353 KiB
Simplicial Sets¶
Introduction¶
A simplicial set (Hilton & Wylie, Homology theory, Cambridge University Press, 1967) $K$ is a union $K={\displaystyle{\bigcup_{q\ge 0}{K^q}}}$, where the $K^q$ are disjoints sets, together with functions
$$\partial_i^q: K^q \longrightarrow K^{q-1},\quad q>0,\quad i=0,\ldots , q,$$
$$\eta_i^q: K^q \longrightarrow K^{q+1},\quad q\ge 0,\quad i=0,\ldots ,q,$$
subject to the relations
$$ \begin{aligned} \partial_i^{q-1}\partial_j^q & = & \partial_{j-1}^{q-1}\partial_i^q, & i < j \cr \eta_i^{q+1}\eta_j^q & = & \eta_j^{q+1}\eta_{i-1}^q, & i > j \cr \partial_i^{q+1}\eta_j^q & = & \eta_{j-1}^{q-1}\partial_i^q, & i < j \cr \partial_i^{q+1}\eta_i^q & = & \partial_{i+1}^{q+1}\eta_i^q & Identity, \cr \partial_i^{q+1}\eta_j^q & = & \eta_j^{q-1}\partial_{i-1}^q, & i > j+1.\cr \end{aligned} $$
An element of $K^q$ is an (abstract) $q$-simplex of $K$ and the functions $\partial$ and $\eta$ are respectively the face operators and the degeneracy operators. Formally, their action on a simplex is very simple. For instance, if a $q$--simplex is an ordered set of vertices, like $\lbrace v_0, v_1, \ldots,v_i,\ldots, v_q \rbrace$, the rules are the following (we omit the indication of the dimension of the simplex):
$$ \begin{aligned} \partial_i \lbrace v_0, v_1, \ldots,v_i,\ldots, v_q \rbrace & = & \lbrace v_0, v_1, \ldots,{\bf v_{i-1}}, {\bf v_{i+1}},\ldots, v_q \rbrace, \\ \eta_i \lbrace v_0, v_1, \ldots,v_i,\ldots, v_q \rbrace & = & \lbrace v_0, v_1, \ldots,{\bf v_i},{\bf v_i},\ldots, v_q \rbrace. \end{aligned} $$
The operators $\partial_i^q$ will be used hereafter to define the boundary operator $d^q$ for the $q$-component of the associated chain complex:
$$d^q = \sum_{i=0}^q { (-1)^i \partial_i^q}.$$
The image of a simplex under some $\eta$ is called degenerate, because it is not directly implied in the realization of the simplicial set.
Example¶
Let us take the small example diabolo from the chapter 1. For a better understanding, it is convenient to use a list representation for the simplices. So the set $K^0$ of 0-simplices (vertices) is:
$$\lbrace (0), (1),\ldots (5) \rbrace.$$
The set $K^1$ of 1-simplices contains the set of regular simplices:
$$\lbrace (0\ 1), (0\ 2), (1\ 2), (2\ 3), (3\ 4), (3\ 5), (4\ 5) \rbrace,$$
but also the degenerate 1-simplices:
$$\lbrace (0\ 0), (1\ 1), (2\ 2),\ldots,(5 \ 5) \rbrace.$$
The set $K^2$ of 2-simplices contains the regular singleton:
$$\lbrace ( 3\ 4\ 5) \rbrace$$
but also the degenerate 2-simplices:
$$\lbrace (0\ 0\ 0), \ldots, (5\ 5\ 5), (0\ 0\ 1), (0\ 0\ 2), (1\ 1\ 2),\ldots, (4\ 5\ 5)\rbrace.$$
The sets $K^q$ ($q >2$) contain only degenerate simplices. With this naming of the elements of a $q$-simplex, the action of $\partial$ and $\eta$ is now clear. For instance
$$\partial_0 (0\ 1) = (1),\ \partial_1 (0\ 1) = (0),$$
$$\eta_0 (0\ 1) = (0\ 0\ 1),\ \eta_1 (1\ 2)= (1\ 2\ 2),\ \eta_2(3\ 4\ 5)=(3\ 4\ 5\ 5). $$
By the way, we see that a simplex is non-degenerate if the list is strictly increasing.
Notion of abstract simplex in this Lisp implementation¶
In this implementation, on account of the essential following property: any simplex may be expressed in an unique way as a possibly iterated degeneracy of a non--degenerate simplex, we call an abstract simplex, (in short absm), a pair consisting of:
- a (possibly iterated) degeneracy operator,
- a "geometric" simplex, i.e. a non--degenerate simplex. For instance, if $\sigma$ is a non-degenerate simplex, and $\sigma'$ is the degenerate simplex $\eta_3\eta_1\sigma$, the corresponding abstract simplices are respectively $\lbrace \emptyset, \sigma \rbrace$ and $\lbrace \eta_3\eta_1, \sigma \rbrace$.
An abstract simplex is represented internally in the system by the following lisp object: $$\texttt{(:absm}\quad\mathit{dgop}\quad\texttt{.}\quad\mathit{gmsm}\texttt{)}$$ where,
- dgop is a non-negative integer coding a strictly decreasing integer list. This strictly decreasing list of integers represents a sequence of $\eta$ operators and is coded as a unique integer with the following convention: a number $i$ in the sequence is the $i+1$--th binary bit of a machine word. The null list is coded as the number 0 whereas the list
(0), i.e. the $\eta_0$ operator, is coded as the integer1, etc. Two functions are provided to help the user for the transformation in both directions. - gmsm is a geometric simplex, i.e. any kind lisp object modelizing a non-degenerate simplex. For practical reason, a type
GMSM, (any kind of lisp object), has been defined.
The simplest constructor is the macro (absm (dgop-ext-int dgop-list) gmsm), where dgop-list is a strictly decreasing integer list. A type ABSM has been defined. The associated printing method prints such an object under the form:
$$\texttt{<AbSm}\quad\mathit{ext-dgop}\quad\mathit{gmsm}{\texttt >}$$
where ext-dgop shows clearly the sequence of the operators $\eta_i$ (see the examples). The accessor functions for the components are the macros dgop and gmsm.
Simple functions for abstract simplices¶
dgop-ext-int ext-dgop [Function]
Code on an integer the valid list representing the degeneracy operator ext-dgop.
dgop-int-ext dgop [Function]
Give the list representing a degeneracy operator from the integer dgop.
absm dgop gmsm [Macro]
Create an abstract simplex, using directly the coded representation, dgop, of the degeneracy operator.
dgop absm [Macro]
Get the integer code of the internal representation of the degeneracy operator of the abstract simplex absm.
gmsm absm [Macro]
Get the basic non-degenerate simplex part of the abstract simplex
absm.
absm-p object [Predicate]
Test if object is an abstract simplex.
Examples¶
Let us suppose that :sigma "is" a geometric simplex, we may construct the abstract simplices of the beginning of the section.
(dgop-ext-int '(0))
1
(dgop-ext-int '(4 0))
17
(dgop-int-ext 63)
(5 4 3 2 1 0)
(dgop-ext-int '(2 2))
SIMPLE-ERROR: In DGOP-EXT-INT, the external dgop (2 2) is not decreasing.
NIL
(def asigma (absm 0 :sigma))
<AbSm - SIGMA>
(def asigma-prime (absm (dgop-ext-int '(3 1)) :sigma))
<AbSm 3-1 SIGMA>
(absm-p asigma-prime)
T
The following command gives the sixth degeneracy of the vertex 0 represented as (0).
(def deg-6 (absm (dgop-ext-int '(5 4 3 2 1 0)) '(0)))
<AbSm 5-4-3-2-1-0 (0)>
(dgop deg-6)
63
(gmsm deg-6)
(0)
Representation of a simplicial set¶
A simplicial set is implemented as an instance of the class SIMPLICIAL-SET, subclass of the class COALGEBRA.
commonlisp
(DEFCLASS SIMPLICIAL-SET (coalgebra)
((face :type face :initarg :face :reader face1)))
This class inherits also from the class CHAIN-COMPLEX and has one slot of its own:
face, a lisp function computing any face of any geometric simplex of the simplicial set. This is a function with 3 arguments: a face index (a non--negative integer,indx), a simplex dimension (a positive integer,dmns) and a geometric simplex (gmsm), the $indx$-th face of which is looked for. Whatever the face simplex is, degenerate or not, the simplex returned by this function must be an abstract simplex, i.e. anABSMobject.
