Journal of Homotopy and Related Structures, vol. 2(2), 2007, pp.187–216
THIRD MAC LANE COHOMOLOGY VIA CATEGORICAL RINGS
M. JIBLADZE AND T. PIRASHVILI
(communicated by George Janelidze)
Abstract
A notion of categorical ring is introduced. Categorical rings are
proved to be classifiable by the third Mac Lane cohomology group.
Introduction
In the fifties, Saunders Mac Lane invented a cohomology theory of rings using the cubical construction introduced earlier by Eilenberg and himself to calculate stable homology
of Eilenberg-Mac Lane spaces. As shown in [9], this theory coincides with the topological
Hochschild cohomology for Eilenberg-Mac Lane ring spectra. In particular, the third dimensional cohomology group is expected to provide classification of 2-types of ring spectra. Some algebraic models for such 2-types have been constructed in [1]. In this paper
we consider one such algebraic model of different kind which in our opinion is especially
straightforwardly related to 3-cocycles in Mac Lane cohomology.
This is the notion of categorical ring—a category carrying the structure of a ring up
to some natural isomorphisms satisfying certain coherence conditions. Our axioms for the
categorical ring present a slightly modified version of the notion of Ann-category due to
Quang [10]. Axioms we use reflect defining relations of Mac Lane 3-cocycles. Our main
result is Theorem 4.4 which asserts that for any ring R and any R-bimodule B there is a
bijection
H 3 (R; B) ≈ Crext(R; B)
between the third Mac Lane cohomology group of R with coefficients in B and equivalence
classes of categorical rings R with π0 (R) = R, π1 (R) = B, and the induced bimodule
structure coinciding with the original one.
In [10], Ann-categories of a particular kind—the so called regular ones—are considered. These correspond to the ring spectra whose underlying spectrum splits into a product
of Eilenberg-Mac Lane spectra. It is shown in [10] that regular Ann-categories are classified
by the third Shukla cohomology group [11]. The latter is the Barr-Beck-Quillen cohomology group for the category of associative rings [2].
Difference between the above two cases is quite subtle. In fact, Shukla and Mac Lane
cohomologies are isomorphic up to dimension 2 (in dimensions 0 and 1 both also coincide
with the Hochschild cohomology; this coincidence extends to dimension 2 if the underlying abelian group of R is free). The third Shukla cohomology group embeds into the
Received August 31, 2006, revised March 3, 2007; published on December 27, 2007.
2000 Mathematics Subject Classification: 16E40, 18D10.
Key words and phrases: Mac Lane cohomology, topological Hochschild cohomology, categorification, categorical
rings.
c 2007, M. Jibladze and T. Pirashvili. Permission to copy for private use granted.
Journal of Homotopy and Related Structures, vol. 2(2), 2007
188
third Mac Lane cohomology group, and failure of isomorphism is measured by an explicit
obstruction furnished by the following exact sequence (see [7, 8], [5], [2]):
3
0 → HShukla
(R; B) → H 3 (R; B) → H 0 (R; 2 B).
It can be shown that in terms of categorical rings, the above map H 3 (R; B) → H 0 (R; 2 B)
sends an element of H 3 (R; B) represented by a categorical ring R to the element
{0,0}
0 → 0 + 0 −−−→ 0 + 0 → 0
of AutR (0) = B, where 0 is the neutral object with respect to the additive structure, {x, y} :
x + y → y + x is the commutativity constraint, and the isomorphisms between 0 and 0 + 0
are the canonical ones.
1.
Recollections on symmetric categorical groups
Let us begin by recalling
(1.1) Definition. A categorical group A is a groupoid equipped with a monoidal
structure—i. e. a bifunctor + : A × A → A , an object 0 ∈ A and natural isomorphisms
ha, b, ci : (a + b) + c → a + (b + c), λ(a) : 0 + a → a and ρ(a) : a + 0 → a satisfying
the Mac Lane coherence conditions—together with a choice, for each object a ∈ A , of
another object −a ∈ A and of an isomorphism ι(a) : −a + a → 0.
A braiding on a categorical group A is a collection of isomorphisms {a, b} : a + b →
b + a turning A into a braided monoidal category; a symmetric categorical group is a
braided one whose monoidal structure is symmetric, i. e. the inverse of {a, b} is equal to
{b, a} for any objects a, b.
Categorical groups are also known in the literature under the name of Picard categories.
There is much literature on them, see e. g. references in [13]. The fact that categorical
groups are classified by third group cohomology was discovered in [12]. As one more
example of relatively early fundamental work on the topic one could name e. g. [4].
We will need some specific facts and auxiliary notation concerning symmetric categorical groups. More on braided and symmetric categorical groups can be found e. g. in [6].
It is easy to see that for any monoidal functor f = (f, f+ , f0 ) : A → A 0 to a categorical group, the canonical isomorphism f0 : f (0) → 0 is determined by the rest of the
structure. Namely, f0 is equal to the composite
f (0)
λ(f (0))−1
ι(f (0))
0 + f (0)
ι(f (0))−1 +f (0)
0O
−f (0) + f (0)
O
−f (0)+f (λ(0))
(−f (0) + f (0)) + f (0)
−f (0) + f (0 + 0)
<
FF
FF
xx
FF
xx
x
FF
x
h−f (0),f (0),f (0)i
−f (0)+f+ (0,0)
xx
"
−f (0) + (f (0) + f (0))
(1.2)
Journal of Homotopy and Related Structures, vol. 2(2), 2007
189
We will use the well-known fact that a monoidal functor f : A → A 0 between categorical groups is an equivalence if and only if it induces an isomorphism on π0 and π1 . Here,
π1 of a categorical group is defined to be the automorphism group of its neutral object, and
the homomorphism f# : π1 (A ) → π1 (A 0 ) induced by a monoidal functor f assigns to
α : 0A → 0A the composite
0A 0
f0−1
/ f (0A )
f (α)
/ f (0A )
/ 0A 0 .
f0
It is equally well known that in a categorical group A , hom(x, y) has a structure of a
bitorsor under π1 (A ) for any isomorphic objects x, y of A . In more detail, hom(x, y) has
compatible left hom(y, y)- and right hom(x, x)-actions which are principal, i. e. for any
ϕ, ϕ0 : x → y there is a unique χ : x → x with ϕ0 = ϕχ (namely χ = ϕ−1 ϕ0 ) and a
unique υ : y → y with ϕ0 = υϕ (namely, υ = ϕ0 ϕ−1 ). And on the other hand for any
categorical group A the maps π1 (A ) →hom(x, x) sending α : 0 → 0 to the composite
λ(x)−1
λ(x)
α+x
x −−−−→ 0 + x −−−→ 0 + x −−−→ x
are group isomorphisms for any object x, which allows to transfer the above bitorsor structures from hom(x, x), resp. hom(y, y), to π1 (A ).
In particular, for any two parallel morphisms ϕ, ϕ0 : x → y there exists a unique α ∈
π1 (A ) making the diagram
0+x
α+ϕ
/ 0+y
ϕ0
/y
λ(x)
x
λ(y)
commute. It will be more convenient for us to depict such circumstances by a diagram of
the form
ϕ
x
α
$
: y.
ϕ0
In such cases we will also write α = ϕ0 ÷ ϕ.
Note that exchanging order of ϕ and ϕ0 introduces a sign, i. e. one has
ϕ0
ϕ
x
α
ϕ0
#
;y
⇐⇒
x
−α
$
: y;
ϕ
in the diagrams that we will encounter, this order is not specified as it can be unambiguously
recovered from the context.
We will need the following simple fact concerning this formalism.
(1.3) Proposition. Let f = (f, f+ ) : A → A 0 be a monoidal functor between categorical
Journal of Homotopy and Related Structures, vol. 2(2), 2007
190
groups. Then for any parallel arrows ϕ, ϕ0 : x → y in A one has
f (ϕ)
ϕ
x
α
#
;y
⇒
f (x)
f# (α)
(
6
f (y).
f (ϕ0 )
ϕ0
Proof. There is a commutative diagram
f# (α)+f (ϕ)
0 + f (x)
f0−1 +f (x)
f (0) + f (x)
/ 0 + f (y)
f0−1 +f (y)
f (α)+f (ϕ)
/ f (0) + f (y)
f+ (0,x)
f+ (0,y)
f (0 + x)
f (α+ϕ)
/ f (0 + y)
f (ϕ0 )
/ f (y).
f (λ(x))
f (λ(y))
f (x)
By the aforementioned uniqueness, the proposition follows.
(1.4) Corollary. For any parallel arrows ϕi , ϕ0i : xi → yi , i = 1, 2, in a braided (in
particular, symmetric) categorical group A one has
ϕ1
x1
α1
ϕ01
%
ϕ2
9 y1 ,
x2
α2
ϕ02
%
ϕ1 +ϕ2
9 y2
⇒
x1 + x2
α1 +α2
)
y5 1 + y2 .
ϕ01 +ϕ02
Proof. It is well known that for a braided category A the functor + : A × A → A
acquires a monoidal structure (in fact it is known that for any monoidal category there is
a one-to-one correspondence between braidings and monoidal functor structures on +).
Since obviously +# (α1 , α2 ) = α1 + α2 for any α1 , α2 ∈ π1 (A ), the statement follows
from (1.3).
For any four objects a, b, c, d of a symmetric categorical group A , by
a b
c d : (a + b) + (c + d) → (a + c) + (b + d)
Journal of Homotopy and Related Structures, vol. 2(2), 2007
191
will be denoted the composite canonical isomorphism in the commutative diagram
(a + b) + (c + d)
OOO
OOO
ooo
o
o
OOO
oo
o
OOO
o
o
o
−1
OOO
o
ha,b,c+di
ha+b,c,di
o
o
OO'
wooo
((a + b) + c) + d
a + (b + (c + d))
??

??

??ha,b,ci+d
a+hb,c,di−1 
??

??

?


ha,b+c,di
/ a + ((b + c) + d)
(a + (b + c)) + d
(a+{b,c})+d
a+({b,c}+d)
/ a + ((c + b) + d)
(a + (c + b)) + d
??
ha,c+b,di

??

??


?


