Download REVIEW OF MONOIDAL CONSTRUCTIONS 1. Strict monoidal

REVIEW OF MONOIDAL CONSTRUCTIONS
RUSTAM SADYKOV
1. Strict monoidal categories
A strict monoidal category is a category C together with a strictly associative
product ⊗ : C × C → C and a strict two sided identity u ∈ C. The strict
assiciativity means the equality of the functors
⊗ ◦ (1 × ⊗) = ⊗ ◦ (⊗ × 1) : C × C × C → C,
while the strict identity axiom means the equality
⊗(x, u) = x = ⊗(u, x)
for any object x ∈ C. For example, every monoid M determines a discrite strict monoidal category. Another example (borrowed from SaundersMacLane) is the strict monoidal category End(C) where C is an arbitrary
category. Objects of End(C) are endomorphisms of the category, while
morhpisms are natural transformations. It is a strict monoidal category
with product given by the composition of functors.
Proposition 1. The classifying space of a strict monoidal category C is a
loop space provided that π0 BC is a group.
Proof. Let C be a strict monoidal category. Then its classifying space BC is
itself a topological monoid. Indeed, the classifying space BC is a realization
of the simplicial space with simplicies (c1 , ..., ck ) given by k-tuples of composable morphisms. On the level of simplicial sets, the monoidal operation
is defined by
Nk × Nk −→ Nk ,
(c1 , ..., ck ) × (c01 , ..., c0k ) 7→ (c1 ⊗ c01 , ..., ck ⊗ c0k ).
Let B 2 C be the classifying space of the topological monoid BC. If C is a
strict monoidal category with π0 BC a group, then BC = ΩB 2 C.
2. Monoidal categories
A monoidal category is a category C together with an associative tensor
product F : C × C → C and a unit u ∈ C. The value F (x, y) of the functor
If C and D are two categories, then C × D is the category of pairs (c, d) of objectys c ∈ C
and d ∈ D. A morphism in C × D is a pair of morphisms in C and D.
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RUSTAM SADYKOV
F is often denoted by x ⊗ y. Associativity is assumed in its relaxed form. It
means that the functor F ◦ (id ×F ) is naturally isomorphic to F ◦ (F × id):
in particular, for any triple x, y, z of objects in C, there is a distinguished
isomorphism
αx,y,z : x ⊗ (y ⊗ z) 7→ (x ⊗ y) ⊗ z;
and for any four objects x, y, w, z in C, there is a commutative diagram of
distinguished isomorphisms
x ⊗ (y ⊗ (w ⊗ z))
/ (x ⊗ y) ⊗ (w ⊗ z)
/ ((x ⊗ y) ⊗ w) ⊗ z
O
/ (x ⊗ (y ⊗ w) ⊗ z)
x ⊗ ((y ⊗ w) ⊗ z)
We say that u ∈ C is a unit if the functors F (u, −) and F (−, u) are naturally
isomorphic to the identity functors: in particular, there are distinguished
isomorphisms
λx : u ⊗ x 7→ x,
ρx : x ⊗ u 7→ x,
and for any objects x, y, there is a commutative diagram of distinguished
isomorphisms
/ (x ⊗ u) ⊗ y
x ⊗ (u ⊗ y)
&
x
x⊗y
and also λu = ρu : u ⊗ u 7→ u.
Given two monoidal categories C, D, a strict monoidal functor ϕ : C → D is
a functor that respects the tensor product and the unit and the distinguished
isomorphism; in particular, ϕ(x ⊗ y) = ϕ(x) ⊗ ϕ(y) and ϕ(u) = u. There is a
more flexible notion of monoidal functors. For a monoidal functor we require
that there are natural equivalences with components F (x)⊗F (y) 7→ F (x⊗y)
and uD 7→ F (uC ) that agree with α, ρ and λ.
It is known that every monoidal category is monoidally equivalent to a strict
monoidal category. In particular, if C is a monoidal category, then BC is a
loop space provided that π0 BC is a group.