A printing method has been associated to the class SIMPLICIAL-SET and the external representation of an instance is a string like [Kn Simplicial-Set], where $n$ is the number plate of the Kenzo object.
The function build-smst¶
To facilitate the construction of instances of the class SIMPLICIAL-SET, the software Kenzo provides the function build-smst defined with keyword parameters:
build-smst
:cmprcmpr:basisbasis:bspnbspn:faceface
:face*face$$*:intr-bndrintr-bndr:bndr-strtbndr-strt
:intr-dgnlintr-dgnl:dgnl-strtdgnl-strt:orgnorgn
The returned value is an object of type SIMPLICIAL-SET. This function assigns an identification number idnm to the Kenzo object instance in a sequential way and pushes this object on the list of already created simplicial sets instances, *smst-list*.
The keywords parameters are:
- cmpr, a comparison function for generators.
- basis, if the simplicial set is effective, this argument must be the lisp function returning the list of non-degenerate simplices of the simplicial set, in a given dimension; if the simplicial set is locally effective, this argument must be the keyword
:locally-effective. - bspn, the lisp representation of the base point (an item of type
GMSM). This is in fact the slotbsgnof the associated chain complex. - face, a lisp function for the face operators.
- $face*$, a lisp function for the face operators, returning the symbol
:degenerateif the corresponding face is degenerate. - intr-bndr, a lisp function for the differential of the associated chain complex.
- bndr-strt, the strategy (
:cmbnor:gnrt) for the function intr-bndr. - intr-dgnl, a lisp function for the coproduct of the coalgebra.
- dngl-strt, the strategy (
:cmbnor:gnrt) for the function intr-dgnl. - org, the comment list, adequately chosen.
The three arguments $face*$, intr-bndr and intr-dgnl are not mandatory. Only the function face is necessary to define precisely the simplicial set. From the function face, we know that we may construct:
- The differential for the associated chain complex, according to the formula: $$d^q = \sum_{i=0}^q { (-1)^i \partial_i^q}.$$
- The coproduct of the underlying coalgebra (the Alexander-Whitney diagonal, $\Delta$) according to the formula: $$\Delta(\sigma)= \sum_{i=0}^{n}{{\tilde{\delta}}^{(i)}\sigma \otimes \delta_0^{(i)} \sigma},$$ where the power $(i)$ denotes the $i$-th composition of the operator and $\tilde{\delta}$ is the last face operator, so that $${\tilde{\delta}}^{(i)}=\delta_{n-i+1}\circ \cdots \circ \delta_{n-1}\circ \delta_n.$$
But, as the differential ignore degenerate faces, it may be interesting for the user to provide a face function, weaker than the function face but sufficient and more efficient to compute the boundary of a simplex. It is the raison d'etre of the parameter $face*$ which may be used as an alternative to face. In both cases, the strategy is set to :gnrt by the system. It may also arrive that the user knows a particularly simple form for the differential, he may then use the parameter intr-bndr. In the computation of differentials in the associated chain complex, this function will be used instead of the canonical differential computed from the face function. In this last case, the user is required to provide the strategy. Same remark for the parameter intr-dgnl.
Examples¶
As first example, we give the construction of the simplicial set $\Delta^n$ corresponding to the
standard $n$-simplex. This is given only as example. The program Kenzo provides a function
for the same purpose which will be described later. For every simplex, we shall use a list representation as in our first example diabolo. The various keyword arguments for the call to build-smst, are:
- bspn, the list
(0), i.e. the representation of the vertex 0. - cmpr, the lisp function
#'l-cmpr, since this function is adequate to compare the lists representing the simplices. - basis, the lisp function returning the list of all possible non-degenerate simplices in some dimension. Here, this is the combinatory function returning the set of all the lists of $p+1$ objects taken among $n+1$. Such a function is for instance the function (
delta-injp n) whose definition is given hereafter. So, the:basiskeyword argument is simply: ``lisp
'(lambda(dmn) (delta-inj dmn n))¶
``.
Note that here, n is a global parameter.
4. face, the lisp function for the face operators (face-delta-n). It consists simply in removing the $i$-th element of the list describing a simplex ($i$ starting at 0).
5. org will be a comment.
(defun delta-inj (m n) (declare (type fixnum m n)) (cond ((> m n) +empty-list+) ((zerop m) (mapcar #'list (<a-b> 0 n))) (t (mapcan #'(lambda (list) (declare (type list list)) (mapcar #'(lambda (k) (declare (type fixnum k)) (cons k list)) (<a-b< 0 (first list)))) (delta-inj (1- m) n)))))
DELTA-INJ
(delta-inj 0 3)
((0) (1) (2) (3))
(delta-inj 1 3)
((0 1) (0 2) (1 2) (0 3) (1 3) (2 3))
(delta-inj 3 3)
((0 1 2 3))
(def face-delta-n #'(lambda (i dmn gsm) (declare (ignore dmn)) (absm 0 (append (subseq gsm 0 i) (subseq gsm (1+ i))) )))
SIMPLE-STYLE-WARNING:
#<SB-INT:SIMPLE-STYLE-WARNING "undefined ~(~A~): ~S" {10057A8B93}>
#<FUNCTION (LAMBDA (I DMN GSM) :IN "/root/cl-jupyter-master/cl-jupyter.lisp") {1005798B3B}>
To create $\Delta^n$ for any order $n$, we embed the call of build-smst in a function having $n$ as parameter.
(defun delta-n(n) (declare (type fixnum n)) (build-smst :cmpr #'l-cmpr :basis #'(lambda(dmn)(delta-inj dmn n)) :bspn '(0) :face face-delta-n :orgn `(delta-n ,n)))
DELTA-N
(def delta4 (delta-n 4))
[K1 Simplicial-Set]
(bspn delta4)
(0)
(basis delta4 0)
((0) (1) (2) (3) (4))
(basis delta4 3)
((0 1 2 3) (0 1 2 4) (0 1 3 4) (0 2 3 4) (1 2 3 4))
A second example is the contruction of the simplicial complex $\Delta^\mathbb{N}$ freely generated by the positive integers. The construction of the SIMPLICIAL-SET instance to implement this simplicial set is quite similar to that of $\Delta^n$, but in this case the :basis keyword argument is the keyword :locally-effective, since the sets $K^q$ are infinite. Whence, the definition of $\Delta^\mathbb{N}$:
(def delta-infty (build-smst :cmpr #'l-cmpr :basis :locally-effective :bspn '(0) :face face-delta-n :orgn `(delta-infinity)))
[K6 Simplicial-Set]
A third example is the construction of the simplicial set corresponding to diabolo. This simplicial set is a sub-complex of $\Delta^2$. The :face function is exactly the same as in the examples above and the :basis function returns the sub-simplices of diabolo, by enumeration.