−1
a+hc,b,di ??
??
 ha,c,bi +d


((a + c) + b) + d
a + (c + (b + d))
OOO
o
OOO
ooo
Oha+c,b,di
ha,c,b+di−1ooo
OOO
oo
OOO
ooo
OOO
o
o
O'
wooo
(a + c) + (b + d)
2.
Categorical rings
Our algebraic models for 2-types of ring spectra are certain bimonoidal categories which
we call categorical rings. They can be called “rings up to coherent isomorphisms”, in the
sense that (a) isomorphism classes of objects of a categorical ring form an associative ring;
and (b) the structure of a categorical ring on a category is an equivalence invariant, i. e.
any equivalence between a categorical ring and another category allows one to transfer the
categorical ring structure along it.
(2.1) Definition. A categorical ring is a symmetric categorical group R together with a
bifunctor (denoted by juxtaposition) R × R → R, an object 1 ∈ R, and natural isomorphisms
[r, s, t] : (rs)t → r(st)
(associativity),
λ.(r) : 1r → r, ρ.(r) : r1 → r
(left and right unitality),
[r ss01 i : r(s0 + s1 ) → rs0 + rs1
Journal of Homotopy and Related Structures, vol. 2(2), 2007
192
(left distributivity),
h rr01 s] : (r0 + r1 )s → r0 s + r1 s
(right distributivity).
It is required that the [, , ] together with λ. and ρ. constitute a monoidal structure (i. e. the
appropriate pentagonal and triangular coherence diagrams commute for it) and moreover
the following diagrams commute for all possible objects of R:
E
h
r s tt0
1
/ r(st0 + st1 ) h
r(s(t0 + t1 ))
E
<
GG
st
x
GG r st01
[r,s,t0 +t1 ]
xx
G
x
GG
xx
GG
xx
GG
x
x
G#
xx
r(st0 ) + r(st1 )
(rs)(t0 + t1 )
SSS
5
SSS
kkk
k
k
SSS
k
k
k
SSS
kkk
SSS
kkk
S
k
E
h
S
k
SSS
k [r,s,t0 ]+[r,s,t1 ]
rs tt0
)
kkk
1
(rs)t0 + (rs)t1
(rs0 + rs1 )t
NNN
8
rs
[r ss01 it qqqq
NNN h rs01 t]
q
N
q
N
NNN
qq
NNN
qqq
q
q
N&
qq
(r(s0 + s1 ))t
[r,s0 +s1 ,t]
(rs0 )t + (rs1 )t
[r,s0 ,t]+[r,s1 ,t]
r((s0 + s1 )t)
r(s0 t) + r(s1 t)
MMM
q8
MMM
qqq
q
MMM
q
qq
MMM
qqq h
E
MMM
q
s0
q
s0 t
r h s1 t]
rs
&
qq
1t
r(s0 t + s1 t)
Journal of Homotopy and Related Structures, vol. 2(2), 2007
193
h rr01 ss t]
/ (r0 s)t + (r1 s)t
(r0 s + r1 s)t
II
y<
II [r0 ,s,t]+[r1 ,s,t]
y
II
yy
y
II
y
y
II
yy
II
y
I$
yy
((r0 + r1 )s)t
r0 (st) + r1 (st)
5
RRR
RRR
jjjj
j
j
RRR
j
RRR
jjj
RRR
jjjj
j
j
R
j
RRR
[r0 +r1 ,s,t]
h rr01 s]t
)
jjjj
(r0 + r1 )(st)
h rr01 s]t
[1 rr01 i
/ 1r0 + 1r1
1(r0 + r1 )
LLL
r
LLL
rrr
r
r
LLL
r
λ.(r0 +r1 )
L&
xrrr λ.(r0 )+λ.(r1 )
r0 + r1fL
LLLρ.(r0 )+ρ.(r1 )
r8
ρ.(r0 +r1 ) rrr
LLL
r
r
r
LLL
rrr
/ r0 1 + r1 1
(r0 + r1 )1
h rr01 1]
r(s00 + s01 ) + r(s10 + s11 )
QQQ
mm6
QQQ
s10
mmm
QQQ [r ss00
m
m
10
11
m
QQQ 01 i+[r s11 i
m
m
m
Q
QQQ
mm
QQQ
mmm
Q(
mmm
r((s00 + s01 ) + (s10 + s11 ))
(rs00 + rs01 ) + (rs10 + rs11 )
E
h
00 +s01
rs
s +s
s s01
r h s00
i
10 s11
00
h rs
rs10
rs01
rs11 i
r((s00 + s10 ) + (s01 + s11 ))
(rs00 + rs10 ) + (rs01 + rs11 )
QQQ
mm6
QQQ
mmm
m
QQQ
m
QQQ
mmm
h
E
QQQ
mmm [r s00 i+[r s01 i
m
s00 +s10
m
Q
r s +s
s10
s11
QQQ
mm
01
11
(
mmm
r(s00 + s10 ) + r(s01 + s11 )
Journal of Homotopy and Related Structures, vol. 2(2), 2007
194
(r0 + r1 )(s0 + s1 )
TTTT
j
TTTT [r0 +r1 ss01 i
h rr01 s0 +s1 ] jjjjjj
TTTT
j
j
j
TTTT
jj
j
j
j
uj
)
r0 (s0 + s1 ) + r1 (s0 + s1 )
(r0 + r1 )s0 + (r0 + r1 )s1
[r0 ss01 i+[r1 ss01 i
(r0 s0 + r0 s1 ) + (r1 s0 + r1 s1 )
h rr10 ss00 rr10 ss11 i
h rr01 s0 ]+h rr01 s1 ]
/ (r0 s0 + r1 s0 ) + (r0 s1 + r1 s1 )
(r00 + r01 )s + (r10 + r11 )s
4
TTTT
r
TTTT h rr00
s + 10 s
jjjj
j
j
j
TTTT01 ] h r11 ]
j
j
j
TTTT
jj
T*
jjjj
(r00 s + r01 s) + (r10 s + r11 s)
((r00 + r01 ) + (r10 + r11 ))s
D
h rr00
10
r00 +r01 s
r10 +r11
i
r01
r11 is
s r01 s
h rr00
i
10 s r11 s
(r00 s + r10 s) + (r01 s + r11 s)
((r00 + r10 ) + (r01 + r11 ))s
TTTT
jj4
TTTT
jjjj
j
j
T
j
T
TTTT
D
i
jjjj h r00 s]+h r01 s]
r00 +r10
TT*
r11
r10
jjjj
r01 +r11 s
(r00 + r10 )s + (r01 + r11 )s
It can be shown that the notion of categorical ring is essentially the same as the one of
Ann-category from [10], in the sense that a categorical ring satisfies coherence conditions
from [10] and vice versa. Note however that the notion of regular Ann-category is strictly
stronger; it corresponds to categorical rings whose symmetry morphisms {a, a} are equal
to the identity of a for all objects a.
Morphisms of categorical rings are defined as follows:
(2.2) Definition. A 2-homomorphism f : R → R 0 is a quadruple (f, f+ , f· , f1 ) where f
is a functor from R to R 0 , f+ , f· are natural morphisms of the form
f+ (r0 , r1 ) : f (r0 ) + f (r1 ) → f (r0 + r1 ),
f· (r, s) : f (r)f (s) → f (rs)
and f1 : f (1R ) → 1R0 is a morphism such that (f, f+ , f0 ) and (f, f· , f1 ) are monoidal
functor structures with respect to the monoidal structures corresponding to + and · respec-
Journal of Homotopy and Related Structures, vol. 2(2), 2007
195
tively, with f0 as in (1.2), and moreover the diagrams
f· (r,s0 )+f· (r,s1 )
f (r)f (s0 ) + f (r)f (s1 ) / f (rs0 ) + f (rs1 )
QQQ f+ (rs0 ,rs1 )
mm6
QQQ
mmm
QQQ
m
m
QQQ
mm
m
m
Q(
mm
f (r)(f (s0 ) + f (s1 ))
f (rs0 + rs1 )
QQQ
mm6
QQQ
mmm
QQQ
m
m
QQQ
mmm
Q(
mmm f ([r s0 i)
f (r)f+ (s0 ,s1 )
s1
/ f (r(s0 + s1 ))
f (r)f (s0 + s1 )
f (r)
f (s0 )
f (s1 )
f· (r,s0 +s1 )
and
f· (r0 ,s)+f· (r1 ,s)
f (r0 )
f (s)
f (r1 )
f (r0 )f (s) + f (r1 )f (s)
m6
mmm
m
m
m
mmm
mmm
(f (r0 ) + f (r1 ))f (s)
QQQ
QQQ
QQQ
QQQ
Q(
f+ (r0 ,r1 )f (s)
f (r0 + r1 )f (s)
/ f (r0 s) + f (r1 s)
QQQ f+ (r0 s,r1 s)
QQQ
QQQ
QQQ
Q(
f (r0 s + r1 s)
m6
mmm
m
m
m
mmm
mmm f (h r0 s])
r0
/ f ((r0 + r1 )s)
f· (r0 +r1 ,s)
commute for all possible objects involved.
We will need the following fact in what follows:
(2.3) Proposition. In any categorical ring R one has