Show that the forgeting functor F from the category of R-modules to the category of
abelian groups is not a monoidal functor, in fact, F (M ⊗R N ) is not isomorphic in general
to F (M ) ⊗Z F (N ).
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3. Permutative categories
A symmetric monoidal category is a monoidal category with isomorphisms
σx,y : x ⊗ y 7→ y ⊗ x for each pair of objects x, y, which is natural in x and
y and such that the composition σx,y ◦ σy,x is the identity morphism and
ρx = λx ◦ σx,e and
x ⊗ (y ⊗ z)
x ⊗ (z ⊗ y)
/ (x ⊗ y) ⊗ z
/ z ⊗ (x ⊗ y)
/ (x ⊗ z) ⊗ y
/ (z ⊗ x) ⊗ y.
It is known (Isbell theorem) that every symmetric monoidal category is
equivalent to a strict symmetric monoidal category, which is also called permutative. In other words, a permutative category is a symmetric monoidal
category for which the distinguished isomorphisms α, λ, ρ are in fact the
identity morphisms.
Example 2. Let I denote the category of finite sets n whose morphisms
are the injective maps. The product ⊗ is the concatenation m ⊗ n = m + n,
while the inicial element is u = 0. Finally, the isomorphism σ : m ⊗ n →
n ⊗ m is a shuffle that moves the last n elements to the front.
In order to explain the relation between permutative categories with infinite
loop spaces, let me recal the notion of Segal Γ-spaces which model infinite
loop spaces.
4. Example
A topological abelian group A (with non-degenerate unit) determines a category with one object and A morphisms. Define its classifying space BA to
be the fat realization of the nerve of the category A. Every homomorphism
of abelian groups A → B defines a map of nerves and a map of classifying
spaces BA. In particular, the product homomorphism A × A → A defines a
map BA × BA → BA.
Exercise 3. Why do we insist that A is abelian? Is A × A → A a homomorphism in general. Show that if BA × BA → BA induced by the product
map exists, then for any two elements a1 , a2 ∈ A, there is a 2-disc in BA
−1
bounding a1 a2 a−1
1 a2 .
Every element g of A corresponds to a 1-simplex g in BA. Let A → ΩBA be
the map that takes g ∈ A to a loop traversing with velocity one the 1-simplex
g. Its adjoint map is ΣA → BA. It follows that BA is an abelian group,
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RUSTAM SADYKOV
and the constuction can be iterated. Thus we get a sequence of spaces B n A
together with maps ΣB n−1 A → B n A. In other words, we get a spectrum
and a generalized homology theory.
Exercise 4. Show that the generalized homology theory for A = Z2 coincides with the homology theory H∗ (−; Z2 ).
Segal greatly generalized this construction.
5. Segal Γ-spaces
For a set T , let P (T ) denote the set of all subsets of T . The category Γ is the
category of sets n = (1, 2, ..., n) and morphisms given by maps θ : S → P (T )
such that θ(α) is disjoint from θ(β) for all α 6= β. A Γ-space is a functor
Γop → Top such that Γ(0) is contractible, and
i∗1 × · · · × i∗n : Γ(n) −→ Γ(1) × · · · Γ(1)
is a weak homotopy equivalence.
Recall that a semisimplicial space is a functor Λop → Top. There is a
functor Λ → Γ defined by n 7→ n and so that δi maps to the morphism
{1, ..., n} 7→ {2, ..., n + 1}
if i = 0,
{1, ..., n} 7→ {1, ..., {i, i + 1}, n + 1}
if 0 < i < n,
{1, ..., n} 7→ {1, ..., n}
if i = n.
Thus, every Γ-space A is a semisimplicial space, and its geometric realization
|A| is well-defined.
