just above, works respectively with 2 arguments (a degree and a generator) or only one (a combination) and must return a combination, image of the generator or the combination argument. \n",
"6. `???-clnm`, an integer updated by the system for statistics (number of times the morphism has been called on combinations - Internal use)\n",
@ -841,9 +835,9 @@
"\n",
"To facilitate the construction of instances of the class `MORPHISM` and to free the user to call the standard constructor `make-instance`, the software provides the function `build-mrph`\n",
"\n",
"```lisp\n",
"build-mrph :sorc sorc :trgt trgt :degr degr :intr intr :strt strt :orgn orgn\n",
"```\n",
"`build-mrph`\n",
"> `:sorc` *sorc* `:trgt` *trgt* `:degr` *degr* `:intr` *intr* `:strt` *strt* `:orgn` *orgn*\n",
"\n",
"\n",
"defined with keyword parameters. The returned value is an instance of the class `MORPHISM`. The keyword arguments of `build-mrph` are:\n",
"\n",
@ -858,7 +852,7 @@
"\n",
"#### The differential homomorphism in a chain complex instance\n",
"\n",
"In a chain complex instance, the differential homomorphism is defined as a morphism with identical source and target, and degree -1. The user must know that the function `build-chcm` calls internally the function `build-mrph` and passes it the keyword argument `pure-dffr`. The function `build-mrph` builds the morphism instance which will be then inserted into the slot `:dffr` of the chain complex instance to be constructed.\n",
"In a chain complex instance, the differential homomorphism is defined as a morphism with identical source and target, and degree -1. The user must know that the function `build-chcm` calls internally the function `build-mrph` and passes it the keyword argument `intr-dffr`. The function `build-mrph` builds the morphism instance which will be then inserted into the slot `:dffr` of the chain complex instance to be constructed.\n",
"\n",
"##### Example\n",
"\n",
@ -951,53 +945,53 @@
"`(cat-init)` *[Function]*\n",
"> Clear in particular `*mrph-list*`, the list of user created morphisms and reset the global counter to 1.\n",
"\n",
"`(mrph n)` *[Function]*\n",
"`(mrph` *n*`)` *[Function]*\n",
"> Retrieve in the list `*mrph-list*` the morphism instance whose identification is `n`. If it does not exist, return `NIL`. \n",
"\n",
"`(zero-mrph chcm1 &optional (chcm2 chcm1) (degr 0))` *[Function]*\n",
"`(zero-mrph` *chcm1* `&optional (`*chcm2 chcm1*`) (`*degr* `0))` *[Function]*\n",
"> Construct the null morphism between the chain complexes `chcm1` and `chcm2` of degree `degr`. The parameters `chcm2` and `degr` are optional and if omitted the default values (respectively `chcm1` and 0) are taken.\n",
"\n",
"`(idnt-mrph chcm)` *[Function]*\n",
"`(idnt-mrph` *chcm*`)` *[Function]*\n",
"> Construct the identity morphism (degree 0) between the chain complex `chcm` and itself.\n",
"\n",
"`(opps mrph)` *[Function]*\n",
"> Construct the opposite morphism of `mrph`, i.e. $-1 \\times mrp$; such a function, as well as the following ones, installs the right source and target.\n",
"`(opps` *mrph*`)` *[Function]*\n",
"> Construct the opposite morphism of `mrph`, i.e. $-1 \\times mrph$; such a function, as well as the following ones, installs the right source and target.\n",
"\n",
"`(cmps mrph1 mrph2 &optional strt)` *[Method]*\n",
"`(cmps` *mrph1 mrph2* `&optional` *strt*`)` *[Method]*\n",
"> Construct the composite of the morphisms, i.e. $mrph_1 \\circ\\, mrph_2$. Of course, the target of $mrph_2$ must be the same as the source of $mrph_1$, otherwise the system signals an error. The new instance inherits its source slot from $mrph_2$ and its target slot from $mrph_1$. This function optimizes the compositions in which appear zero morphisms or identity morphisms. Unless the user gives explicitly the strategy (`strt`), the resulting strategy is determined by the respective strategy of the two morphisms.\n",
"\n",
"`(cmps chcm1 chcm2 &optional strt)` *[Method]*\n",