ϕ
x
α
ϕ0
rϕ
#
;y
⇒
rx
rα
&
8 ry,
rϕ0
ϕr
xr
αr
&
8 yr.
ϕ0 r
Proof.
− It follows
from the definition of categorical ring that for any r ∈ R the morphisms
r − , resp. −
− r , constitute a structure of a monoidal functor on the endofunctor r·, resp.
·r : R → R of R (with respect to the additive monoidal structure). The proposition is thus
particular case of (1.3).
3.
Third Mac Lane cohomology group
We refer to [7, 8] for the original construction of Mac Lane cohomology. Here we will
only explicitate the definition of the third cohomology group as this is all that we need.
To make expressions shorter, we will need the cross-effect notation. Recall that for a map
f : A → B between abelian groups, its first cross-effect is a map
(− | −)f : A × A → B
Journal of Homotopy and Related Structures, vol. 2(2), 2007
196
given by
(x | y)f = f (x) + f (y) − f (x + y).
(3.1) Definition. For a ring R and an R-bimodule B, the group C 3 (R; B) of Mac Lane
3-cochains of R with coefficients in B consists of quadruples (ϕ· , ϕ·+ , ϕ+· , ϕ+ ), of maps
ϕ· , ϕ·+ , ϕ+· : R3 → B
and
ϕ+ : R 4 → B
which are normalized in the sense that ϕ· , ϕ·+ and ϕ+· take zero values if one of their
arguments is zero, and moreover ϕ+ satisfies
ϕ+ ( r00 r01 ) = ϕ+ r00 r01 = ϕ+ rr01 00 = ϕ+ 00 rr01 = ϕ+ r00 r01 = 0
for all r0 , r1 ∈ R. The group structure is given by valuewise addition of functions.
The subgroup Z 3 (R; B) ⊆ C 3 (R; B) of 3-cocycles is singled out by the following
equations:
rϕ· (s, t, u) − ϕ· (rs, t, u) + ϕ· (r, st, u)
−ϕ· (r, s, tu) + ϕ· (r, s, t)u =0,
rϕ·+ (s, t0 , t1 ) − ϕ·+ (rs, t0 , t1 ) + ϕ·+ (r, st0 , st1 ) = (t0 | t1 )ϕ· (r,s,−) ,
ϕ·+ (r, s0 t, s1 t) − ϕ·+ (r, s0 , s1 )t =ϕ+· (rs0 , rs1 , t) − rϕ+· (s0 , s1 , t),
ϕ+· (r0 s, r1 s, t) − ϕ+· (r0 , r1 , st) + ϕ+· (r0 , r1 , s)t = − (r0 | r1 )ϕ· (−,s,t) ,
00
ϕ+ ( rs
rs10
rs01
rs11
) − rϕ+ ( ss00
10
s01
s11
) = ((s00 , s10 ) | (s01 , s11 ))ϕ·+ (r,−,−)
− ((s00 , s01 ) | (s10 , s11 ))ϕ·+ (r,−,−) ,
ϕ+ ( rr01 ss00 rr01 ss11
) = (r0 | r1 )ϕ·+ (−,s0 ,s1 )
− (s0 | s1 )ϕ+· (r0 ,r1 ,−) ,
s r01 s
ϕ+ ( rr00
)
10 s r11 s
−
r01
ϕ+ ( rr00
10 r11
) s = ((r00 , r10 ) | (r01 , r11 ))ϕ+· (−,−,s)
− ((r00 , r01 ) | (r10 , r11 ))ϕ+· (−,−,s) ,
(( rr000
010
r001
r011
r001
− (( rr000
100 r101
r010
+ (( rr000
100 r110
) | ( rr100
110
r101
r111
))ϕ+
)|
r011
( rr010
110 r111
))ϕ+
)|
r011
( rr001
101 r111
))ϕ+ =0.
The subgroup B 3 (R; B) ⊆ Z 3 (R; B) of 3-coboundaries consists of those quadruples
(ϕ· , ϕ·+ , ϕ+· , ϕ+ ) for which there exist maps γ· , γ+ : R2 → B such that
ϕ· (r, s, t) =rγ· (s, t) − γ· (rs, t) + γ· (r, st) − γ· (r, s)t,
ϕ·+ (r, s0 , s1 ) =rγ+ (s0 , s1 ) − γ+ (rs0 , rs1 ) + (s0 | s1 )γ· (r,−) ,
ϕ+· (r0 , r1 , s) =γ+ (r0 s, r1 s) − γ+ (r0 , r1 )s − (r0 | r1 )γ· (−,s) ,
ϕ+ ( rr00
10
r01
r11
) = ((r00 , r01 ) | (r10 , r11 ))γ+ − ((r00 , r10 ) | (r01 , r11 ))γ+
Journal of Homotopy and Related Structures, vol. 2(2), 2007
197
for all r, ... ∈ R.
Finally, we define
H 3 (R; B) := Z 3 (R; B)/B 3 (R; B).
4.
Characteristic class of a categorical ring
Suppose given a categorical ring R as in 2.1. Then the set R = π0 (R) of isomorphism
classes of objects of R is a ring, and the group B = π1 (R) of automorphisms of the zero
object of R has a canonical structure of a π0 (R)-bimodule, with rα, resp. αr, for any
r ∈ π0 (R) and any α : 0 → 0, given by
r̄·−1
r̄α
r̄·
·r̄ −1
αr̄
·r̄
0
0
0,
0 −−−
→ r̄0 −→ r̄0 −−→
resp.
0
0 −−0−→ 0r̄ −→ 0r̄ −−→
0,
where r̄ is any object from the isomorphism class r, and the morphisms r̄·0 , resp. ·r̄0 , come
from the monoidal functor structures on r̄·, resp. ·r̄ indicated in the proof of 2.3.
We are going to assign to R a cohomology class
hRi ∈ H 3 (π0 (R); π1 (R)).
For that, we arbitrarily choose an object r̄ of R in each isomorphism class r ∈ π0 (R);
moreover we arbitrarily choose morphisms
σ· (r, s) : r̄s̄ → rs
and
σ+ (r0 , r1 ) : r̄0 + r̄1 → r0 + r1 .
These morphisms give rise to several not necessarily commutative diagrams. They define
elements of π1 (R) as in section 1.
In particular, for any r, s, t ∈ π0 (R) the diagram
r̄σ· (s,t)
/ r̄st
r̄(s̄t̄)
>
@@
~
@
[r̄,s̄,t̄] ~~
~
σ· (r,st)
~
@@
~
@
~~
ϕ· (r,s,t)
(r̄s̄)t̄ P
7 rst
PPP
nnn
n
P
n
n
σ· (rs,t)
σ· (r,s)Pt̄
n
PPP
n
n
PP'
nnn
rs t̄
produces an element ϕ· (r, s, t) ∈ π1 (R) measuring deviation from its commutativity. Sim-
Journal of Homotopy and Related Structures, vol. 2(2), 2007
198
ilarly for any r, s0 , s1 ∈ π0 (R) deviation from commutativity of the diagram
r̄s̄0 + r̄s̄1
p8
p
p
s̄0
r̄
ppp
s̄1 ppp
pp
r̄(s̄0 + s̄1 )
NNN
N
r̄σ+ (s0 ,s
NN1N)
N&
r̄s0 + s1
σ· (r,s0 )+σ· (r,s1 )
/ rs + rs
0
1N
NNN
σ+ (rs0 ,rs
N 1)
NNN
&
rs0 + rs1
ppp
p
p
ppp
ppp
/ r(s0 + s1 )
ϕ·+ (r,s0 ,s1 )
σ· (r,s0 +s1 )
is measured by an element ϕ·+ (r, s0 , s1 ) ∈ π1 (R); for any r0 , r1 , s ∈ π0 (R) the diagram
r̄0 s̄ + r̄1 s̄
pp8
p
r̄ 0
p
p
s̄
r̄ 1
ppp
ppp
(r̄0 + r̄1 )s̄
NNN
N
σ+ (r0 ,rN
1 )s̄
NNN
&
r0 + r1 s̄
σ· (r0 ,s)+σ· (r1 ,s)
/ r0 s + r1 s
NNN
N
σ+ (r0 s,r
N1 s)
ϕ+· (r0 ,r1 ,s)
σ· (r0 +r1 ,s)
NNN
&
r0 s + r1 s
p
ppp
p
p
p
ppp
/ (r0 + r1 )s
gives ϕ+· (r0 , r1 , s) ∈ π1 (R); and for any r00 , r01 , r10 , r11 ∈ π0 (R) the diagram
kkk
kkkk
(r̄00 + r̄01 ) + (r̄10 + r̄11 )
r00 + r01 + r10 + r11
SSSS
5
kkk
SSS
kkkk
σ+ (r00 ,r01 )+σ+ (r10 ,r11 )
r̄ 00 r̄ 01
r̄ 10 r̄ 11
(r̄00 + r̄10 ) + (r̄01 + r̄11 )
SSSS
SSS
r01
r11
SSSS
S)
r00 + r01 + r10 + r11
r r01
ϕ+ ( r00
)
10 r11
σ+ (r00 ,r10 )+σ+
SS(r01 ,r11 )
gives ϕ+ ( rr00
10
σ+ (r00 +r01 ,rS
10
S+r11 )
+ 00
SSSS
kkk
S)
kkkk
r00 + r10 + r01 + r11
σ (r
r00 + r10 + r01 + r11