Given a Γ-space A, the classifying space BA of A is the Γ-space for which
BA(S) = |T 7→ A(S × T )|, the geometric realization of the Γ-space T 7→
A(S × T ). It follows that A, BA, B 2 A, ... forms a spectrum, and therefore
A defines a generalized cohomology theory.
A Γ-category A is a fuctor Γop → Cat to the category of small categories
such that D(0) is equivalent to the trivial category, and the map D(S t
For S = {a1 , a2 , a3 } and T = {b1 , ..., b6 },
θ : (a1 , a2 , a3 ) 7→ ({b1 , b2 , b4 }, {b3 , b6 }, ∅)
is an example of a morphism S → T . Let A be an abelian group, and define Γ(n) to be
A × · · · × A. Then it is a Γ-space, where, for example, for above θ, we have
θ∗ (b1 , ..., b6 ) = (b1 + b2 + b4 , b3 + b6 , 0).
Recall that a semisplicial category Λ consists of sets n = {0, 1, ..., n} and strictly increasing
maps m → n. Note that every morphism in Λ is a composition of the morphisms δi : n →
n + 1 defined by {0, ..., n} 7→ {0, ..., î, ..., n + 1}.
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5
T ) → D(S) × D(T ) induced by the inclusions of S and T into S t T is an
equivalence. Note that given a Γ-category, composing the fucntor Γop →
Cat with the geometric realization functor, we get a Γ-space.
6. Γ-categories vs permutative categories
Let C be a permutative category. We will define the corresponding Γcategory ΓC. Put ΓC(n) = C n , and for a morphism θ : m → n, define
θ∗ : (c1 , ..., cn ) 7→ (d1 , ..., dm )
by dk = ⊗j cj where j ranges over indices in θ(k). Composing ΓC with
the geometric realization functor, we obtain a Γ-space n 7→ (BC)n . Its
geometric realization is B 2 C.
Thus every permutative category defines a spectrum and a generalized homology theory.
7. Symmetric monoidal categories
Now let me give an exposition of the paper ”Homotopy colimits in the category of small categories” by Thomason. In this paper, Thomason shows
that symmetric monoidal categories define infinite loop spaces. The construction of Thomason is direct, without reduction to the permutative categories.
7.1. Grothendieck construction.
Given a functor F : D → Cat, Grothendieck
R
defines a new category F , which, informally, is a bundle of categories
Fd = F (d) indexed by the objects of the base diagram D. More precisely,
the Grothendieck category consists of pairs (d, x) where x ranges over objects in Fd . A morphism from x1 ∈ Fd1 to x0 ∈ Fd0 is defined to be a
choice of a morphism k : d1 → d0 in the diagram category D and a morphism F (k)x1 → x0 in the category Fd0 . The homotopy colimit diagram
identifies the classifying space of the Grothendieck category:
Z
| F | ' hocolim|Fd |.
Here hocolim is the homotopy colimit of topological spaces |Fd | = BFd ; the
homotopy colimit is taken over the diagram D. In fact, Thomason prefers to
Example 5. If A is an abelian group, then it defines a semisimplicial space A; 1-simplexes
in A correspond to elements in the abelian group a, while 2-simplexes correspond to pairs
(a1 , a2 ); these are glued along its sides to a1 , a2 and a1 + a2 .
Now let A denote the Γ-space determined by the abelian group A. There are three maps
1 → 2, and the maps A × A → A corresonding to these three maps are the projections to
the first factor, the last factor, and taking the sum a1 + a2 . Thus the geometric realization
of the Γ-space A coincides with the classifying space of the category A.
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RUSTAM SADYKOV
work with simplicial sets and spaces, and formulates the homotopy colimit
diagram as follows:
Z
N ( F ) ' hocolim N Fd .