(def diabolo-ss (build-smst :cmpr #'l-cmpr :basis #'(lambda (dmn) (case dmn (0 '((0)(1)(2)(3)(4)(5))) (1 '((0 1)(0 2)(1 2)(2 3)(3 4)(3 5)(4 5))) (2 '((3 4 5))))) :bspn '(0) :face face-delta-n :orgn '(simplicial set for diabolo)))
[K11 Simplicial-Set]
A first set of helpful functions on simplicial sets¶
We give now a first set of functions, methods or macros acting on simplicial set instances.
cat-init [Function]
Clear in particular
*smst-list*, the list of user created simplicial sets and reset the global counter to 1.
smst n [Function]
Retrieve in the list
*smst-list*the simplicial set instance whose the Kenzo identification is $n$. If it does not exist, returnNIL.
cmpr smst absm1 absm2 [Macro]
Compare the abstract simplices absm1 and absm2 with the comparison function associated with the simplicial set smst.
bspn smst [Macro]
Return the base point of the simplicial set smst.
bndr smst &rest [Macro]
Identical to the macro
dffr(see the chain complex chapter). For simplicial sets, it is more traditional to use the term boundary.
dgnl smst &rest [Macro]
Identical to the macro
cprd(see the coalgebra chapter). For simplicial sets, this recalls the Alexander-Witney diagonal.
face smst i k gmsm-or-absm [Macro]
Versatile macro to apply the face operator $\partial_i^k$ on the simplex gmsm-or-absm of the simplicial set smst. The name of the simplex parameter shows clearly that it may be either a geometric simplex or an abstract simplex. With only one argument (smst), return the function face. In any other case, we recall that this application returns always an object of type
ABSM.
degenerate-p absm [Macro]
Return
tif the abstract simplex asm is degenerate,nilotherwise.
non-degenerate-p absm [Macro]
Return
tif the abstract simplex asm is non-degenerate,nilotherwise.
check-smst smst dmns1 &optional dmns2 (1+ dmns1) [Function]
Return
tif the fundamental relations between the face operators $\partial_i^k$ with $k=dmns1,\ldots, dmns2-1$, applied on the effective simplicial set smst, are satisfied, otherwisenil. If the optional parameter dmns2 is omitted, the test is done only for $k=dmns1$.
Examples¶
After having created instances of the standard simplicial sets $\Delta^2$ and $\Delta^3$ ($\Delta^\mathbb{N}$ has been defined previously), let us test some simple functions. Note that, since the class of simplicial sets is a sub-class of the class of coalgebras, itself subclass of chain complexes, the functions defined on these classes are available.
(def delta2 (delta-n 2))
[K16 Simplicial-Set]
(def delta3 (delta-n 3))
[K21 Simplicial-Set]
(degenerate-p (absm 0 '(1 2 3)))
NIL
(degenerate-p (absm 5 '(0 1 2 3 4)))
T
(basis delta3 2)
((0 1 2) (0 1 3) (0 2 3) (1 2 3))
(basis delta2 0)
((0) (1) (2))
The set of simplices in any dimension for the simplicial set $\Delta^\mathbb{N}$ is infinite:
(basis delta-infty 1)
SIMPLE-ERROR: The object [K6 Simplicial-Set] is locally-effective.
NIL
Let us test the face function. We see that, though delta-infty is locally effective, nevertheless, we may work with its simplices:
(face delta-infty 2 4 '(0 1 2 3 4))
<AbSm - (0 1 3 4)>
(face delta-infty 2 4 (absm 0 '(0 1 2 3 4)))
<AbSm - (0 1 3 4)>
(face delta-infty 2 5 (absm (dgop-ext-int '(1)) '(0 1 2 3 4)))
<AbSm - (0 1 2 3 4)>
(face delta-infty 2 5 (absm (dgop-ext-int '(3)) '(0 1 2 3 4)))
<AbSm 2 (0 1 3 4)>
(bspn delta-infty)
(0)
The two following statement show that delta-infty may be considered also as a coalgebra and a chain complex.
(cprd delta-infty 4 '(0 1 2 3 4))
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr (0) (0 1 2 3 4)>>
<1 * <TnPr (0 1) (1 2 3 4)>>
<1 * <TnPr (0 1 2) (2 3 4)>>
<1 * <TnPr (0 1 2 3) (3 4)>>
<1 * <TnPr (0 1 2 3 4) (4)>>
------------------------------------------------------------------------------
(? delta-infty 4 '(0 1 2 3 4))
----------------------------------------------------------------------{CMBN 3}
<1 * (0 1 2 3)>
<-1 * (0 1 2 4)>
<1 * (0 1 3 4)>
<-1 * (0 2 3 4)>
<1 * (1 2 3 4)>
------------------------------------------------------------------------------
The simplicial set diabolo-ss, has also all the properties of a coalgebra and of a chain complex.
(basis diabolo-ss 1)
((0 1) (0 2) (1 2) (2 3) (3 4) (3 5) (4 5))
(basis diabolo-ss 2)
((3 4 5))
(? diabolo-ss 2 (first *))
----------------------------------------------------------------------{CMBN 1}
<1 * (3 4)>
<-1 * (3 5)>
<1 * (4 5)>
------------------------------------------------------------------------------
(? diabolo-ss *)
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
(dgnl diabolo-ss **)
----------------------------------------------------------------------{CMBN 1}
<1 * <TnPr (3) (3 4)>>
<-1 * <TnPr (3) (3 5)>>
<1 * <TnPr (4) (4 5)>>
<1 * <TnPr (3 4) (4)>>
<-1 * <TnPr (3 5) (5)>>
<1 * <TnPr (4 5) (5)>>
------------------------------------------------------------------------------
The function build-finite-ss¶
The function build-finite-ss is a powerful function simplifying the creation of a finite simplicial set. From the lisp point of view, it accepts only one argument, the structured list of the simplices and their faces. In this list, every non--degenerate simplex must be coded as a symbol given by the user. The structure of the list is the following:
- 0 (in fact, optional)
- $<$Sequence of the vertices, the first item being the base point$>$,
- 1
- $<$Sequence of description of simplices in dimension 1$>$,
- 2
- $<$Sequence of description of simplices in dimension 2$>$,
- ... ... ... ... ... ...
- n
- $<$Sequence of description of simplices in dimension n$>$.
where $<$Sequence of description of simplices in dimension k$>$ is a sequence of the following items:
- A symbol, the symbolic name of the simplex.
- A list of faces of this simplex, describing the faces of the simplex. A face can be described in two ways:
- a symbol, the symbolic name of a non-degenerate simplex composing itself the face.
- or a list of the form ($i_k\ i_{k-1}\ \ldots \ i_1 \ i_0$
symbol) meaning that the degeneracy $\eta_{i_k}\eta_{i_{k-1}}\ldots\eta_{i_1}\eta_{i_0}$ is to be applied to the non-degenerate simplex of namesymbolto produce the actual (possibly degenerate) face. If the simplex face has dimension 0 (a vertex), the name of this vertex is sufficient in the face description because the program computes itself the unique possible degeneracy of the vertex, i.e. $\eta_{k-2}\ldots\eta_0$.
Examples¶
Let us give some examples to clarify the use of build-finite-ss. In the first example, we use build-finite-ss to create a simplicial complex, corresponding to a triangulation of the torus. As the faces are never degenerated, the description of the faces is of simplified form.
(cl-jupyter-user:png-from-file "torus2.png")
The vertices will be named v0, v1, ..., v6, v0 being the base point. The edges
will be named e01, e02, ..., e56, the list of faces of the edge $e_{ij}$ being
$(v_j\ v_i)$. The triangles will be named t013, t015, ..., t456, the list of faces of
the triangle $t_{ijk}$ being $(e_{jk}\ e_{ik}\ e_{ij})$. With this convention of describing the simplices, the call to build-finite-ss is in simplified form:
(def torus2 (build-finite-ss '(v0 v1 v2 v3 v4 v5 v6 1 e01 (v1 v0) e02 (v2 v0) e03 (v3 v0) e04 (v4 v0) e05 (v5 v0) e06 (v6 v0) e12 (v2 v1) e13 (v3 v1) e14 (v4 v1) e15 (v5 v1) e16 (v6 v1) e23 (v3 v2) e24 (v4 v2) e25 (v5 v2) e26 (v6 v2) e34 (v4 v3) e35 (v5 v3) e36 (v6 v3) e45 (v5 v4) e46 (v6 v4) e56 (v6 v5) 2 t013 (e13 e03 e01) t015 (e15 e05 e01) t024 (e24 e04 e02) t026 (e26 e06 e02) t036 (e36 e06 e03) t045 (e45 e05 e04) t125 (e25 e15 e12) t126 (e26 e16 e12) t134 (e34 e14 e13) t146 (e46 e16 e14) t234 (e34 e24 e23) t235 (e35 e25 e23) t356 (e56 e36 e35) t456 (e56 e46 e45) )))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {10068618AB}>