5
kkk
kkkk
+r10 ,r01 +r11 )
) ∈ π1 (R).
Thus the above diagrams give rise to a 3-cochain ϕ in the Mac Lane complex of π0 (R)
Journal of Homotopy and Related Structures, vol. 2(2), 2007
199
with coefficients in π1 (R). Explicitly, it is defined by
ϕ· (r, s, t) = σ· (rs, t) ◦ σ· (r, s)t̄ ÷ σ· (r, st) ◦ r̄σ· (s, t) ◦ r̄, s̄, t̄ ,
E
h
s̄
ϕ·+ (r, s0 , s1 ) = σ· (r, s0 + s1 ) ◦ r̄σ+ (s0 , s1 ) ÷ σ+ (rs0 , rs1 ) ◦ (σ· (r, s0 ) + σ· (r, s1 )) ◦ r̄ s̄0 ,
D 1i
r̄
ϕ+· (r0 , r1 , s) = σ· (r0 + r1 , s) ◦ σ+ (r0 , r1 )s̄ ÷ σ+ (r0 s, r1 s) ◦ (σ· (r0 , s) + σ· (r1 , s)) ◦ r̄0 s̄ ,
1
E
D
r̄ 00 r̄ 01
00 r01
)
=
σ
(r
+
r
,
r
+
r
)
◦
(σ
(r
,
r
)
+
σ
(r
,
r
))
◦
ϕ+ ( rr10
+ 00
10 01
11
+ 00 10
+ 01 11
r11
r̄
r̄
10
11
÷ σ+ (r00 + r01 , r10 + r11 ) ◦ (σ+ (r00 , r01 ) + σ+ (r10 , r11 )).
We have
(4.1) Proposition. The above cochain is a cocycle.
Proof. We have to check the eight equalities from (3.1). These equalities are deducible
from considering eight diagrams below. In all of these diagrams, all quadrangles commute
by naturality, the inner pentagons are filled by the indicated elements of π1 (R) using (1.4)
and (2.3) as needed, whereas the outer perimeters commute since each of them coincides
with a coherence diagram from the definition of categorical ring.
[r̄,s̄t̄,ū]
/ r̄((s̄t̄)ū)
(r̄(s̄t̄))ū
I
II
++
u
I
uu
++
r̄(σ· (s,t)ū)
(r̄σ· (s,t))
IIIū
++
u
zuu
$
++
[r̄,st,ū]
/ r̄(stū)
++
ū
(
r̄st)
++
++
σ· (r,st)ū
r̄σ· (st,u)
++
+r̄+ [s̄,t̄,ū]
ϕ· (r,st,u)
[r̄,s̄,t̄]ū ϕ· (r,s,t)u
rϕ
(s,t,u)
·
r̄stu
rstū
++
III
C
6
[
u
66
I
++
uuu
6
σ· (rst,u)
σ· (r,stu)
66
III
++
u
u
66
I$
zuu
++
++
σ· (rs,t)ū
r̄σ· (s,tu)
rstu
6
66
O
++
6
66
++
ϕ
(rs,t,u)
σ
(rs,tu)
ϕ
(r,s,tu)
·
·
·
66
+
r̄(
s̄σ· (t,u))
(σ· (r,s)t̄)ū
o
/
r̄(s̄(t̄ū))
((r̄s̄)t̄)ū
rs tu
r̄(s̄tu)
(rs t̄)ū
KK
II
dIII
s9
u:
u:
KK
s
u
I
u
s
u
I
u
KK
II
ss
uu
rsσ· (t,u)
σ· (r,s)tu
KK
III
ss
uur̄,s̄,tu]
[rs,t̄,ū] III$
KK
ss
uuu
uu [
KK
s
KK
ss
KK
ss
rs(t̄ū)
(r̄s̄)tu
s
KK
s
]::
s
KK
B
ss
::
K
s[sr̄,s̄,t̄ū]
[r̄s̄,t̄,ū] KKK
s
:
KK
ss
:
KK
ss
KKσ· (r,s)(:t̄ū)
(r̄ s̄)σ· (t,u) sss
KK ::
s
KK ::
ssss
KK :
s
KK :
s
%
ss
(r̄s̄)(t̄ū)
.
r̄
s̄t̄0
s̄t̄1
/ r̄(s̄t̄ ) + r̄(s̄t̄ )
r̄(s̄t̄0 + s̄t̄1 )
0
W.. 1
H
KKK
rr
KK
"
+
..
rrr
st
r̄(σ· (s,t0 )+σK· (s,t1 ))
r̄σ· (s,t0 )+r̄σ· (s,t1 )
..
r̄ 0
KKK
r
r
st1
K%
..
rr
x
r
..
/
r̄(st0 + st1 )
r̄st0 + r̄st1
..
..
..
σ· (r,st0 )+σ· (r,st1 )
r̄σ+ (st0 ,st1 )
..
ϕ
(r,st
,st
)
t̄0
·+
0
1
r̄ s̄
. [r̄,s̄,t̄0 ]+[r̄,s̄,t̄1 ]
t̄1
rϕ·+ (s,t0 ,t1 )
ϕ· (r,s,t0 )+ϕ· (r,s,t1 ) ..
r̄s(t0 + t1 )
rst0 + rst1
;
]
LLL
B
..
s
;;
LL
sss
..
;
s
;
σ+ (rst0 ,rst1 )
σ· (r,s(t0 +t
;;
1 ))
L
..
L
s
LLL
;;
ss
s
..
%
y
s
;
..
σ
(rs,t
)+σ
(rs,t
)
r̄σ· (s,t0 +t1 )
·
0
·
1
rs(t
+
t
)
;
0
..
;;
O 1
;;
..
;;
..
σ· (rs,t0 +t1 )
ϕ·+ (rs,t0 ,t1 )
ϕ
(r,s,t
+t
)
·
0
1
;
;;
σ· (r,s)t̄0 +σ· (r,s)t̄1 .
r̄(s̄σ+ (t0 ,t1 ))
/ r̄(s̄t0 + t1 )
(r̄s̄)t̄0 + (r̄s̄)t̄1
r̄(s̄(t̄0 + t̄1 ))
rs t̄0 + rs t̄1 o
rs t0 + t1
7
fMMM
K
e
eKKK
9
K
s
r8
s
K
r
s
ppp
MMM
KKK
r
K
s
p
r
s
p
MMM
KKK
rr
pp
rsσ+ (t0K
,t1 )
σ· (r,s)t0 +t1
KKK
MMM
KKK
rrr
ppp
ss
r
p
s
r
p
s
MMM[r̄,s̄,t0 +t1 ]
r
s
t̄
pp
rs 0
MMM
ppp
t̄1
p
rs(t̄0 + t̄1 )
(r̄s̄)t0 + t1
p
MMM
^==
@
pp
MMM
==
ppp
p
MMM
p
==
pp
MMM
==
ppp t̄0 [r̄,s̄,t̄0 +t̄1 ]
MMM
p
p
r̄ s̄
p
MMM(r̄s̄)σ+ (t=0 ,t1 )
t̄1
σ· (r,s)(t̄0 +t̄1 ) ppp
p
MMM ==
pp
p
MMM ==
p
p
MMM ==
ppp
M=
ppp
(r̄s̄)(t̄0 + t̄1 )
Journal of Homotopy and Related Structures, vol. 2(2), 2007
200
[r̄,s̄0 +s̄1 ,t̄]
/ r̄((s̄ + s̄ )t̄)
(r̄(s̄0 + s̄1 ))t̄
0 + 1
KKK
++
ss
KKK
s
s
++
ss
r̄(σ+ (s0 ,s1 )t̄)
(r̄σ+ (s0 ,s
KK1K))t̄
++
s
KKK
ss
++
[r̄,s0 +s1 ,t̄]
%
ysss
++
/
r̄(s0 + s1 t̄)
(r̄s0 + s1 )t̄
++
++
++
r̄σ+ (s0 +s1 ,t)
σ· (r,s0 +s1 )t̄
++
++ s̄ ϕ
(r,s
+s
,t)
s̄
·
0
1
r̄ 0 t̄
++ r̄ s̄0 t̄
s̄1
ϕ·+ (r,s0 ,s1 )t
rϕ+· (s0 ,s1 ,t)
1
r̄(s0 + s1 )t
r(s0 + s1 ) t̄
++
JJJ
\99
t
B
++
t
J
9
t
JJ
99
++
ttt
99
σ· (r(s0 +s
σ· (r,(s0 +s1 )t)
1 ),t)
J
++
J
t
9
J
t
JJJ
99
tt
++
t
%
y
t
9
++
σ+ (rs0 ,rs1 )t̄
r̄σ+ (s0 t,s
r(s0 + s1 )t
99 1 t)
++
O
99
++
9
++
9
9
σ+ (rs0 t,rs1 t)
ϕ+· (rs0 ,rs1 ,t)
ϕ·+ (r,s0 t,s1 t)
99
++
99
+
r̄(σ· (s0 ,t)+σ· (s1 ,t))
(σ· (r,s0 )+σ· (r,s1 ))t̄ /
o
(r̄s̄0 + r̄s̄1 )t̄
r̄(s̄0 t̄ + s̄1 t̄)
(rs0 + rs1 )t̄
rs0 t + rs1 t
r̄(s0 t + s1 t)
KK
KKK
9
eJJJ
s
t
s
t
tt
KK
J
K
JJ
KKK
ss
tt
tt
KK
s
t
t
s
t
t
KK
KKK
s
σ· (rs0 ,t)+σ· (rs1 ,t)
σ· (r,s0 t)+σJ· (r,s1 t)
tt
KK
JJJ
t
sss"
tt
KK
t
s
*
# KKK
+
t
t
s
t
JJ
KK
KK
rs0
s t
tt
%
ttt
ysss
KK
t̄
r̄ 0
tt
KK
t
rs
s
t
1
1
t
KK
r̄s0 t + r̄s1 t
rs0 t̄ + rs1 t̄
tt
KK
O
O
tt s̄ t̄ KK
ϕ· (r,s0 ,t)+ϕ· (r,s1 ,t)
t
r̄ s̄0
t
KK
t
t̄
r̄ 0
KK
tt
r̄ s̄1
s̄1 t̄
r̄σ· (s0 ,t)+r̄σ· (s1 ,t) ttt
KKσ· (r,s0 )t̄+σ· (r,s1 )t̄
KK
t
KK
tt
%
ytt
/ r̄(s̄ t̄) + r̄(s̄ t̄)
(r̄s̄0 )t̄ + (r̄s̄1 )t̄
0
1
[r̄,s̄0 ,t̄]+[r̄,s̄1 ,t̄]
.
Journal of Homotopy and Related Structures, vol. 2(2), 2007
201
r̄ 0 s̄
t̄
r̄ 1 s̄
/ (r̄ s̄)t̄ + (r̄ s̄)t̄
(r̄0 s̄ + r̄1 s̄)t̄
0
MMM
// 1
H
*
#
M
ppp
//
p
r0 s
σ· (r0 ,s)t̄+σ· (r1 ,s)t̄
t̄
//
(σ· (r0 ,s)+σ·M(rMM1 ,s))t̄
p
p
r1 s
M&
xpp
//
/ r0 s t̄ + r1 s t̄
//
(r0 s + r1 s)t̄
//
//
σ· (r0 s,t)+σ· (r1 s,t)
σ+ (r0 s,r1 s)t̄
//
r̄ 0
ϕ+· (r0 s,r1 s,t)
// [r̄ ,s̄,t̄]+[r̄ ,s̄,t̄]
s̄ t̄
1
0
r̄ 1
ϕ+· (r0 ,r1 ,s)t
ϕ· (r0 ,s,t)+ϕ· (r1 ,s,t) //
(r0 + r1 )s t̄
r0 st + r1 st
/
=
^
A
MMM
q
==
//
M
qqq
==
//
σ+ (r0 st,r1 st)
σ· ((r0 +r1M)s,t)
=
MMM
==
qq
//
q
x
&
q
=
//
σ· (r0 ,st)+σ
σ· (r0 +r1 ,s)t̄
· (r1 ,st)
(r
+
r
)st
=
//
0
1
=
O
==
//
=
==
//
σ· (r0 +r1 ,st)
ϕ+· (r0 ,r1 ,st)
ϕ· (r0 +r1 ,s,t)
=
=
r̄ 0 σ· (s,t)+r̄ 1 σ· (s,t) /
(σ+ (r0 ,r1 )s̄)t̄
/ (r + r s̄)t̄
r̄0 (s̄t̄) + r̄1 (s̄t̄)
((r̄0 + r̄1 )s̄)t̄
r̄0 st + r̄1 st o
r0 + r1 st
0
1
OOO
MMM
8
fMMM
8
n7
q
p