"`(cmps` *chcm1 chcm2* `&optional` *strt*`)` *[Method]*\n",
"Construct the composite of the differential of the chain complexes `chcm1` and `chcm2`, i.e. $d_1 \\circ d_2$.\n",
"\n",
"`(cmps chcm1 mrph2 &optional strt)` *[Method]*\n",
"`(cmps` *chcm1 mrph2* `&optional` *strt*`)` *[Method]*\n",
"> Construct the composite of the differential of the chain complex `chcm1` and the morphism `mrph2`, i.e. $d_1 \\circ mrph_2$.\n",
"\n",
"`(cmps mrph1 chcm2 &optional strt)` *[Method]*\n",
"`(cmps` *mrph1 chcm2* `&optional` *strt*`)` *[Method]*\n",
"> Construct the composite of the morphism `mrph1` and the differential of the chain complex `chcm2`, i.e. $mrph_1 \\circ d_2$.\n",
"\n",
"`(i-cmps mrph1 mrph2 ... mrphk)` *[Macro]*\n",
"`(i-cmps` *mrph1 mrph2 ... mrphk*`)` *[Macro]*\n",
"> Construct the composite of the morphisms, i.e. $mrph_1 \\circ\\, mrph2 \\circ\\, \\cdots\\, \\circ\\, mrph_k$. Of course, the target of $mrph_i$ must be the same as the source of $mrph_{i-1}$, otherwise the system signals an error. The new instance inherits its source slot from $mrph_k$ and its target slot from $mrph_1$. This function optimizes the compositions in which appear zero morphisms or identity morphisms.\n",
"\n",
"`(add mrph1 mrph2 &optional strt)` *[Method]*\n",
"`(add` *mrph1 mrph2* `&optional` *strt*`)` *[Method]*\n",
"> Construct a morphism, sum of the morphisms $mrph_1$ and $mrph_2$. The result of the mapping of the morphism sum is the sum of the results of the mappings of the morphisms. The respective definitions of the morphisms $mrph_1$ and $mrph_2$ must be coherent, in particular they must have the same source, target and degree. The user may impose its strategy, otherwise it is defined in the program according to the respective strategy of the arguments.\n",
"\n",
"`(i-add mrph1 mrph2 ... mrphk)` *[Macro]*\n",
"`(i-add` *mrph1 mrph2 ... mrphk*`)` *[Macro]*\n",
"> Construct a morphism, sum of the morphism $mrph_1$, $mrph_2$, $\\ldots$, $mrph_k$. The result of the mapping of the morphism sum is the sum of the results of the mappings of the $mrph_i$. The respective definitions of $mrph_i$ must be coherent, in particular they must have the same source, target and degree. The macro `i-add` has an undefinite number of arguments. With one argument, the macro returns that argument.\n",
"\n",
"`(sbtr mrph1 mrph2 &optional strt)` *[Method]*\n",
"`(sbtr` *mrph1 mrph2* `&optional` *strt*`)` *[Method]*\n",
"> Construct a morphism, difference of the morphisms $mrph_1$ and $mrph_2$. The conditions of validity are similar to those of the method `add`.\n",
"\n",
"`(i-sbtr mrph1 mrph2 ... mrphk)` *[Macro]*\n",
"`(i-sbtr` *mrph1 mrph2 ... mrphk*`)` *[Macro]*\n",
"> Construct a morphism, difference of the morphisms $mrph_1$, $mrph_2$, $\\ldots$, $mrph_k$, in the sense $mrph_1 - mrph_2 - \\cdots - mrph_k$. The conditions of validity are similar to those of the method `add` for the morphisms. The macro `i-sbtr` must have at least 2 arguments.\n",
"\n",
"`(change-sorc-trgt mrph &key sorc trgt)` *[Function]*\n",
"> Build from the morphism `mrph` a new morphism inheriting from `mrph` the slots `:degr` (degree), `:intr` (mapping) and `:strt` (strategy). The source and target slots of this new morphism are given by the key parameters `sorc` and `trgt`. If any key parameter is omitted, the corresponding slot is inherited from `mrph` (default value).\n",
"`(change-sorc-trgt` *mrph* `&key` *sorc trgt*`)` *[Function]*\n",
"> Build from the morphism `mrph` a new morphism inheriting from `mrph` the slots `:degr` (degree), `:intr` (mapping) and `:strt` (strategy). The source and target slots of this new morphism are given by the key parameters `sorc` and `trgt`. If any key parameter is omitted, the corresponding slot is inherited from `mrph` (default value). This function is most often used when the old and new sources and/or targets are chain complexes with the same underlying graded modules, but with different differentials. Important to have this tool when handling perturbations of differentials.\n",