2. BSGN: V0
3. DFFR: [K27 Morphism (degree -1): K26 -> K26]
4. GRMD: [K26 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 26
7. ORGN: (BUILD-FINITE-SS (V0 V1 V2 V3 V4 V5 V6 1 E01 (V1 V0) ...))
8. CPRD: [K30 Morphism (degree 0): K26 -> K28]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {10068618CB}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
[K26 Simplicial-Set]
After verification of the coherence of the description of the faces, the function build-finite-ss calls adequately the function build-smst to create an instance of the class SIMPLICIAL SET subclass of the class CHAIN COMPLEX. So that, it is easy for the user to obtain the homology groups of the torus, for instance in dimension 1:
(chcm-homology torus2 1)
Computing boundary-matrix in dimension 1. Rank of the source-module : 21. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/21 (dimension 1) : E01 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/21 (dimension 1) : E02 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/21 (dimension 1) : E03 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 4/21 (dimension 1) : E04 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 5/21 (dimension 1) : E05 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 6/21 (dimension 1) : E06 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 7/21 (dimension 1) : E12 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 8/21 (dimension 1) : E13 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 9/21 (dimension 1) : E14 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 10/21 (dimension 1) : E15 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 11/21 (dimension 1) : E16 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 12/21 (dimension 1) : E23 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 13/21 (dimension 1) : E24 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 14/21 (dimension 1) : E25 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 15/21 (dimension 1) : E26 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 16/21 (dimension 1) : E34 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 17/21 (dimension 1) : E35 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 18/21 (dimension 1) : E36 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 19/21 (dimension 1) : E45 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 20/21 (dimension 1) : E46 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 21/21 (dimension 1) : E56 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 14. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/14 (dimension 2) : T013 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/14 (dimension 2) : T015 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/14 (dimension 2) : T024 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 4/14 (dimension 2) : T026 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 5/14 (dimension 2) : T036 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 6/14 (dimension 2) : T045 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 7/14 (dimension 2) : T125 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 8/14 (dimension 2) : T126 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 9/14 (dimension 2) : T134 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 10/14 (dimension 2) : T146 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 11/14 (dimension 2) : T234 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 12/14 (dimension 2) : T235 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 13/14 (dimension 2) : T356 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 14/14 (dimension 2) : T456 End of computing. Homology in dimension 1 : Component Z Component Z ---done---
NIL
The function build-finite-ss calls internally some verification functions mentioned above to verify the coherence of the description of the simplicial set and gives an error message if the description of the simplicial set is incorrect. For instance:
(def mm (build-finite-ss '(s0 s1 s2 s3 1 0-1 (s1 s0) 1-2 (s2 s1) 2-3 (s3 s2) 2 0-1-2 (2-3 1-2 0-1)) ))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {10069D1BCB}>
2. BSGN: S0
3. DFFR: [K32 Morphism (degree -1): K31 -> K31]
4. GRMD: [K31 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 31
7. ORGN: (BUILD-FINITE-SS (S0 S1 S2 S3 1 |0-1| (S1 S0) |1-2| (S2 S1) |2-3| ...))
8. CPRD: [K35 Morphism (degree 0): K31 -> K33]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {10069D1BEB}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
SIMPLE-ERROR: Noncoherent boundary operators detected by CHECK-FACES : Simplex => 0-1-2 del_0 o del_0 => <AbSm - S3> del_0 o del_1 => <AbSm - S2>
NIL
In the following examples, we show the power of build-finite-ss for the description of classical surfaces as simplicial sets, and in each case we verify the well known homology groups of those surfaces. The examples of the torus or the dunce-hat must be compared, as to the simplicity of the definition, to the respective ones in the previous page and in the Homology chapter.
(cl-jupyter-user:png-from-file "cylinss.png")
(def cylinss (build-finite-ss '(v0 v1 1 e0 (v1 v0) e1 (v1 v1) e2 (v0 v0) e3 (v1 v0) 2 t0 (e1 e3 e0) t1 (e0 e3 e2)) ))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {1006B8FB7B}>
2. BSGN: V0
3. DFFR: [K37 Morphism (degree -1): K36 -> K36]
4. GRMD: [K36 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 36
7. ORGN: (BUILD-FINITE-SS (V0 V1 1 E0 (V1 V0) E1 (V1 V1) E2 (V0 V0) E3 ...))
8. CPRD: [K40 Morphism (degree 0): K36 -> K38]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {1006B8FB9B}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
[K36 Simplicial-Set]
(dotimes (i 3) (chcm-homology cylinss i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 0) : V0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 0) : V1 End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 4. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/4 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/4 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/4 (dimension 1) : E2 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 4/4 (dimension 1) : E3 End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 4. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/4 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/4 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/4 (dimension 1) : E2 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 4/4 (dimension 1) : E3 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Homology in dimension 1 : Component Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : ---done---
NIL
(cl-jupyter-user:png-from-file "moebiuss.png")
(def moebiuss (build-finite-ss '(v0 v1 1 e0 (v1 v0) e1 (v0 v1) e2 (v1 v0) e3 (v0 v0) 2 t0 (e1 e3 e0) t1 (e0 e2 e3)) ))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {1006E0FBAB}>
2. BSGN: V0
3. DFFR: [K42 Morphism (degree -1): K41 -> K41]
4. GRMD: [K41 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 41
7. ORGN: (BUILD-FINITE-SS (V0 V1 1 E0 (V1 V0) E1 (V0 V1) E2 (V1 V0) E3 ...))
8. CPRD: [K45 Morphism (degree 0): K41 -> K43]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {1006E0FBCB}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
[K41 Simplicial-Set]
(dotimes (i 3) (chcm-homology moebiuss i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 0) : V0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 0) : V1 End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 4. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/4 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/4 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/4 (dimension 1) : E2 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 4/4 (dimension 1) : E3 End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 4. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/4 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/4 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/4 (dimension 1) : E2 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 4/4 (dimension 1) : E3 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Homology in dimension 1 : Component Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : ---done---
NIL
(cl-jupyter-user:png-from-file "toruss.png")
(def toruss (build-finite-ss '(v0 1 e0 (v0 v0) e1 (v0 v0) e2 (v0 v0) 2 t0 (e1 e2 e0) t1 (e0 e2 e1)) ))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {100707F18B}>
2. BSGN: V0
3. DFFR: [K47 Morphism (degree -1): K46 -> K46]
4. GRMD: [K46 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 46
7. ORGN: (BUILD-FINITE-SS (V0 1 E0 (V0 V0) E1 (V0 V0) E2 (V0 V0) 2 T0 ...))
8. CPRD: [K50 Morphism (degree 0): K46 -> K48]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {100707F1AB}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
[K46 Simplicial-Set]
(dotimes (i 3) (chcm-homology toruss i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/1 (dimension 0) : V0 End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 3. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/3 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/3 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/3 (dimension 1) : E2 End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 3. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/3 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/3 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/3 (dimension 1) : E2 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Homology in dimension 1 : Component Z Component Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : Component Z ---done---
NIL
(cl-jupyter-user:png-from-file "bottless.png")
(def bottless (build-finite-ss '(v0 1 e0 (v0 v0) e1 (v0 v0) e2 (v0 v0) 2 t0 (e1 e2 e0) t1 (e0 e1 e2)) ))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {10072DF6BB}>