q
p
n
M
q
OOO
MMM
nn
q
pp
n
p
n
OOO
MMM
p
σ+ (r0 ,r1M)st
r0 +r1 σ· (s,t)
pp
nnn
MMM
q
OOO
MM&
nnn
ppp r̄0 qqq
OOO[r0 +r1 ,s̄,t̄]
n
n
st
OOO
nn
r̄ 1
(r̄0 + r̄1 )st
r0 + r1 (s̄t̄)
nnn
OOO
n
@
`
?
n
@@
OOO
~~
nnn
@@
OOO
~~
nnn
@@
n
~
OOO
n
~
@
r̄ 0
[r̄0 +r̄1 ,s̄,t̄]
OOO
nnn
~~
n
s̄t̄
n
n
OOσO+ (r0 ,r1@)(s̄t̄)
(r̄ 0 +r̄ 1 )σ· (s,t) n
r̄ 1
~
nn
OOO @@@
~~ nnnnn
OOO @@
~
OOO @@
~~ nn
~~ nnn
'
(r̄0 + r̄1 )(s̄t̄)
.
Journal of Homotopy and Related Structures, vol. 2(2), 2007
202
.
r̄ s̄00 r̄ s̄01
r̄ s̄10 r̄ s̄11
+
r̄
*
s̄00 s̄01
s̄10 s̄11
+
/ (r̄s̄00 + r̄s̄10 ) + (r̄s̄01 + r̄s̄11 )
s U,,
I KKKK
sss
,,
(σ· (r,s00 )+σ· (r,s10 ))
(σ· (r,s00 )+σ· (r,s01 ))
*
+
,,
+(σ· (r,s10 )+σ· (r,s
rs00 rs01
+(σ· (r,s01 )+σ· (r,s11 ))
KKK11 ))
,,
s
s
rs10 rs11
K%
s
ys
,,
/
(rs00 + rs01 ) + (rs10 + rs11 )
(rs00 + rs10 ) + (rs01 + rs11 )
,,
,,
,,
σ+ (rs00 ,rs01 )+σ+ (rs10 ,rs11 )
σ+ (rs00 ,rs10 )+σ+ (rs01 ,rs11 )
rs
,,
rs01 00
ϕ+ rs
"
+ "
+
10 rs11
,, " s̄00 + " s̄01 +
s̄00
s̄10
r̄
+ r̄
ϕ
(r,s
,s
)
ϕ·+ (r,s00 ,s01 ) r(s
,, r̄ s̄10 + r̄ s̄11
s̄01
s̄11
00
10
·+
r(s00 + s10 ) + r(s01 + s11 )
00 + s01 ) + r(s10 + s11 )
KKK
C
[88
+ϕ·+ (r,s01 ,s11 )
+ϕ·+ (r,s10 ,s11 )
,,
s
KKK
88
sss
,,
s
s
88
11 )) σ+ (r(s00 +s10 ),r(s01 +s11 ))
,,
KK+s
8
σ+ (r(s00 +s01 ),r(sK10
s
s
8
KKK
,,
88
sss
%
y
s
8
,,
r(s00 + s01 + s10 + s11 )
σ· (r,s00 +s01 )+σ· (r,s10 +s11 )
σ· (r,s00 +s10 )+σ
88· (r,s01 +s11 )
,,
O
88
,,
8
88
,,
σ
(r,s
+s
+s
+s
)
00
01
10
11 ϕ·+ (r,s00 +s10 ,s01 +s11 )
ϕ·+ (r,s00 +s01 ,s10 +s11 ) ·
88
r̄σ+ (s00 ,s10 )
,,
r̄σ+ (s00 ,s01 )
8
+r̄σ+ (s01 ,s11 )
+r̄σ+ (s10 ,s11 )
r̄(s̄00 + s̄01 )
r̄s00 + s10 o
r̄(s̄00 + s̄10 )
/ r̄s00 + s01
r̄s00 + s01 + s10 + s11
+r̄(s̄10 + s̄11 )
+r̄(s̄01 + s̄11 )
+r̄s10 + s11
+r̄s01 + s11
I
:
d
I
u
eJJ
9
dII
II
u:
uu
JJ
tt
II
II
uu
uu
JJ
tt
I
u
u
t
I
u
JJ
t
r̄σ+ (s00 +s01 ,s10 +s11 ) r̄σ+ (s00 +s10I
,s01 +s11 )
II
uu
JJ
tt
II
uu
uu "
+
JJ " s +s + IIII
tt
I
u
u
t
I
u
u
s
+s
JJ r̄ 00 01
10
I
I
r̄ 00
tt
uu
uu
JJ s10 +s11
s01 +s11 tt
JJ
t
t
r̄(s
r̄(s
+
s
+
s
+
s
)
+
s
+
s
+
s
)
00
01
10
11
00
10
01
11
JJ
t
"
+
O
O
s
JJ
tt " s̄00 +s̄10 +
s
s̄ +s̄01
r̄ϕ+ s00 s01
JJ
r̄ 00
r̄
tt
10 11
t
s̄10 +s̄11
s̄01 +s̄11
JJr̄(σ (s̄ ,s̄ )+σ (s̄ ,s̄ ))
tt
r̄(σ+ (s̄00 ,s̄10 )+σ+ (s̄01 ,s̄11 ))
JJ+ 00 01 + 10 11
t
t
JJ
t
JJ
tt
tt
/ r̄((s̄00 + s̄10 ) + (s̄01 + s̄11 ))
r̄((s̄00 + s̄01 ) + (s̄10 + s̄11 ))
(r̄ s̄00 + r̄ s̄01 ) + (r̄ s̄10 + r̄ s̄11 )
*
Journal of Homotopy and Related Structures, vol. 2(2), 2007
203
.
r̄ 0 s̄0 r̄ 0 s̄1
r̄ 1 s̄0 r̄ 1 s̄1
+
(r̄ 0 + r̄ 1 )(s̄0 + s̄1 )
/ (r̄0 s̄0 + r̄1 s̄0 ) + (r̄0 s̄1 + r̄1 s̄1 )
OOO
W..
G
ooo
..
*
+
(σ· (r0 ,s0 )+σ· (r0 ,s1 ))
(σ· (r0 ,s0 )+σ· (r1 ,s0 ))
r 0 s0 r 0 s1
..
+(σ
(r
,s
)+σ
(r
,s
))
+(σ
(r
,s
)+σ
(r
,s
))
·
·
·
·
1
0
O
1
1
0
1
1
1
r 1 s0 r 1 s1
OO'
wooo
..
/ (r0 s0 + r1 s0 ) + (r0 s1 + r1 s1 )
..
(r0 s0 + r0 s1 ) + (r1 s0 + r1 s1 )
..
..
σ+ (r0 s0 ,r0 s1 )+σ+ (r1 s0 ,r1 s1 )
r s r s σ+ (r0 s0 ,r1 s0 )+σ+ (r0 s1 ,r1 s1 )
.. *
ϕ+ r 0 s 0 r 0 s 1
"
+ "
+
# *
#
1
0
1
1
s̄
s̄
r̄ 0 0 + r̄ 1 0
.. r̄r̄0 s̄0 + r̄r̄0 s̄1
s̄1
s̄1
ϕ+· (r0 ,r1 ,s0 )
ϕ·+ (r0 ,s0 ,s1 ) r0 (s0 + s1 ) + r1 (s0 + s1 )
1
1
(r0 + r1 )s0 + (r0 + r1 )s1
..
@
NNN
^<<
+ϕ·+ (r1 ,s0 ,s1 )
+ϕ+· (r0 ,r1 ,s1 )
p
NN
..
<<
ppp
p
<<
σ+ (r0 (s0 +s1 ),rN1N(s0 +s1 ))σ+ ((r0 +rp1 )s0 ,(r0 +r1 )s1 )
..
<<
NNN
pp
..
p
w
'
p
<
<
..
(r0 + r1 )(s0 + s1 )
σ· (r0 ,s0 +s1 )+σ· (r1 ,s0 +s1 )
σ· (r0 +r1 ,s0 )+σ
· (r0 +r1 ,s1 )
<
O
..
<<
<<
..
r̄0 σ+ (s0 ,s1 )
<<
σ+ (r0 ,r1 )s̄0
σ· (r0 +r1 ,s0 +s1 )
..
ϕ·+ (r0 +r1 ,s0 ,s1 )
ϕ+· (r0 ,r1 ,s0 +s1 )
<
+σ+ (r0 ,r1 )s̄1
+r̄1 σ+ (s0 ,s1 )
r̄ 0 (s̄0 + s̄1 )
r0 + r1 s̄0 o
(r̄ 0 + r̄ 1 )s̄0
/ r̄0 s0 + s1
r 0 + r 1 s0 + s1
+r̄ 1 (s̄0 + s̄1 )
+(r̄ 0 + r̄ 1 )s̄1
+r̄ 1 s0 + s1
+r0 + r1 s̄1
L
8
f
L
r
7
gOOO
LLL
r
fLLL
r8
oo
OOO
LLL
rrr
rrr
ooo
r
σ+ (r0 ,r1 )s0 +s1
r0 +r1 σ+ (s
OOO
o
0 ,s1 )
L
r
L
o
LLL
LLL
OOO *
rr "
rr
#
+
ooo
L
L
OOO r̄0 s0 +s1
rrr
rrr r0 +r1 s̄0 oooo
Or̄O1 O
s̄1oo
r0 + r1 (s̄0 + s̄1 )
(r̄ 0 + r̄ 1 )s0 + s1
OOO
ooo
>
`AA
o
}
OOO
o
AA
}}
OOO
ooo
AA
+
*
#
}}
OOO
ooo "
A
o
}
s̄
r̄ 0
o
A
}
OOO
o
r̄ 0 +r̄ 1 0
s̄ +s̄1
}
o
s̄
r̄ 1 0
o
1
O(Or̄O0 +r̄1 )σ+A(s0 ,s1 )
σ+ (r0 ,r1 )(s̄0 +s̄1 )oo
OOO AA
}} ooooo
}
OOO AA
}} ooo
OOOAA
O A }}o}ooo
(r̄ 0 s̄0 + r̄ 0 s̄1 ) + (r̄ 1 s̄0 + r̄ 1 s̄1 )
*
Journal of Homotopy and Related Structures, vol. 2(2), 2007
204
.
r̄ 00 s̄ r̄ 01 s̄
r̄ 10 s̄ r̄ 11 s̄
+
*
r̄ 00 r̄ 01
r̄ 10 r̄ 11
s̄
+
/ (r̄00 s̄ + r̄10 s̄) + (r̄01 s̄ + r̄11 s̄)
MMM
V-H
qqq
-
*
+
(σ· (r00 ,s)+σ· (r01 ,s))
(σ· (r00 ,s)+σ· (r10 ,s))
-r00 s r01 s
+(σ· (r01 ,s)+σ· (r11 ,s))
+(σ· (r10 ,s)+σ· (rM11
MM,s))
r10 s r11 s
-&
xqqq
-/ (r00 s + r10 s) + (r01 s + r11 s)
(r00 s + r01 s) + (r10 s + r11 s)
-
-σ+ (r00 s,r01 s)+σ+ (r10 s,r11 s)
σ+ (r00 s,r10 s)+σ+ (r01 s,r11 s)
-
r s r s
00
01
ϕ+ r s r s
*
# *
#
-- * r̄ # * r̄ #
10
11
r̄ 00
r̄ 10
01 s̄
s̄ +
s̄
s̄ +
-- r̄00
ϕ+· (r00 ,r01 ,s)
ϕ+· (r00 ,r10 ,s)
r̄ 01
r̄ 11
r̄ 11
10
(r00 + r01 )s + (r10 + r11 )s
(r00 + r10 )s + (r01 + r11 )s
-]::
MMM
+ϕ+· (r01 ,r11 ,s)
A
+ϕ+· (r10 ,r11 ,s)
M
qqq
-σ ((r +r MM)s,(r +r )s) σ ((r +r )s,(r
qqq +r )s) :::
00
10
01
11
+
-MMM 11
::
+ 00 01 10
q
q
MMM
q
:
q
-q
:
x
&
q
:
-
(r00 + r01 + r10 + r11 )s
σ· (r00 +r01 ,s)+σ· (r10 +r11 ,s)
σ· (r00 +r10 ,s)+σ
::· (r01 +r11 ,s)
-O
:
::
-
::
ϕ (r +r ,r +r ,s) σ· (r00 +r01 +r10 +r11 ,s) ϕ (r +r ,r +r ,s)
-
σ
(r
,r
)
s̄
σ
(r
,r
)
s̄
10 01
11
01 10
11
+· 00
+· 00
+ 00 10
::
+σ+ (r00 ,r01 )s̄
+σ+ (r01 ,r11 )s̄
+ 10 11
r
+
r
r
+
r
s̄
s̄
(r̄ 00 + r̄ 01 )s̄
(
r̄
00
01
00
10
00 + r̄ 10 )s̄
/
o
r00 + r01 + r10 + r11 s̄
+(r̄ 10 + r̄ 11 )s̄
+(r̄ 01 + r̄ 11 )s̄
