38 KiB
(ql:quickload "kenzo")
(use-package "CAT")
(cat-init)
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\index{simplicial sets}
$$\partial_i^q: K^q \longrightarrow K^{q-1},\quad q>0,\quad i=0,\ldots , q,$$
$$\eta_i^q: K^q \longrightarrow K^{q+1},\quad q\ge 0,\quad i=0,\ldots ,q,$$
subject to the relations
$$ \begin{align} \partial_i^{q-1}\partial_j^q & = & \partial_{j-1}^{q-1}\partial_i^q, & i < j \cr \eta_i^{q+1}\eta_j^q & = & \eta_j^{q+1}\eta_{i-1}^q, & i > j \cr \partial_i^{q+1}\eta_j^q & = & \eta_{j-1}^{q-1}\partial_i^q, & i < j \cr \partial_i^{q+1}\eta_i^q & = & \partial_{i+1}^{q+1}\eta_i^q & Identity, \cr \partial_i^{q+1}\eta_j^q & = & \eta_j^{q-1}\partial_{i-1}^q, & i > j+1.\cr \end{align} $$
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):
$$ \begin{eqnarray*} \partial_i \lbrace v_0, v_1, \ldots,v_i,\ldots, v_q \rbrace & = & \lbrace v_0, v_1, \ldots,{\bf v_{i-1}}, {\bf v_{i+1}},\ldots, v_q \rbrace, \\ \eta_i \lbrace v_0, v_1, \ldots,v_i,\ldots, v_q \rbrace & = & \lbrace v_0, v_1, \ldots,{\bf v_i},{\bf v_i},\ldots, v_q \rbrace. \end{eqnarray*} $$
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))
(dgop-ext-int '(4 0))
(dgop-int-ext 63)
(dgop-ext-int '(2 2))
(def asigma (absm 0 :sigma))
(def asigma-prime (absm (dgop-ext-int '(3 1)) :sigma))
(absm-p asigma-prime)
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)))
(dgop deg-6)
(gmsm deg-6)
Representation of a simplicial set¶
A simplicial set is implemented as an instance of the class SIMPLICIAL-SET, subclass of the class COALGEBRA.
(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:
#'(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 0 3)
(delta-inj 1 3)
(delta-inj 3 3)
(def face-delta-n
'(lambda(i dmn gsm)¶
(declare(ignore dmn))
(absm 0 (append (subseq gsm 0 i)
(subseq gsm (1+ i))) )))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)))
(def delta4 (delta-n 4))
(bspn delta4)
(basis delta4 0)
(basis delta4 3)
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)))
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)))
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))
(def delta3 (delta-n 3))
(degenerate-p (absm 0 '(1 2 3)))
(degenerate-p (absm 5 '(0 1 2 3 4)))
(basis delta3 2)
(basis delta2 0)
The set of simplices in any dimension for the simplicial set $\Delta^\mathbb{N}$ is infinite:
(basis delta-infty 1)
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))
(face delta-infty 2 4 (absm 0 '(0 1 2 3 4)))
(face delta-infty 2 5 (absm (dgop-ext-int '(1)) '(0 1 2 3 4)))
(face delta-infty 2 5 (absm (dgop-ext-int '(3)) '(0 1 2 3 4)))
(bspn delta-infty)
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))
(? delta-infty 4 '(0 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)
(basis diabolo-ss 2)
(? diabolo-ss 2 (first *))
(? diabolo-ss *)
(dgnl diabolo-ss **)
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 {\bf 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) )))
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)
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)) ))
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)) ))
(dotimes (i 3) (chcm-homology cylinss i))
(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)) ))
(dotimes (i 3) (chcm-homology moebiuss i))
(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)) ))
(dotimes (i 3) (chcm-homology toruss i))
(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)) ))
(dotimes (i 3) (chcm-homology bottless i))
(cl-jupyter-user:png-from-file "ppss.png")
(def ppss (build-finite-ss '( * 1 e (* *) 2 t (e * e)) ))
(dotimes (i 3) (chcm-homology ppss i))
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)) ))
(dotimes (i 3) (chcm-homology duncess i))
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.