2. BSGN: V0
3. DFFR: [K52 Morphism (degree -1): K51 -> K51]
4. GRMD: [K51 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 51
7. ORGN: (BUILD-FINITE-SS (V0 1 E0 (V0 V0) E1 (V0 V0) E2 (V0 V0) 2 T0 ...))
8. CPRD: [K55 Morphism (degree 0): K51 -> K53]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {10072DF6DB}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
[K51 Simplicial-Set]
(dotimes (i 3) (chcm-homology bottless i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/1 (dimension 0) : V0 End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 3. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/3 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/3 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/3 (dimension 1) : E2 End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 3. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/3 (dimension 1) : E0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/3 (dimension 1) : E1 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 3/3 (dimension 1) : E2 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Homology in dimension 1 : Component Z/2Z Component Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/2 (dimension 2) : T0 End of computing. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 2/2 (dimension 2) : T1 End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : ---done---
NIL
(cl-jupyter-user:png-from-file "ppss.png")
(def ppss (build-finite-ss '( * 1 e (* *) 2 t (e * e)) ))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {100753E22B}>
2. BSGN: *
3. DFFR: [K57 Morphism (degree -1): K56 -> K56]
4. GRMD: [K56 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 56
7. ORGN: (BUILD-FINITE-SS (* 1 E (* *) 2 T (E * E)))
8. CPRD: [K60 Morphism (degree 0): K56 -> K58]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {100753E24B}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
[K56 Simplicial-Set]
(dotimes (i 3) (chcm-homology ppss i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/1 (dimension 0) : * End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/1 (dimension 1) : E End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/1 (dimension 1) : E End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/1 (dimension 2) : T End of computing. Homology in dimension 1 : Component Z/2Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 21s. Computing the boundary of the generator 1/1 (dimension 2) : T End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : ---done---
NIL
The user will note that the list of faces for the 2-simplex t, is in simplified form. In particular, as the face 1 of t is the 0-degeneracy of the base point "$*$", it is sufficient to code the face 1 of tby "*" instead of the complete form (0 *).
(cl-jupyter-user:png-from-file "duncess.png")
(def duncess (build-finite-ss '( * 1 e (* *) 2 t (e e e)) ))
The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.
0. CMPR: #<FUNCTION S-CMPR>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {100778E04B}>
2. BSGN: *
3. DFFR: [K62 Morphism (degree -1): K61 -> K61]
4. GRMD: [K61 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 61
7. ORGN: (BUILD-FINITE-SS (* 1 E (* *) 2 T (E E E)))
8. CPRD: [K65 Morphism (degree 0): K61 -> K63]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {100778E06B}>
>
Checking the 0-simplices...
Checking the 1-simplices...
Checking the 2-simplices...
[K61 Simplicial-Set]
(dotimes (i 3) (chcm-homology duncess i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 0) : * End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 1) : E End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 1) : E End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 2) : T End of computing. Homology in dimension 1 : ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 2) : T End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : ---done---
NIL
Special simplicial sets¶
The system provides useful functions to create interesting simplicial sets of constant usage. In particular, the user is advised from now to use the following version of the standard simplex implemented in Kenzo and not the one given previously for simplicity reason.
The standard simplex¶
delta dmns [Function]
Create the standard simplex $\Delta^n$ with vertices $0,1,\ldots,n$, where dmns is the parameter for $n$. This simplicial set is so important in the applications that the implementor has decided to code the simplices in a way similar to the coding of the degeneracy operators. An increasing sequence of non-negative integers describing a simplex, is coded on a binary integer with the following convention: a number $i$ representing the vertex $i$ is the $(i+1)$--th binary bit of a machine word. The base point $0$ is the bit 1 in position 1 of the word. This representation is very efficient for saving memory space but somehow awkward to read.
delta-infinity [Function]
Create the locally effective standard simplex $\Delta^\mathbb{N}$ freely generated by the positive integers.
deltab [Function]
Create the locally effective reduced simplicial set $\bar \Delta$, built from $\Delta^\mathbb{N}$ by identifying all the vertices to the base point.
deltab2 [Function]
Create the locally effective 1-reduced simplicial set $\Delta_2^\mathbb{N}$, obtained from the above $\bar \Delta$, by identification of all the edges with the base point. This is the efficient version of the coalgebra that is used as example in the cobar chapter.
dlop-ext-int ext-dlop [Function]
Code on an integer the valid list (increasing order) representing the simplex ext-dlop of the standard simplex.
dlop-int-ext dgop [Function]
Give the list representing a simplex of the standard simplex from the integer dgop.
vertex-i absm i [Function]
Give the $i$-th vertex of the (coded, degenerate or not) abstract simplex
absmbelonging to a simplicial set of the $\Delta$ family.
absm-ext-int vlist [Function]
Create a valid abstract simplex of the $\Delta$ family from the simplex vlist written as a non-decreasing list of non-negative integers (i.e coding of any degenerate or non--degenerate simplex of the $\Delta$ family).
absm-int-ext absm [Function]
From an internal coded form of an abstract simplex of the $\Delta$ family, create the non-decreasing list of non--negative integers representing the canonical external form of such a simplex.
soft-delta dmns [Function]
Create another version of the standard simplex $\Delta^n$ designed for a better clarity in the printing of the results at testing time. More precisely, the simplices are represented internally by a list of the form
(:deltbinary-code). With this representation, the system is able to recognize a coded simplex and to print it in a readable form. For the user, the only requirement is to write a simplex coded $n$ as(dn), wheredis the macro building the internal representation (see the examples). Attention: the above conversion functions do not work with this representation.
soft-delta-infinity [Function]
Examples¶
(def d3 (delta 3))
[K66 Simplicial-Set]
The simplices $2$, $4$ and $8$ are the coded representation of the vertices $1$, $2$ and $3$:
(cmpr d3 2 4)
:LESS
(cmpr d3 4 4)
:EQUAL
(cmpr d3 8 4)
:GREATER
The list returned by the following statement getting the basis of $\Delta^3$ in dimension 2, namely (7 11 13 14) is in fact the coded list of the following simplices ((0 1 2) (0 1 3) (0 2 3) (1 2 3)).
(basis d3 2)
(7 11 13 14)
In the following statement, the integer $21$ represents the simplex in dimension $2$, (0 1 4), $1$ represents the operator $\partial_1$ and the face is therefore (0 4) represented as the integer $17$.
(face d3 1 2 21)
<AbSm - 17>
We may obtain the same result, using:
(face d3 1 2 (dlop-ext-int '(0 1 4)))
<AbSm - 17>
Let us test the conversion functions.
(vertex-i (absm 0 1) 0)
0
(vertex-i (absm 1 1) 0)
0
(vertex-i (absm 1 1) 1)
0
(vertex-i (absm 0 7) 2)
2
(absm-ext-int '(0 0 0 1 2 3 3 3))
<AbSm 6-5-1-0 15>
(absm-ext-int '(0 1 1 1 2))
<AbSm 2-1 7>
(absm-int-ext (absm-ext-int '(0 0 0 1 2 3 3 3)))
(0 0 0 1 2 3 3 3)
The following call to the macro ? returns the boundary of the simplex (0 2 3) of $\Delta^3$, the integers $5$, $9$ and $12$ representing respectively the simplices (0 2), (0 3) and (2 3). The application of the coproduct dgnl to the simplex (0 1 2 3), coded $15$, is also easily interpreted:
(? d3 2 13)
----------------------------------------------------------------------{CMBN 1}
<1 * 5>
<-1 * 9>
<1 * 12>
------------------------------------------------------------------------------
(dgnl d3 3 15)
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr 1 15>>
<1 * <TnPr 3 14>>
<1 * <TnPr 7 12>>
<1 * <TnPr 15 8>>
------------------------------------------------------------------------------
We may show now some examples with the soft version of the standard simplex. Using the macro d and the function dlop-ext-int the user may work with a readable form of the simplices and the degeneracy operators.