+r10 + r11 s̄
+r01 + r11 s̄
K
9
e
K
s
fLLL
KKK
eKK
9
r8
ss
KK
ss
LLL
rrr
sss+r )s̄ σ (r +rK ,r +r )s̄
KK
ss
r
s
σ
(r
+r
,r
LLL
r
KK
01 10
11
10 K
01
s
+ 00
+ 00
KKK 11
rr
KK
ss
LLL *
#
sss
KK
ss * r00 +r10 # rrrr
KK
s
s
s
LLL r00 +r01 r̄
s
s
s̄
rr
r01 +r11
LLrL10 +r11 (r + r + r + r )s̄
(r00 + r10 + r01 + r11 )s̄
rr*r
00
01
10
11
LLL
r
*
#
#
O
O
r
r
r
r̄ 00 +r̄ 01
r̄ 00 +r̄ 10
LLL
rr
ϕ+ r00 r01 s̄
s̄
s̄
r
10 11
r
r̄ 10 +r̄ 11
r̄
+
r̄
L(σ
r
01
11
LL+L(r̄00 ,r̄01 )+σ+ (r̄10 ,r̄11 ))s̄
(σ+ (r̄ 00 ,r̄ 10 )+σ+ (r̄ 01 ,r̄ 11 ))r
s̄r
LLL
rr
r
r
r
/ ((r̄00 + r̄10 ) + (r̄01 + r̄11 ))s̄
((r̄ 00 + r̄ 01 ) + (r̄ 10 + r̄ 11 ))s̄
(r̄ 00 s̄ + r̄ 01 s̄) + (r̄ 10 s̄ + r̄ 11 s̄)
*
Journal of Homotopy and Related Structures, vol. 2(2), 2007
205
.
011
110
r̄ 000 +r̄ 010 r̄ 001 +r̄ 011
r̄ 100 +r̄ 110 r̄ 101 +r̄ 111
+
010
*
r̄ 000 r̄ 001
r̄ 100 r̄ 101
111
*
+
+
r̄ 010 r̄ 011
r̄ 110 r̄ 111
010
+
110
011
111
((r̄ 000 + r̄ 010 ) + (r̄ 100 + r̄ 110 ))
/ +((
r̄ 001 + r̄ 011 ) + (r̄ 101 + r̄ 111 ))
I
JJJ
++
ttt
+
(σ (r000 ,r010 )+σ (r001 ,r011 ))
(σ+ (r000 ,r010 )+σ+ (r100 ,r110 ))+
+
+
+ (r100 ,r110 )+σ+ (r101 ,r111 )) * r000 +r010 r001 +r011 + +(σ+ (r001 ,r011 )+σ+ (r101 ,r111 ))++
+(σ
JJJ
++
r100 +r110 r101 +r111
$
zttt
++
(r
+
r
+
r
(r000 + r010 + r001 + r011 )
000
010
100 + r110 )
/
+(r100 + r110 + r101 + r111 )
+(r001 + r011 + r101 + r111 )
++
++
σ+ (r000 +r010 ,r001 +r011 )
σ+ (r000 +r010 ,r100 +r110 )
*
+
*
+
++
r̄ 000 r̄ 001
r̄ 000 r̄ 010
+σ+ (r100 +r110 ,r101 +r111 )
+σ+ (r001 +r011 ,r101 +r111 )
+
r̄ 010 r̄ 011
r̄ 100 r̄ 110
r
+
*
+
*
+
r001 r
++
ϕ+ rr000
P
P
ϕ+ r000 r010
r̄ 100 r̄ 101
r̄ 001 r̄ 011
010 r011
100 110
r0ij
rij0
r000 +r010 r001 +r011
+
+
P
P
ϕ+
+
+ϕ r100 r101
r001 r011
r̄ 110 r̄ 111
r̄ 101 r̄ 111
r100 +r110 r101 +r111
+
r
+
r
+ϕ+ r
1ij
ij1
++
+ r110 r111
101 r111
KKK
[88
C
tt
++
8
K
t
P
P
P
P
t
8
σ+ ( rij0 , rij1 )
σ+ ( r0ij , r1ij )
88
++
K
K
KKK
88
tt
++
K%
ttt
y
t
++
P
σ+ (r000 +r001 ,r010 +r011 )
σ+ (r000 +r100 ,r010 +r110 )
rijk
+
+σ+ (r100 +r101 ,r110 +r111 )
+σ+ (r001 +r101 ,r
+r111 )
O
011
8
88 σ+ (r000 , r100 ) ++
σ+ (r000 , r001 ) 88
+
+σ (r010 , r110 )
P
P
r000 +r100 r001 +r101
+σ+ (r010 , r011 ) ϕ+ rr000 +r
001 r010 +r011 σ+ ( ri0j , ri1j ) ϕ
+
88 σ+
+ r
++
(r100 , r101 )
010 +r110 r011 +r111
100 +r101 r110 +r111
+ (r001 , r101 )
+
+ +σσ+
8
+σ+ (r011 , r111 )
+ (r110 , r111 ) P
r
((r̄ 000 + r̄ 001 ) + (r̄ 010 + r̄ 011 ))
(r000 + r100 + r010 + r110 )
(r000 + r001 + r010 + r011 )
((r̄ 000 + r̄ 100 ) + (r̄ 010 + r̄ 110 ))
/
o
P i0j
+((r̄ 100 + r̄ 101 ) + (r̄ 110 + r̄ 111 ))
+((r̄ 001 + r̄ 101 ) + (r̄ 011 + r̄ 111 ))
+(r100 + r101 + r110 + r111 )
+ ri1j
+(r001 + r101 + r011 + r111 )
JJ
II
:
dIII
u:
I
JJ
uu:
uu
u
uu
u
+
*
+
*
u
JJ
σ
(r
+r
,r
+r
)
σ
(r
+r
,r
+r
)
u
r000 +r001 r010 +r011 + 000
001 100
101 + 000
100 001
101 r000 +r100 r001 +r101
JJ
uu
) + (r010 +r110 ,r
+r111 ) r010 +r110 r011 +r111
JJ r100 +r101 r110 +rII111+σ+ (r010 +r011 ,r110 +r111+σ
uu
II011
u
u
u
JJ
I
I
u
u
u
u
$
u
r
JJ
uu
000 r100
JJ
(r000 + r100 + r001 + r101 )
(r
+ r001 + r100 + r101 ) ϕ+ r001 r101
uu
u
JJ +(r000
u
r010 r110 +(r010 + r110 + r011 + r111 )
010 + r011 + r110 + r111 ) +ϕ
*
+ J
+
O
O
+ r011 r111
uu*
JJ
r̄ 000 +r̄ 001 r̄ 010 +r̄ 011
uu r̄000 +r̄100 r̄001 +r̄101
J
u
J
r̄ 100 +r̄ 101 r̄ 110 +r̄ 111
r̄ 010 +r̄ 110 r̄ 011 +r̄ 111
JJ (σ+ (r000 ,r001 )+σ+ (r100 ,r101 )) (σ+ (r000 ,r100 )+σ+ (r001 ,r101 )) uuu
JJ+ (r010 ,r011 )+σ+ (r110 ,r111 )) +(σ+ (r010 ,r110 )+σ+ (r011 ,r111u))u
+(σ
JJ
u
uu
$
((r̄ 000 + r̄ 001 ) + (r̄ 100 + r̄ 101 ))
((r̄ 000 + r̄ 100 ) + (r̄ 001 + r̄ 101 ))
/
+((r̄
+ r̄
) + (r̄
+ r̄
))
+((r̄
+ r̄
) + (r̄
+ r̄
))
((r̄ 000 + r̄ 010 ) + (r̄ 001 + r̄ 011 ))
+((r̄ 100 + r̄ 110 ) + (r̄ 101 + r̄ 111 ))
*
Journal of Homotopy and Related Structures, vol. 2(2), 2007
206
Journal of Homotopy and Related Structures, vol. 2(2), 2007
207
To show that the above rule indeed determines an assignment of a cohomology class to
each categorical ring, we must also show
(4.2) Proposition. A different choice of representative objects r 7→ r̃ and morphisms
e
σ·0 (r, s) : r̃s̃ → rs,
0
σ+
(r0 , r1 ) : r̃0 + r̃1 → r^
0 + r1
leads to a cocycle ϕ0 which is cohomologous to ϕ.
Proof. By (3.1), 3-cocycles ϕ and ϕ0 are cohomologous if and only if there exist maps
γ· , γ+ : R × R → B such that the following four equalities
ϕ0· (r, s, t)
0
ϕ·+ (r, s0 , s1 )
=ϕ· (r, s, t) + rγ· (s, t) − γ· (rs, t) + γ· (r, st) − γ· (r, s)t,
=ϕ·+ (r, s0 , s1 ) + rγ+ (s0 , s1 ) − γ+ (rs0 , rs1 ) + (s0 | s1 )γ· (r,−) ,
ϕ0+· (r0 , r1 , s) =ϕ+· (r0 , r1 , s) + γ+ (r0 s, r1 s) − γ+ (r0 , r1 )s − (r0 | r1 )γ· (−,s) ,
ϕ0+ ( rr00
10
r01
r11
) =ϕ+ ( rr00
10
r01
r11
) + ((r00 , r01 ) | (r10 , r11 ))γ+ − ((r00 , r10 ) | (r01 , r11 ))γ+
are satisfied for all possible elements r, ... of R.
Let us then choose arbitrary morphisms
r̂ : r̄ → r̃
for all r ∈ R and define the maps γ+ and γ· , as above for ϕ, to measure deviation from
commutativity of the diagrams
σ· (r,s)
/ rs
r̄s̄
r̂ ŝ
γ· (r,s)
/ rs
e
r̃s̃
b
rs
σ·0 (r,s)
and
σ+ (r0 ,r1 )
r̄0 + r̄1
r̂ 0 +r̂ 1
r̃0 + r̃1
/ r0 + r 1
γ+ (r0 ,r1 )
r\
0 +r1
/ r^
0 + r1 .
0
σ+
(r0 ,r1 )
That is, we define
b ◦ σ· (r, s)
γ· (r, s) = σ·0 (r, s) ◦ r̂ŝ ÷ rs
Journal of Homotopy and Related Structures, vol. 2(2), 2007
208
and
0
γ+ (r0 , r1 ) = σ+
(r0 , r1 ) ◦ (r̂0 + r̂1 ) ÷ r\
0 + r1 ◦ σ+ (r0 , r1 ).
The above four equalities then follow from considering the following four diagrams, in
view, as before, of (1.4) and (2.3), where “” marks the strictly commuting quadrangles:
r̄σ· (s,t)
/ r̄st
? EE
<<
~
E
~
<<
~