"\n",
"`(dstr-change-sorc-trgt mrph &key sorc trgt)` *[Function]*\n",
"`(dstr-change-sorc-trgt` *mrph* `&key` *sorc trgt*`)` *[Function]*\n",
"> Modify **destructively** the morphism `mrph`. The source and target slots of the first argument are replaced respectively by the key parameters `sorc` and `trgt`.\n",
"\n",
"`(add chcm perturbation &optional strt)` *[Method]*\n",
"> Create from the chain complex `chcm` a new chain complex inheriting from `chcm` the slots `cmpr` and `basis`. The boundary morphism attached to this new chain complex is the sum of the boundary morphism $d$ of `chcm` (slot `dffr`) and a perturbation morphism $\\delta$ represented by the `MORPHISM` instance `perturbation`. Of course, the new boundary operator must verify $(d+\\delta)\\circ (d+\\delta)=0$. The user will note that this method does not create a morphism but a chain complex.\n",
"`(add` *chcm perturbation* `&optional` *strt*`)` *[Method]*\n",
"> Create from the chain complex `chcm` a new chain complex inheriting from `chcm` the slots `cmpr` and `basis`. The boundary morphism attached to this new chain complex is the sum of the boundary morphism $d$ of `chcm` (slot `dffr`) and a perturbation morphism $\\delta$ represented by the `MORPHISM` instance `perturbation`. Of course, the new boundary operator must satisfy $(d+\\delta)\\circ (d+\\delta)=0$.\n",
"\n",
"#### Examples\n",
"\n",
@ -1351,7 +1345,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us compose `upper-shift` and the differential in both ways. Recall that the differential is a morphism structure, and may be obtained from the `dffr` slot of the chain complex `ccn` by the reader accessor function `dffr1`. This morphism has been built by `build-chcm` from the lisp function `ccn-boundary`. One can see that the operators do not commute."
"Let us compose `upper-shift` and the differential in both ways. Recall that the differential is a morphism structure, and may be obtained from the `dffr` slot of the chain complex `ccn` by the reader accessor function `dffr1`. This morphism has been built by `build-chcm` from the lisp function `ccn-boundary`. One can see that the operators do not commute; in other words, `upper-shift` is not a chain complex morphism."
]
},
{
@ -1445,16 +1439,16 @@
"### Accessing `Kenzo` objects\n",
"\n",
"Up to now, we have seen two kinds of `Kenzo` objects stored in specific lists and retrievable by a number, namely the chain complexes and the morphisms. The retrieval functions are specific to the object: `chcm` for a chain complex, `mrph` for a morphism. The same scheme will be applied for others kinds of objects: reductions, homotopy equivalences, \n",
"coalgebras, algebras, simplicial morphism, etc, each having its specific list. But, in fact the numbering is independent\n",
"coalgebras, algebras, simplicial morphisms, etc., each having its specific list. But, in fact the numbering is independent\n",
"of the type of the object and is incremented each time an object is created. Three general functions are at the disposal of the user to get information about the `n`-th `Kenzo` object: `K`, `KD`, and `KD2`. They may be useful for debugging purpose.\n",
"\n",
"`(k n)` *[Function]*\n",
"`(k` *n*`)` *[Function]*\n",
"> Get the `n`-th `Kenzo` object.\n",
"\n",
"`(kd n)` *[Function]*\n",
"`(kd` *n*`)` *[Function]*\n",
"> Give the type of the `Kenzo` objet number `n` and print the comment list (slot `:orgn`) of the object.\n",
"\n",
"`(kd2 n)` *[Function]*\n",
"`(kd2` *n*`)` *[Function]*\n",
"> Give the type of the `Kenzo` objet number `n`, print the comment list (slot `:orgn`) of the object and recursively, give the same kind of informations about all the `Kenzo` objects of the same type in relation with this `n`-th object. Return the list of numbers of all those objects. See in the following example, the case of composition of morphisms.\n",
"\n",
"#### Examples"