(def d3 (soft-delta 3))
[K71 Simplicial-Set]
(cmpr d3 (d 2) (d 4))
:LESS
(basis d3 1)
((:DELT . 3) (:DELT . 5) (:DELT . 6) (:DELT . 9) (:DELT . 10) (:DELT . 12))
(dgnl d3 3 (d (dlop-ext-int '(0 1 2 3))))
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr (DELT . 1) (DELT . 15)>>
<1 * <TnPr (DELT . 3) (DELT . 14)>>
<1 * <TnPr (DELT . 7) (DELT . 12)>>
<1 * <TnPr (DELT . 15) (DELT . 8)>>
------------------------------------------------------------------------------
(face d3 1 2 (d (dlop-ext-int '(0 2 4))))
<AbSm - (DELT . 17)>
(? d3 2 (d (dlop-ext-int '(0 2 3))))
----------------------------------------------------------------------{CMBN 1}
<1 * (DELT . 5)>
<-1 * (DELT . 9)>
<1 * (DELT . 12)>
------------------------------------------------------------------------------
Spheres, Moore spaces and projective spaces¶
sphere n [Function]
Create a simplicial set, a model for the sphere of dimension $n$, ($n \geq 0$). This is a typical example where the differential is known a priori to be null, so the
:intr-bndrkeyword parameter is set to functionzero-pure-dffr. This function generates a nameSn for the unique simplex of dimension $n$ whose faces are the degeneracies of the base point labelled "*".
sphere-wedge dmns1 ... dmnsn [Function]
Create a simplicial set for the wedge of spheres. Here, the $dmns_i$ are integers, namely the dimensions of the spheres to be wedged. The differential is null. In the representation created by the software, the 0-simplex (base point) is labelled "
*" and in dimension $p$ the simplexes are labelledSp-1, ...,Sp-s, where s is the number of spheres of dimension $p$ in the wedge. (See the example).
moore p n [Function]
Construct a simplicial set, a model for ${\rm Moore}(\mathbb{Z}/{p\mathbb{Z}},n)$. The integers $p$ and $n$ must satisfy the conditions $p>1,\ n> 2p-4$, otherwise the result is undefined. ${\rm Moore}(\mathbb{Z}/{p\mathbb{Z}},n)$ is a space, whose the only non-null homology groups are $H_0=\mathbb{Z}$ and $H_n=\mathbb{Z}/{p\mathbb{Z}}$. A Moore space has only three non-degenerate simplices, namely in dimension $0$, $n$ and $n+1$. A number $p$ of faces of the $(n+1)$-simplex are identified with the $n$-simplex, the others faces being contracted on the base point. In the representation created by the software, the 0-simplex (base point), the $n$-simplex and the $(n+1)$-simplex are respectively labelled "
*",M$n$ andN$n'$, where $n'=n+1$.
R-proj-space &optional k l [Function]
If $k=1$ or omitted, build a simplicial set model of $K(\mathbb{Z}_2,1)= P^\infty\mathbb{R}$. In dimension $n$, this simplicial set has only one non-degenerate simplex, namely the integer $n$. The faces of this non-degenerate simplex $n$ are given by the following formulas: $\partial_0 n = \partial_n n = n-1$ and for $i \not= 0$ and $i \not=n$, $\partial_i n= \eta_{i-1} (n-2)$. If $k >1$, build an analogous simplicial set but with no simplices in dimensions $1 \leq m < k$. If in addition to $k$ the argument $l$, ($l \geq k$), is provided, build an analogous simplicial set but with no simplices in dimensions $m \geq l$.
Examples¶
Let us define first, an auxiliary function show-structure with 2 arguments ss and dmn, to show the structure (i.e. generators and faces) of the simplicial set ss, from the dimension 0 up to the dimension dmn included.
commonlisp
(defun show-structure (ss dmn)
(dotimes (i (1+ dmn))
(format t "~2%Dimension = ~D :" i)
(case i
(0 (format t "~2%~8TVertices : ~8T~A" (basis ss 0)))
(otherwise
(dolist (s (basis ss i))
(format t "~2%~8TSimplex : ~A~2%~16TFaces : ~A"
s (mapcar #'(lambda (j) (face ss j i s))
(<a-b> 0 i)))))
)))
In these elementary examples, we show the structure of some simplicial sets built from the functions above. Let us begin with $S^2$.
(def s2 (sphere 2))
[K76 Simplicial-Set]
(bspn s2)
*
Applying the function basis, we see that the only non-null simplices are in dimension 0 and 2.
(dotimes (i 4) (print (basis s2 i)))
(*) NIL (S2) NIL
NIL
In dimension 2, the 3 faces of the simplicial set are all the degeneracy $\eta_0$ of the base point.
(mapcar #'(lambda(i)(face s2 i 2 's2)) '(0 1 2))
(<AbSm 0 *> <AbSm 0 *> <AbSm 0 *>)
(face s2 5 7 (absm (dgop-ext-int '(5 3 1 0)) 's2))
<AbSm 3-1-0 S2>
(face s2 2 7 (absm (dgop-ext-int '(5 3 1 0)) 's2))
<AbSm 4-2-0 S2>
The differential of the simplex s2 in dimension 2 is of course the null combination of degree 1:
(? s2 2 's2)
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Now, let us see the stucture of $S^3$, using our auxiliary function show-structure.
(def s3 (sphere 3))
[K81 Simplicial-Set]
(show-structure s3 3)
Dimension = 0:
Vertices : (*)
Dimension = 1:
Dimension = 2:
Dimension = 3:
Simplex : S3
Faces : (<AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *>)
NIL
The space ${\rm Moore}(2,1)$ is the projective plane:
(def p2 (moore 2 1))
[K86 Simplicial-Set]
(show-structure p2 2)
Dimension = 0:
Vertices : (*)
Dimension = 1:
Simplex : M1
Faces : (<AbSm - *> <AbSm - *>)
Dimension = 2:
Simplex : N2
Faces : (<AbSm - M1> <AbSm 0 *> <AbSm - M1>)
NIL
In the following example, note the identification of $p$ faces of the $(n+1)$-simplex with the $n$-simplex.
(def sp2r (moore 2 2))
[K91 Simplicial-Set]
(show-structure sp2r 3)
Dimension = 0:
Vertices : (*)
Dimension = 1:
Dimension = 2:
Simplex : M2
Faces : (<AbSm 0 *> <AbSm 0 *> <AbSm 0 *>)
Dimension = 3:
Simplex : N3
Faces : (<AbSm - M2> <AbSm 1-0 *> <AbSm - M2> <AbSm 1-0 *>)
NIL
Let us see an example of a wedge of spheres:
(def w (sphere-wedge 3 2 3))
[K96 Simplicial-Set]
(show-structure w 5)
Dimension = 0:
Vertices : (*)
Dimension = 1:
Dimension = 2:
Simplex : S2-1
Faces : (<AbSm 0 *> <AbSm 0 *> <AbSm 0 *>)
Dimension = 3:
Simplex : S3-1
Faces : (<AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *>)
Simplex : S3-2
Faces : (<AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *>)
Dimension = 4:
Dimension = 5:
NIL
(cmpr w 's3-1 's3-2)
:LESS
(face w 2 3 's3-1)
<AbSm 1-0 *>
(? w 3 's3-2)
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Let us show now some examples with the simplicial sets generated by the function R-proj-space.
