23 KiB
23 KiB
In [1]:
(ql:quickload :kenzo)
To load "kenzo":
Load 1 ASDF system:
kenzo
; Loading "kenzo"
Out[1]:
(:KENZO)
In [2]:
(cat:cat-init)
---done---
Out[2]:
NIL
2.2.1 Representation of a reduction¶
In [3]:
(documentation 'cat:reduction 'type)
Out[3]:
"----------------------------------------------------------------[class-doc]
REDUCTION
Slots: (tcc bcc f g h idnm orgn)
Instances of this class represent reductions of chain complexes.
^ h s ^ f ° g = 1
C -----> C C
^ h ° d + d ° h = 1 - g ° f
| | ^
f | | g C
| | f ° h = 0
v h ° g = 0
C h ° h = 0
The class has 7 slots:
^
1. TCC, an object of type CHAIN-COMPLEX, the top chain complex C. Its reader
function is TCC1.
2. BCC, an object of type CHAIN-COMPLEX, the bottom chain complex C. Its reader
function is BCC1.
3. F, an object of type MORPHISM representing the morphism f. Its reader
function is F1.
4. G, an object of type MORPHISM representing the morphism g. Its reader
function is G1.
5. H, an object of type MORPHISM representing the morphism h. Its reader
function is H1.
9. IDNM, an integer, a system-generated identifier for this object. Its reader
function is IDNM.
10. ORGN, a list containg a comment indicating the origin of the object. The
should be unique (per session), since it is used in the implementation for
caching purposes. Its reader function is ORGN.
-----------------------------------------------------------------------------"
2.2.2 The function $\texttt{build-rdct}$¶
In [4]:
(documentation 'cat:build-rdct 'function)
Out[4]:
"--------------------------------------------------------------[function-doc]
BUILD-RDCT
Args: (&key f g h orgn)
Returns an instance of the class REDUCTION. The keyword arguments are as
follows:
:F F, an object of type morphism representing the morphism f of a reduction
:G G, an object of type morphism representing the morphism g of a reduction
:H H, an object of type morphism representing the morphism h of a reduction
:ORGN ORGN, a list containing a relevant and carefully chosen comment about
the origin of the chain complex. This comment should be unique
for a Kenzo session (between calls of CAT-INIT), as it is used
for caching purposes.
Use this function instead of creating instances via the standard constructor
MAKE-INSTANCE.
------------------------------------------------------------------------------"
2.2.4 Verification functions¶
In [5]:
(documentation 'cat:pre-check-rdct 'function)
Out[5]:
"--------------------------------------------------------------[function-doc]
PRE-CHECK-RDCT
Args: (rdct)
Assigns the following Lisp global variables, using morphisms from RDCT and the
differentials and identity morphisms of the underlying chain complexes,
according to these formulas:
*TDD* = d ° d
^ ^
C C
*BDD* = d ° d
C C
*ID-FG* = Id - f ° g
C
*ID-GF-DH-HD* = Id - g ° f - (d ° h + h ° d )
^ ^ ^
C C C
*HH* = h ° h
*FH* = f ° h
*HG* = h ° g
*DF-FD* = d ° f - f ° d
C ^
C
*DG-GD* = d ° g - g ° d
^ C
C
------------------------------------------------------------------------------"
In [6]:
(documentation 'cat:check-rdct-no-wait 'function)
Out[6]:
"--------------------------------------------------------------[function-doc] CHECK-RDCT-NO-WAIT Args: () Maps combinations of the top and bottom chain complexes of a reduction using the morphisms created by PRE-CHECK-RDCT. Having no parameters, the combinations must be provided via the Lisp global variables *TC* and *BC*, for the top and bottom chain complex, respectively. If the morphisms are coherent, the result of each mapping is a null combination. ------------------------------------------------------------------------------"
Example¶
In [7]:
(defun cdelta (dmns) (cat:build-chcm :cmpr #'cat:l-cmpr :basis :locally-effective :bsgn '(0) :intr-dffr #'(lambda (degr gmsm) (cat:make-cmbn :degr (1- degr) :list (do ((rslt cat:+empty-list+ (cons (cons sign (append (subseq gmsm 0 nark) (subseq gmsm (1+ nark)))) rslt)) (sign 1 (- sign)) (nark 0 (1+ nark))) ((> nark degr) rslt)))) :strt :gnrt :orgn `(locally effective version of C_* delta ,dmns)))
Out[7]:
CDELTA
In [8]:
(defun make-f (tdmns bdmns) (cat:build-mrph :sorc (cdelta tdmns) :trgt (cdelta bdmns) :degr 0 :intr #'(lambda (degr gmsm) (let ((pos (position-if #'(lambda (vertex) (>= vertex bdmns)) gmsm))) (if pos (if (< pos degr) (cat:zero-cmbn degr) (cat:cmbn degr 1 (nconc (butlast gmsm) (list bdmns)))) (cat:cmbn degr 1 gmsm)))) :strt :gnrt :orgn `(projection delta ,tdmns => delta ,bdmns)))
Out[8]:
MAKE-F
In [9]:
(defun make-g (tdmns bdmns) (cat:build-mrph :sorc (cdelta bdmns) :trgt (cdelta tdmns) :degr 0 :intr #'identity :strt :cmbn :orgn `(injection delta ,bdmns => delta ,tdmns)))
Out[9]:
MAKE-G
In [10]:
(defun make-h (tdmns bdmns) (cat:build-mrph :sorc (cdelta tdmns) :trgt (cdelta tdmns) :degr +1 :intr #'(lambda (degr gmsm) (let ((pos (position-if #'(lambda (vertex) (>= vertex bdmns)) gmsm))) (if pos (if (member bdmns gmsm) (cat:zero-cmbn (1+ degr)) (cat:cmbn (1+ degr) (cat:-1-expt-n pos) (append (subseq gmsm 0 pos) (list bdmns) (subseq gmsm pos)))) (cat:zero-cmbn (1+ degr))))) :strt :gnrt :orgn `(homotopy for delta ,tdmns => ,bdmns)))
Out[10]:
MAKE-H
In [11]:
(defun make-rdct (tdmns bdmns) (let ((rdct (cat:build-rdct :f (make-f tdmns bdmns) :g (make-g tdmns bdmns) :h (make-h tdmns bdmns) :orgn `(reduction delta ,tdmns ,bdmns)))) rdct))
Out[11]:
MAKE-RDCT
In [12]:
(cat:def rdct (make-rdct 6 3))
Out[12]:
[K8 Reduction K1 => K3]
In [13]:
(cat:orgn rdct)
Out[13]:
(REDUCTION DELTA 6 3)
In [14]:
(cat:pre-check-rdct rdct)
---done---
Out[14]:
NIL
In [15]:
(setf cat:*tc* (cat:cmbn 2 1 '(0 1 2) 10 '(1 2 3) 100 '(1 2 4) 1000 '(2 3 4)))
Out[15]:
----------------------------------------------------------------------{CMBN 2}
<1 * (0 1 2)>
<10 * (1 2 3)>
<100 * (1 2 4)>
<1000 * (2 3 4)>
------------------------------------------------------------------------------
In [16]:
(setf cat:*bc* (cat:cmbn 3 4 '(0 1 2 3)))
Out[16]:
----------------------------------------------------------------------{CMBN 3}
<4 * (0 1 2 3)>
------------------------------------------------------------------------------
In [17]:
(cat:check-rdct-no-wait)
*TC* =>
----------------------------------------------------------------------{CMBN 2}
<1 * (0 1 2)>
<10 * (1 2 3)>
<100 * (1 2 4)>
<1000 * (2 3 4)>
------------------------------------------------------------------------------
*BC* =>
----------------------------------------------------------------------{CMBN 3}
<4 * (0 1 2 3)>
------------------------------------------------------------------------------
Checking *TDD* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *BDD* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Checking *DF-FD* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Checking *DG-GD* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *ID-FG* = 0
Result:
----------------------------------------------------------------------{CMBN 3}
------------------------------------------------------------------------------
Checking *ID-GF-DH-HD* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *HH* = 0
Result:
----------------------------------------------------------------------{CMBN 4}
------------------------------------------------------------------------------
Checking *FH* = 0
Result:
----------------------------------------------------------------------{CMBN 3}
------------------------------------------------------------------------------
Checking *HG* = 0
Result:
----------------------------------------------------------------------{CMBN 4}
------------------------------------------------------------------------------
---done---
Out[17]:
NIL
In [18]:
(cat:def trdct (make-rdct 6 4))
Out[18]:
[K35 Reduction K1 => K30]
In [19]:
(cat:def brdct (make-rdct 4 3))
Out[19]:
[K39 Reduction K30 => K3]
In [20]:
(setf rdct (cat:cmps brdct trdct))
Out[20]:
[K45 Reduction K1 => K3]
In [21]:
(cat:pre-check-rdct rdct)
---done---
Out[21]:
NIL
In [22]:
(cat:check-rdct-no-wait)
*TC* =>
----------------------------------------------------------------------{CMBN 2}
<1 * (0 1 2)>
<10 * (1 2 3)>
<100 * (1 2 4)>
<1000 * (2 3 4)>
------------------------------------------------------------------------------
*BC* =>
----------------------------------------------------------------------{CMBN 3}
<4 * (0 1 2 3)>
------------------------------------------------------------------------------
Checking *TDD* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *BDD* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Checking *DF-FD* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Checking *DG-GD* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *ID-FG* = 0
Result:
----------------------------------------------------------------------{CMBN 3}
------------------------------------------------------------------------------
Checking *ID-GF-DH-HD* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *HH* = 0
Result:
----------------------------------------------------------------------{CMBN 4}
------------------------------------------------------------------------------
Checking *FH* = 0
Result:
----------------------------------------------------------------------{CMBN 3}
------------------------------------------------------------------------------
Checking *HG* = 0
Result:
----------------------------------------------------------------------{CMBN 4}
------------------------------------------------------------------------------
---done---
Out[22]:
NIL