c

rγ
(s,t)
r̂(ŝt̂)
<<
r̂
st
·
EE

~

E
<<
~

~
~
"
[r̄,s̄,t̄] 
<<σ· (r,st)
/ r̃st
e
γ· (r,st)
r̃σ·0 (s,t)

r̃(
s̃
t̃)
<<

<
@

@
z
<<
@
z

z
<<

σ·0 (r,st)
[r̃,s̃,t̃]

@@
<<

z
@

z
z

ϕ0· (r,s,t)
/ (r̃s̃)t̃ Q
d
f o
(r̂ ŝ)t̂
rst
(r̄ s̄)t̄ P
rst
rst
7
7
n
o
QQQ
PPP
nn
oo
n
QQ
o
PPP
n
o
n
PPP
oo
σ·0 (r,st)
σ 0 (rs,t)
QQQ
PPP
ooo
nn·
Q
o
n
Q
n
o
Q
PPP
Q(
nn
oo
PPP γ· (r,s)t
e t̃
γ· (rs,t)
rs
oσo·o(rs,t)
o
PPP
σ· (r,s)t̄
O
o
PPP
oo
PPP
ooo
c t̂
o
o
PPP rs
oo
PP(
ooo
r̄(s̄t̄)
rst̄
σ· (r,s0 )+σ· (r,s1 )
/ rs0 + rs1
@ ==
==
=
==
==
d +rs
d
γ· (r,s0 )+γ· (r,s1 )
r̂ ŝ0 +r̂
ŝ1
rs
=
0
1
"
+
=
==
s̄
=
r̄ 0
0
0
== σ+ (rs0 ,rs1 )
σ· (r,s0 )+σ· (r,s1 )
s̄1
rγ (s ,s )
/
==
g
g
γ
(rs
,rs
)
r̃ s̃0 + r̃ s̃1
+ 0 1
rs0 + rs1 + 0 1
==
@
<<
<
"
+
==
s̃0
0
==
σ+ (rs0 ,rs1 )
r̃
<<
s̃1
==
<
r̂(ŝ0 +ŝ1 )
ϕ0·+ (r,s0 ,s1 )
\
r(s
0 +s1 )
/ r̃(s̃0 + s̃1 )
o
r̄(s̄0 + s̄1 )
r(s0 + s1 )
r(s^
0 + s1 )
NNN
MMM
p8
q8
MM
NNN
qqq
ppp
q
p
0
0
NNN
p
rσ+ (s0 ,s1 )
σ+ (r,s0 +s1 )
MMM
pp
NNN
qq
MM&
ppp
NNN
p
qqq
p
NNN
pp
γ· (r,s0 +s1 )
r̃ s^
0 + s1
pσp·p(r,s0 +s1 )
r̄σ+ (s0 ,s1 ) NN
p
O
NNN
p
p
NNN
ppp
p
NNN r̂s\
p
0 +s1
pp
NN&
ppp
r̄ s̄0 + r̄ s̄1
r̄s0 + s1
Journal of Homotopy and Related Structures, vol. 2(2), 2007
209
σ· (r0 ,s)+σ· (r1 ,s)
/ r0 s + r1 s
==
@ ===
==
==
d
γ· (r0 ,s)+γ· (r1 ,s)
r̂ 0 ŝ+r̂
*
#
rd
=
1 ŝ
0 s+r1 s
r̄ 0
==
==
s̄
0 (r ,s)+σ 0 (r ,s)
σ
r̄ 1
== σ+ (r0 s,r1 s)
· 0
· 1
γ (r ,r )s
/
==
f γ+ (r0 s,r1 s)
r̃ 0 s̃ + r̃ 1 s̃
+ 0 1
rf
0 s + r1 s
==
@
<<
<
#
*
==
r̃ 0
0
==
σ+ (r0 s,r1 s)
s̃
<
r̃
<<
==
1
ϕ0+· (r0 ,r1 ,s)
\
(r0
+r1 )s
(r̂0 +r̂1 )ŝ
o
/
(r̄ 0 + r̄ 1 )s̄
(r̃ 0 + r̃ 1 )s̃
(r0 + r1 )s
(r0^
+ r1 )s
NNN
8
MMM
q8
MM
NNN
ppp
qqq
p
q
p
0
0
NNN
σ+ (r0 ,r1 )s
σ+ (r0 +r1 ,s)
pp
MMM
NNN
qq
ppp
MM&
p
NNN
p
qqq
pp
NNN
γ· (r0 +r1 ,s)
ppp
r^
0 + r1 s̃
NNN p
p
O
σ+ (r0 ,r1 )s̄
NNN
pp σ· (r0 +r1 ,s)
NNN
ppp
p
r\
+r1 ŝ
p
0
NNN
pp
N&
ppp
r̄ 0 s̄ + r̄ 1 s̄
r0 + r1 s̄
n6
nnn
n
n
nnn
nnn
\
)+σ (r ,r )
@@
@@
@@
@@
@@
\
σ+ (r00 ,r01
+ 10 11 r00 +r01 +r10 +r11
@@
nn
n
n
@@
nn
n
γ
(r
,r
)+γ
(r
,r
)
n
@@
+ 00 01
+ 10 11
n
@@ σ (r +r ,r +r )
nnn
@@ + 00 01 10 11
^
^
(r̄ 00 + r̄ 01 ) + (r̄ 10 + r̄ 11 )
r00
+ r01 + r10
+ r11
OOO
>>
@@
o7
o
OO
o
>
o
>γ>+ (r00 +r01 ,r10 +r11@)@@
0
0
(r̂ 00 +r̂ 01 )+(r̂O
+r̂ 11 )σ+ (r00 ,r01 )+σ+ (r10 ,r11 )
10
>>
OOO
@@
o
O'
>
@@
ooo
0
@@
σ+ (r00 +r01 ,r10 +r11 )
(r̃ 00 + r̃ 01 ) + (r̃ 10 + r̃ 11 )
>>
@@
>>
@@
>>
@@
>>
P
@
\
*
*
+
+
rij
r
P
r̄ 00 r̄ 01
r̃ 00 r̃ 01
P
r
^
rij
ϕ0+ r00 r01
rij o
10 11
r̄ 10 r̄ 11
r̃ 10 r̃ 11
@
?





0 (r +r ,r +r )
σ+
(r̃ 00 + r̃ 10 ) + (r̃ 01 + r̃ 11 )
00
10 01
11


7
O
OOO

oo
ooo

0
0 (r ,r )
γ+ (r00 +r10 ,r01 +r11
)
(r̂ 00 +r̂ 10 )+(r̂ 01 +r̂ 11 )σ+ (r00 ,r10 )+σ+
01 11
O
OOO

oo
'
ooo


^
^
(r̄ 00 + r̄ 10 ) + (r̄ 01 + r̄ 11 )