(def p1 (R-proj-space))
[K101 Simplicial-Set]
(dotimes (i 7)(print(basis p1 i)))
(0) (1) (2) (3) (4) (5) (6)
NIL
(show-structure p1 5)
Dimension = 0:
Vertices : (0)
Dimension = 1:
Simplex : 1
Faces : (<AbSm - 0> <AbSm - 0>)
Dimension = 2:
Simplex : 2
Faces : (<AbSm - 1> <AbSm 0 0> <AbSm - 1>)
Dimension = 3:
Simplex : 3
Faces : (<AbSm - 2> <AbSm 0 1> <AbSm 1 1> <AbSm - 2>)
Dimension = 4:
Simplex : 4
Faces : (<AbSm - 3> <AbSm 0 2> <AbSm 1 2> <AbSm 2 2> <AbSm - 3>)
Dimension = 5:
Simplex : 5
Faces : (<AbSm - 4> <AbSm 0 3> <AbSm 1 3> <AbSm 2 3> <AbSm 3 3> <AbSm - 4>)
NIL
(dotimes (i 5)(chcm-homology p1 i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 0) : 0 End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 1) : 1 End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 1) : 1 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 2) : 2 End of computing. Homology in dimension 1 : Component Z/2Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 2) : 2 End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 3) : 3 End of computing. Homology in dimension 2 : ---done--- Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 3) : 3 End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 4) : 4 End of computing. Homology in dimension 3 : Component Z/2Z ---done--- Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 4) : 4 End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 5) : 5 End of computing. Homology in dimension 4 : ---done---
NIL
(def p2 (R-proj-space 2))
[K106 Simplicial-Set]
(show-structure p2 5)
Dimension = 0:
Vertices : (0)
Dimension = 1:
Simplex : 1
Faces : (<AbSm - 0> <AbSm - 0>)
Dimension = 2:
Dimension = 3:
Dimension = 4:
Dimension = 5:
NIL
(dotimes (i 5)(chcm-homology p2 i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 0) : 0 End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 1) : 1 End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 1) : 1 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Homology in dimension 1 : Component Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : ---done--- Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Computing boundary-matrix in dimension 4. Rank of the source-module : 0. Homology in dimension 3 : ---done--- Computing boundary-matrix in dimension 4. Rank of the source-module : 0. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Homology in dimension 4 : ---done---
NIL
(def pr3 (R-proj-space 3))
[K111 Simplicial-Set]
(show-structure pr3 4)
Dimension = 0:
Vertices : (0)
Dimension = 1:
Simplex : 1
Faces : (<AbSm - 0> <AbSm - 0>)
Dimension = 2:
Simplex : 2
Faces : (<AbSm - 1> <AbSm 0 0> <AbSm - 1>)
Dimension = 3:
Dimension = 4:
NIL
(dotimes (i 7) (print (? pr3 i i)))
----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
----------------------------------------------------------------------{CMBN 1}
<2 * 1>
------------------------------------------------------------------------------
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
----------------------------------------------------------------------{CMBN 3}
<2 * 3>
------------------------------------------------------------------------------
----------------------------------------------------------------------{CMBN 4}
------------------------------------------------------------------------------
----------------------------------------------------------------------{CMBN 5}
<2 * 5>
------------------------------------------------------------------------------
NIL
Let us use now the parameter l, allowing the truncation in the upper dimensions.
(def p12 (R-proj-space 1 2))
SIMPLE-ERROR: The assertion (AND (TYPEP K (QUOTE FIXNUM)) (PLUSP K) (OR (EQ CAT::L :INFINITY) (AND (TYPEP CAT::L #) (<= K CAT::L)))) failed.
NIL
(show-structure p12 2)
TYPE-ERROR: #<TYPE-ERROR expected-type: CAT::SIMPLICIAL-SET datum: NIL>
NIL
Setting $k=l$ does not creates something amazing!
(def p22 (R-proj-space 2 2))
[K116 Simplicial-Set]
(show-structure p22 2)
Dimension = 0:
Vertices : (0)
Dimension = 1:
Dimension = 2:
NIL
(def p47 (R-proj-space 4 7))
SIMPLE-ERROR: The assertion (AND (TYPEP K (QUOTE FIXNUM)) (PLUSP K) (OR (EQ CAT::L :INFINITY) (AND (TYPEP CAT::L #) (<= K CAT::L)))) failed.
NIL
(show-structure p47 8)
TYPE-ERROR: #<TYPE-ERROR expected-type: CAT::SIMPLICIAL-SET datum: NIL>
NIL
Cartesian product of simplicials sets¶
Let $X$ and $Y$ be two simplicial sets, the construction of $X\times Y$ is based on the very definition $(X\times Y)_n =X_n\times Y_n$ where $X_n$, $Y_n$ and $(X\times Y)_n$ are the possibly degenerate simplices of $X$, $Y$ and $X\times Y$ respectively. A simplex of the product $X\times Y$ is characterized by its projections on the factors $X$ and $Y$.
The non-degenerate simplices of $X\times Y$ are represented internally in the system by a lisp object of the form:
(:crpr (dgop1.gmsm1).(dgop2.gmsm2))
where,
dgop1is an integer representing a coded degeneracy operator.gmsm1is a non-degenerate simplex of $X$, to which is applied the degeneracy operatordgop1.dgop2is an integer representing a coded degeneracy operator.gmsm2is a non-degenerate simplex of $Y$, to which is applied the degeneracy operatordgop2.
This object must be a non-degenerate simplex of $X \times Y$, that is to say, the degeneracy operators dgop1 and dgop2 must not have a common $\eta_j$, and therefore their list representations must have a void intersection.
The coded representation by binary bit positions, has the same property. The corresponding type is CRPR.
To construct such an object, one may use the macro crpr. As usual, a printing method has been defined to reflect the structure of the product under the form:
<CrPrext-dgop1 gmsm1 ext-dgop2 gmsm2>
where the sequence of the operators $\eta_i$ is printed in explicit form (this is the meaning of ext-dgop).
If this sequence of $\eta_i$ is void, i.e. if the simplex gmsm is not degenerate, the symbol - is printed instead.
Functions and macros for the product of simplicial sets¶
crpr dgop1 gmsm1 dgop2 gmsm2 [Macro]
Build an object of type
CRPR, using directly the integer coding for the degeneracy operators. The arguments gmsm1 and gmsm2 are non-degenerate simplices.
crpr absm1 absm2 [Macro]
Build an object of type
CRPR, using two abstract simplices absm1 and absm2. If these abstract simplices are degenerate, the degeneracy operators must verify the condition of the definition, i.e. no common $\eta_i$.
2absm-acrpr absm1 absm2 [Function]
Build an abstract simplex, i.e. object of type
ABSM, cartesian product of both abstract simplices absm1 and absm2. In contrast with the previous function, there is no condition upon the degeneracy operators of the abstract simplices. Of course, the function2absm-acrprreturns a legal abstract simplex.
crpr-p object [Predicate]
Test if object is of type
CRPR.
dgop1 crpr [Macro]
Select the degeneracy operator dgop1 from the object
crpr.
gmsm1 crpr [Macro]
Select the geometric simplex gmsm1 from the object
crpr.
dgop2 crpr [Macro]
Select the degeneracy operator dgop2 from the object
crpr.
gmsm2 crpr [Macro]
Select the geometric simplex gmsm2 from the object
crpr.
absm1 crpr [Macro]
Build the abstract simplex
(absmdgop1 gmsm1)from crpr.
absm2 crpr [Macro]
Build the abstract simplex
(absmdgop2 gmsm2)from crpr.
crts-prdc smst1 smst2 [Function]
Build the simplicial set, cartesian product $smst1 \times smts2$.
Examples¶
In the first example, note that the coded representations of $\eta_1$ and $\eta_2$ are respectively 2 and 4, but these degeneracy operators appear clearly as 1 and 2 in the printed result. In the second statement, the integer 28 is the coded representation of the degeneracy (4 3 2).
(crpr 2 'a 4 'b)
<CrPr 1 A 2 B>
(crpr 0 '(0 1 2 3 4 5) 28 '(0 1 2))
<CrPr - (0 1 2 3 4 5) 4-3-2 (0 1 2)>
For more clarity, one may also write:
(crpr 0 '(0 1 2 3 4 5) (dgop-ext-int '(4 3 2)) '(0 1 2))
<CrPr - (0 1 2 3 4 5) 4-3-2 (0 1 2)>
In the following example, we see that both functions crpr and 2absm-acrpr return the same geometric simplex,
(crpr (absm 4 'a)(absm 3 'b))
<CrPr 2 A 1-0 B>
(2absm-acrpr (absm 4 'a)(absm 3 'b))
<AbSm - <CrPr 2 A 1-0 B>>
whereas, in the following call to crpr, the condition upon the degeneracy operator is not respected and the result is an illegal cartesian product. The function 2absm-acrpr returns the correct answer.
(crpr (absm 5 'a)(absm 3 'b))
<CrPr 2-0 A 1-0 B>
(2absm-acrpr (absm 5 'a) (absm 3 'b))
<AbSm 0 <CrPr 1 A 0 B>>
The following example is particularly instructive. The simplicial set whose the realization is homeomorphic to the square is built by using the product of two segments. A call like (delta n) builds the standard $n$-simplex. So first, we generate two copies X and Y of the standard 1-simplex and we list the abstract simplices up to dimension 2 for a better understanding of the components of the product XY.
(def X (delta 1))
[K121 Simplicial-Set]
(def Y X)
[K121 Simplicial-Set]
(cl-jupyter-user:png-from-file "ssquare.png")
In list representation (non-coded simplices), the abstract simplices of $X$ up to the dimension 2, including the degenerate ones are:
- Dimension 0:
<AbSm - (0)>, <AbSm - (1)>. - Dimension 1:
<AbSm 0 (0)>, <AbSm 0 (1)>, <AbSm - (0 1)>. - Dimension 2:
<AbSm 1-0 (0)>, <AbSm 1-0 (1)>, <AbSm 0 (0 1)>, <AbSm 1 (0 1)>.
Let us look first at the non-degenerate simplices of the simplicial set $X \times Y$. Those simplices are objects of type CRPR. In dimension 1, we see that some projections are degeneracies of vertices and, in dimension 2, the projections are all degeneracies of the 1-simplex (0 1) coded as 3. Nevertheless, all the listed simplices are geometric simplices (i.e. non-degenerate ones) of the product XY. The diagram above shows the relationship between the 3 simplicial sets.
(def XY (crts-prdc X Y))
[K126 Simplicial-Set]
(show-structure XY 2)
Dimension = 0:
Vertices : (<CrPr - 1 - 1> <CrPr - 1 - 2> <CrPr - 2 - 1> <CrPr - 2 - 2>)
Dimension = 1:
Simplex : <CrPr - 3 - 3>
Faces : (<AbSm - <CrPr - 2 - 2>> <AbSm - <CrPr - 1 - 1>>)
Simplex : <CrPr - 3 0 1>
Faces : (<AbSm - <CrPr - 2 - 1>> <AbSm - <CrPr - 1 - 1>>)
Simplex : <CrPr - 3 0 2>
Faces : (<AbSm - <CrPr - 2 - 2>> <AbSm - <CrPr - 1 - 2>>)
Simplex : <CrPr 0 1 - 3>
Faces : (<AbSm - <CrPr - 1 - 2>> <AbSm - <CrPr - 1 - 1>>)
Simplex : <CrPr 0 2 - 3>
Faces : (<AbSm - <CrPr - 2 - 2>> <AbSm - <CrPr - 2 - 1>>)
Dimension = 2:
Simplex : <CrPr 0 3 1 3>
Faces : (<AbSm - <CrPr - 3 0 2>> <AbSm - <CrPr - 3 - 3>> <AbSm - <CrPr 0 1 - 3>>)
Simplex : <CrPr 1 3 0 3>
Faces : (<AbSm - <CrPr 0 2 - 3>> <AbSm - <CrPr - 3 - 3>> <AbSm - <CrPr - 3 0 1>>)
NIL
To be more readable we may translate the coded representations of the degeneracy operators and of the simplices in the list representation:
Dimension 0 :
Vertices : (<CrPr - (0) - (0)>
<CrPr - (0) - (1)>
<CrPr - (1) - (0)>
<CrPr - (1) - (1)>)
Dimension 1 :
Simplex : <CrPr 0 (0) - (0 1)>
Faces : (<AbSm - <CrPr - (0) - (1)> >
<AbSm - <CrPr - (0) - (0)> >)
Simplex : <CrPr 0 (1) - (0 1)>
Faces : (<AbSm - <CrPr - (1) - (1)> >
<AbSm - <CrPr - (1) - (0)> >)
Simplex : <CrPr - (0 1) 0 (0)>
Faces : (<AbSm - <CrPr - (1) - (0)> >
<AbSm - <CrPr - (0) - (0)> >)
Simplex : <CrPr - (0 1) 0 (1)>
Faces : (<AbSm - <CrPr - (1) - (1)> >
<AbSm - <CrPr - (0) - (1)> >)
Simplex : <CrPr - (0 1) - (0 1)>
Faces : (<AbSm - <CrPr - (1) - (1)> >
<AbSm - <CrPr - (0) - (0)> >)
Dimension 2 :
Simplex : <CrPr 0 (0 1) 1 (0 1)>
Faces : (<AbSm - <CrPr - (0 1) 0 (1)> >
<AbSm - <CrPr - (0 1) - (0 1)> >
<AbSm - <CrPr 0 (0) - (0 1)> >)
Simplex : <CrPr 1 (0 1) 0 (0 1)>
Faces : (<AbSm - <CrPr 0 (1) - (0 1)> >
<AbSm - <CrPr - (0 1) - (0 1)> >
<AbSm - <CrPr - (0 1) 0 (0)> >)
In an analogous way, let us build the torus as the cartesian product of two 1-spheres. We recall that the base point of the sphere is named *.
(def s1xs1 (crts-prdc (sphere 1) (sphere 1)))
[K136 Simplicial-Set]
(show-structure s1xs1 2)
Dimension = 0:
Vertices : (<CrPr - * - *>)
Dimension = 1:
Simplex : <CrPr - S1 - S1>
Faces : (<AbSm - <CrPr - * - *>> <AbSm - <CrPr - * - *>>)
Simplex : <CrPr - S1 0 *>
Faces : (<AbSm - <CrPr - * - *>> <AbSm - <CrPr - * - *>>)
Simplex : <CrPr 0 * - S1>
Faces : (<AbSm - <CrPr - * - *>> <AbSm - <CrPr - * - *>>)
Dimension = 2:
Simplex : <CrPr 0 S1 1 S1>
Faces : (<AbSm - <CrPr - S1 0 *>> <AbSm - <CrPr - S1 - S1>> <AbSm - <CrPr 0 * - S1>>)
Simplex : <CrPr 1 S1 0 S1>
Faces : (<AbSm - <CrPr 0 * - S1>> <AbSm - <CrPr - S1 - S1>> <AbSm - <CrPr - S1 0 *>>)
NIL
Let us verify that $S^1\times S^1$ has the same homology groups as the torus.
(dotimes (i 3)(chcm-homology s1xs1 i))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/1 (dimension 0) : <CrPr - * - *> End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 3. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/3 (dimension 1) : <CrPr - S1 - S1> End of computing. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 2/3 (dimension 1) : <CrPr - S1 0 *> End of computing. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 3/3 (dimension 1) : <CrPr 0 * - S1> End of computing. Homology in dimension 0 : Component Z ---done--- Computing boundary-matrix in dimension 1. Rank of the source-module : 3. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/3 (dimension 1) : <CrPr - S1 - S1> End of computing. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 2/3 (dimension 1) : <CrPr - S1 0 *> End of computing. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 3/3 (dimension 1) : <CrPr 0 * - S1> End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/2 (dimension 2) : <CrPr 0 S1 1 S1> End of computing. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 2/2 (dimension 2) : <CrPr 1 S1 0 S1> End of computing. Homology in dimension 1 : Component Z Component Z ---done--- Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 1/2 (dimension 2) : <CrPr 0 S1 1 S1> End of computing. ;; Clock -> 2016-03-11, 23h 47m 22s. Computing the boundary of the generator 2/2 (dimension 2) : <CrPr 1 S1 0 S1> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : Component Z ---done---
NIL
Of course, we may generate more complex products, whose basis are composed of rather long
lists of simplices as shown below. The user will also pay attention to the fact that the composition of the macro crpr is not associative. In the following example, the simplicial sets located respectively by p1 and p2 are different objects, though being canonically isomorphic.
(def p1 (crts-prdc (sphere 2) (crts-prdc (sphere 2) (sphere 3))))
[K146 Simplicial-Set]
(def p2 (crts-prdc (crts-prdc (sphere 2) (sphere 2)) (sphere 3)))
[K156 Simplicial-Set]
(dotimes (i 8) (print (length (basis p1 i))))
1 0 3 22 138 390 480 210
NIL
Lisp files concerned in this chapter¶
simplicial-sets.lisp, delta.lisp, specials-smsts.lisp, cartesian-products.lisp.
[classes.lisp, macros.lisp, various.lisp].