r00 + r10 + r01 + r11
 σ+ (r00 +r10 ,r01 +r11 )
PPP
O

PPP

γ+ (r00 ,r10 )+γ+ (r01 ,r11 )
PPP

PPP


\
\
σ+ (r00 ,r10 )+σP
+r10 +r01
+r11 
+ (r01 ,r11 ) r00
PPP

PPP
PPP


PP(

r00 + r01 + r10 + r11
r00 + r10 + r01 + r11
Journal of Homotopy and Related Structures, vol. 2(2), 2007
210
We have thus obtained a map from the set of all categorical rings R with π0 (R) = R,
π1 (R) = B and matching bimodule structure to the group H 3 (R; B). Let us next show
that this map factors through a quotient of the former set to yield a map
h−i : Crext(R; B) → H 3 (R; B),
where Crext(R; B) denotes the set of equivalence classes of categorical rings with π0
equal to R and π1 equal to B, two such being considered equivalent if there exists a 2homomorphism between them inducing identities on R and B.
Indeed, in the same way as in (4.2) we more generally have:
(4.3) Proposition. For any categorical rings R and R 0 such that there exists a 2-homomorphism R → R 0 inducing identity maps on π0 and π1 , one has hRi = hR 0 i.
Proof. Given a 2-homomorphism f : R → R 0 , let us choose ¯ and σ· , σ+ for R as above,
and then choose the corresponding maps for R 0 as follows:
0
r̄ = f (r̄),
f· (r̄,s̄)
f σ· (r,s)
σ·0 (r, s) = f r̄f s̄ −−−−→ f (r̄s̄) −−−−−→ f rs ,
f+ (r̄ ,r̄ )
f σ+ (r0 ,r1 )
0
σ+
(r0 , r1 ) = f r̄0 + f r̄1 −−−−0−−1→ f (r̄0 + r̄1 ) −−−−−−−→ f r0 + r1 .
In view of (1.3), (1.4), since f induces identity on π1 , i. e. f# is the identity map, one
has the diagrams
/ f r̄f (s̄t̄) f r̄f σ· (s,t) / f r̄f st
==
==
==
==f· (r̄,st)
==
==
=
=
f· (r̄,s̄t̄)
f (r̄σ· (s,t))
/ f (r̄st)
f (r̄(s̄t̄))
@
==
f [r̄,s̄,t̄] ==f σ· (r,st)
==
=
ϕ· (r,s,t)
(f r̄f s̄)f t̄
f ((r̄s̄)t̄)
f (rst)
NNN
NNN
p8
p8
p
p
N
NNN
p
p
N
NNN
pp
pp
NNN
NNN
ppp
ppp
NN&
f· (r̄,s̄)f t̄
f (σ· (r,s)t̄)
ppp f σ· (rs,t)
&
ppp f· (r̄s̄,t̄)
f (r̄s̄)f t̄
f (rst̄)
NNN
p8
p
NNN
p
pp
NNN
ppfp(rs,t̄)
NN&
p
f σ· (r,s)f t̄
p
·
p
f r̄(f s̄f t̄)
@
[f r̄,f s̄,f t̄] f r̄f· (s̄,t̄)
f rsf t̄
/ f rs + f rs
0
88 1
88
88
88 f+ (rs0 ,rs1 )
88
f+ (r̄ s̄0 ,<r̄ s̄1 )
<<
88
<<
88
<<
8
f (σ· (r,s0 )+σ· (r,s1 ))
/ f (rs0 + rs1 )
f (r̄s̄0 + r̄s̄1 )
A
99
99
s̄
f r̄ 0
99 f σ (rs ,rs )
s̄1
+
0
1
99
99
99
99
9
ϕ·+ (r,s0 ,s1 )
f r̄(f s̄0 + f s̄1 )
f (r̄(s̄0 + s̄1 ))
f (r(s0 + s1 ))
LLL
??
s9
~>
~
??
L
ss
~
L
s
L
~
s
??
LLL
ss
~~
??
L
ss
~~
s
??
s
f· (r̄,s̄0 +s̄1 )
f (r̄σ+ (s0L,s1 ))
??
ss
LLL
ss f σ· (r,s0 +s1 )
~~
??
s
~
L
f r̄f+ (s̄0 ,s̄1 )
s
LLL
~
s
??
~~
ss
LL%
?
ss
~~
f r̄f (s̄0 + s̄1 )
f (r̄s0 + s1 )
@@
9
rr
@@
r
r
r
@@
rr
@@
rrr
@@
r
r
r
@@
rrr f (r̄,s +s )
@@
r
f r̄f σ+ (s0 ,s1 )
r
·
0
1
@@
rr
@
rrr
f r̄f s0 + s1
f r̄f s̄0 + f r̄f s̄1
G
f s̄0
f r̄
f s̄1
f σ· (r,s0 )+f σ· (r,s1 )
/ f (r̄s̄ ) + f (r̄s̄ )
0
<< 1
<<
<<
<
f· (r̄,s̄0 )+f· (r̄,s̄1 )
Journal of Homotopy and Related Structures, vol. 2(2), 2007
211
f σ· (r0 ,s)+f σ· (r1 ,s)
f r0 + r1 f s̄
/ f (r̄ s̄) + f (r̄ s̄)
/ f r0 s + f r1 s
f r̄0 f s̄ + f r̄1 f s̄
0
11 1
F
11
1
11
1
11
11
11 f+ (r0 s,r1 s)
f+ (r̄ 0 s̄,1 r̄ 1 s̄)
11
1
11
11
1
11
f r̄ 0
f (σ· (r0 ,s)+σ· (r1 ,s)) 1
f s̄
f r̄ 1
/ f (r0 s + r1 s)
f (r̄0 s̄ + r̄1 s̄)
F
11
11
r̄ 0
11
f
s̄
11 f σ+ (r0 s,r1 s)
r̄ 1
11
11
11
ϕ+· (r0 ,r1 ,s)
(f r̄0 + f r̄1 )f s̄
f ((r̄0 + r̄1 )s̄)
f ((r0 + r1 )s)
=
CC
CC
{=
CC
{{
CC
{{
CC
{{
CC
{
{
CC
{
CC
{{
{{
CC
{{
{
f
(
r̄
+
r̄
,
s̄)
f
(σ
(r
,r
)
s̄)
·
CC
+ 0C1
{{
CC
{0 1
CC
{{ f σ· (r0 +r1 ,s)
CC
{{
f+ (r̄ 0 ,r̄ 1 )f s̄
CC
{
{
{
C
CC
{
CC
{{
{{
!
{{
!
f (r̄0 + r̄1 )f s̄
f (r0 + r1 s̄)
CC
{=
CC
{{
CC
{
CC
{{
CC
{{
{
CC
{{
CC
{{ f· (r0 +r1 ,s̄)
f σ+ (r0 ,r1 )f s̄
CC
{
CC
{{
{{
!
f· (r̄ 0 ,s̄)+f· (r̄ 1 ,s̄)
Journal of Homotopy and Related Structures, vol. 2(2), 2007
212
Journal of Homotopy and Related Structures, vol. 2(2), 2007
213
f r00 + r01 + f r10 + r11
//
oo7
//
ooo
o
o
o
//
ooo
//
/f/ + (r00 +r01 ,r10 +r11 )
f (r̄00 + r̄01 ) + f (r̄10 + r̄11 )
/
7
//
o
//
f+ (r̄ 00 ,r̄ 01 )+f+ (r̄ 10 ,r̄ 11 )
ooo
//
o
/
o
//
oo
/
ooo
/
(f r̄00 + f r̄01 ) + (f r̄10 + f r̄11 ) f+ (r̄00 +r̄01/,r̄10 +r̄11 )
f (r00 + r01 + r10 + r11 )
//
..
o7
//
ooo
.
/f/ (σ+ (r00 ,roo01 )+σ+ (r10 ,r11 )) ..
..
oo
..
f ((r̄00 + r̄01 ) + (r̄10 + r̄11 ))
.f. σ+ (r00 +r01 ,r10 +r11 )
..
..
X
r̄ 00 r̄ 01
f r̄ 00 f r̄ 01
r00 r01
rij
f
f
ϕ+ ( r10 r11 )
r̄ 10 r̄ 11
f r̄ 10 f r̄ 11
06i,j61
H
f σ (r +r ,r +r )
f ((r̄00 + r̄10 ) + (r̄01 + r̄11 ))
+ 00 10 01 11
OOO
G
O
f (σ+ (r00 ,r10 )+σO+OO(r01 ,r11 ))
O'
(f r̄00 + f r̄10 ) + (f r̄01 + f r̄11 ) f+ (r̄00 +r̄10 ,r̄01 +r̄11 )
f (r00 + r10 + r01 + r11 )
OOO
G
OOO
OOO
OO'
f+ (r̄ 00 ,r̄ 10 )+f+ (r̄ 01 ,r̄ 11 )
f (r̄00 + r̄10 ) + f (r̄01 + r̄11 )
f+ (r00 +r10 ,r01 +r11 )
OOO
OOO
OOO
OO'
f σ+ (r00 ,r10 )+f σ+ (r01 ,r11 )
f σ+ (r00 ,r01 )+f σ+ (r10 ,r11 )
f r00 + r10 + f r01 + r11
where unlabeled polygons strictly commute by coherence of f and naturality. These diagrams show that the cocycles ϕ and ϕ0 representing characteristic classes of, respectively,
R and R 0 , are cohomologous.
We have thus defined a map
h−i : Crext(R; B) → H 3 (R; B).
Let us now construct a map in the opposite direction.
For a 3-cocycle ϕ = (ϕ· , ϕ·+ , ϕ+· , ϕ+ ) of R with coefficients in B let Rϕ be the
following categorical ring. The set of objects of Rϕ is R. The set of morphisms is B × R,
where (b, r) is a morphism from r to r. Identities are morphisms of the form (0, r), and
composition is given by (b, r) ◦ (b0 , r) = (b + b0 , r). Categorical group structure is as
Journal of Homotopy and Related Structures, vol. 2(2), 2007
214
follows. Addition of objects and morphisms is given by
(b0 , r0 ) + (b1 , r1 ) = (b0 + b1 , r0 + r1 );
the neutral object is (0, 0), with the neutrality constraints given by λ(r) = ρ(r) = (0, r),
the associativity constraint is given by
hr, s, ti = (ϕ+ ( 0r st ) , r + s + t) ,
and the symmetry by
{r, s} = (ϕ+ ( 0s 0r ) , r + s) .
Note that the latter two equalities are equivalent to the equality


X
r01
r01
),
rij  .
i = ϕ+ ( rr00
h rr00
10 r11
10 r11
06i,j61
Next, we define multiplication of objects and morphisms by
(b, r)(b0 , r0 ) = (br0 + rb0 , rr0 ),
the unit by 1, the associativity constraint for the multiplication by
[r, s, t] = (ϕ· (r, s, t), rst)
and the unitality constraints by
λ.(r) = (−ϕ· (1, 1, r), r),
ρ.(r) = (ϕ· (r, 1, 1), r).
Moreover, we define the distributivity constraints by the equalities
[r ss01 i = (ϕ·+ (r, s0 , s1 ), r(s0 + s1 ))
and
h rr01 s] = (ϕ+· (r0 , r1 , s), (r0 + r1 )s).
It then turns out that commutativity of the coherence diagrams necessary for Rϕ to be a
categorical ring correspond precisely to the equations expressing the cocycle condition for
ϕ.
Now suppose we are given two cohomologous 3-cocycles ϕ, ϕ0 , i. e. there is a 2-cochain
γ = (γ· , γ+ ) satisfying the required equalities. We then define a 2-homomorphism f =
(f, f+ , f· , f1 ) : Rϕ → Rϕ0 as follows. Since the underlying categories of Rϕ and Rϕ0 are
identical, we can define f to be the identity functor. Moreover we define
f+ (r0 , r1 ) = (γ+ (r0 , r1 ), r0 + r1 ),
f· (r, s) = (γ· (r, s), rs)
and
f1 = (γ· (1, 1), 1).
Then again it is straightforward to verify that the coherence conditions for f to be a 2homomorphism precisely amount to the equalities expressing the fact that ϕ0 differs from
ϕ by the coboundary of γ.
Journal of Homotopy and Related Structures, vol. 2(2), 2007
215
We have thus obtained a well-defined map
R− : H 3 (R; B) → Crext(R; B)
in the opposite direction.
Now it is obvious that constructing the characteristic class hRϕ i we can choose the maps
σ· , σ+ in the beginning of section 4 to be identities, which will produce the cocycle ϕ back.
Thus hRϕ i is equal to the cohomology class of ϕ. So one composite of our maps (from H 3
to itself) is in fact identity. For the other composite to be also the identity, it thus remains
to construct, for any categorical ring R, a 2-homomorphism between R and Rϕ for some
cocycle ϕ, inducing identity on π0 and π1 . For this, let us return to the construction of
the characteristic class of R; for that construction, we have chosen an object r̄ of R in
each isomorphism class r ∈ π0 (R) = R and morphisms σ· , σ+ , which then produced
the cocycle ϕ representing hRi. Obviously these choices can be made in such a way that
0̄ = 0, 1̄ = 1, σ+ (0, r) = λ(r̄), σ+ (r, 0) = ρ(r̄), σ· (1, r) = λ.(r̄), and σ· (r, 1) = ρ.(r̄).
Let us then use these data to define a functor f : Rϕ → R. On objects this functor is given
by f (r) = r̄ and on morphisms by
λ(r̄)−1
b+r̄
λ(r̄)
f (b, r) = r̄ −−−−→ 0 + r̄ −−→ 0 + r̄ −−−→ r̄.
This f then extends to a 2-homomorphism f = (f, f+ , f· , f1 ) : Rϕ → R with f· = σ· ,
f+ = σ+ and f1 = identity of 1.
Summarizing all of the above, we have thus proved
(4.4) Theorem. For any ring R and any R-bimodule B there is a bijection
H 3 (R; B) ≈ Crext(R; B)
between the third Mac Lane cohomology of R with coefficients in B and equivalence
classes of categorical rings R with π0 (R) = R, π1 (R) = B and the resulting bimodule structure coinciding with the original one.
References
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Cambridge Phil. Soc. (to appear).
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arxiv:math.KT/0401158.
[3] Samuel Eilenberg and Saunders Mac Lane, Homology theories for multiplicative systems, Trans. Amer.
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http://www.emis.de/ZMATH/
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This article may be accessed via WWW at http://jhrs.rmi.acnet.ge
M. Jibladze
jib@rmi.acnet.ge
A. Razmadze Mathematical Institute,
1 M. Alexidze Street, 0193 Tbilisi,
Georgia
T. Pirashvili
tp59@le.ac.uk
Department of Mathematics,
University of Leicester,
Leicester, UK