7.2. Generalization of the Grothendieck construction. It is known
that the Grothendieck construction works well not only for functors, but
also for op-lax functors F : D → Cat. By definition an op-lax factor F is
a collection of data. To each d ∈ D, it associates a category Fd ; to each
morphism k : i → j in D, it associates a functor F (k) : Fi → Fj . It also
associates a natural transform
f (k1 , k2 ) : F (k1 k2 ) −→ F (k1 )F (k2 )
to each composition k1 k2 of morphisms in D, and a natural transform
f (d) : F (1d ) → 1Fd to each object d. These data is required to satisfy two
properties. Namely, for
k
k
k
1
2
3
d0 ←−
d1 ←−
d2 ←−
d3
we have
F (k1 )f (k2 , k3 ) ◦ f (k1 , k2 k3 ) = f (k1 , k2 )F (k3 ) ◦ f (k1 k2 , k3 ).
Here both natural transformations are applied to the functor F (k1 , k2 , k3 );
and for every functor F , the natural transformation F is just the identity
trasformation of F . The second mentioned property is that for every morphism k : d1 → d0 in D, we have
f (d0 )Fd1 ◦ f (1, k) = 1F (k) = f (k, 1) ◦ F (k)f (d1 ).
Here, for example, the left hand side transformation is a representation of
the identity transformation of the functor F (k) : Fd1 → Fd0 :
F (k) = F (1d0 ◦ k) −→ F (1d0 ) ◦ F (k) −→ 1Fd0 ◦ F (k) = F (k)
R
For an op-lax functor F , the Grothendiek construction F is a category
as in the original Grothendieck construction for functors, except that the
composition is defined differently. In the original Grothendieck construction,
Delooping a strict monidal category C Thomason associates with C a simplicial
category
W C : ∆op −→ CAT.
Qn
It is defined by W C(n) = 1 C and for di : n → n − 1,
W C(di ) : (c1 , ..., cn ) 7→ (c1 , ..., ci ⊗ ci+1 , ci+2 , ..., cn );
the maps W C(d0 ) and W C(dn ) forget the first and the last factors respectively. Thomason
proves (Proposition 2.1.3) that
Z
|
W C| ' B 2 C.
REVIEW OF MONOIDAL CONSTRUCTIONS
7
for a morphism (k, s) and (k 0 , s0 ) we defined the composition to be (kk 0 , s ◦
F (k)s0 ). It can be pictured as follows:
x2 −−−−→ F (k 0 )x2 −−−−→ F (kk 0 )x2




F (k)s0 y
s0 y
x1
−−−−→ F (k)x1


sy
x0 .
Now in F the composition of the two morphisms is given by (s ◦ F (k)s0 ) ◦
f (k, k 0 ). It is a morphism from F (kk 0 )x2 to x0 , and in more details it is
R
f (k,k0 )
F (k)s0
s
F (kk 0 )x2 −→ F (k)F (k 0 )x2 −→ F (k)x1 −→ x0 .
8. The category of op-lax functors
Maps of Grothendieck constructions over a fixed category D correspond
to natural transformations of op-lax functors. A natural transformation
between two op-lax functors F, G : D → Cat associates with each object d
in D a functor d : Fd → Gd , and with each morphism k : d1 → d0 a natrual
transformation G(k)d1 → d0 F (k) for which certain commutative diagrams
take place. The op-lax functors together with natural transformations form
a category.
Thomason recalls (the R. Street construction) that for
R every
R op-lax functor
F , one can construct a genuine functor F̃ such that F ' F̃ .
R
Namely, let F : D → Cat beR an op-lax functor. Then F is a genuine
category, and the projection F → D is a genuine functor with which we
R
associate a comma category C ' F . The functor F̃ : D → Cat takes d
to the subcategory C whose objects are (d0 , l0 , c0 ) with d0 = d, and whose
morphisms are (1, m). If k is a morphism in D, then the morphism F̃ (d) is
defined by
(d1 , l, c1 ) 7→ (d0 , k ◦ l, c1 ),
(1, m) 7→ (1, m).
R
R
There is an obvious map F̃ → C ' F , and it is a weak equivalence.
Review. Recall that every map f : X → Y can be made into a fibration by replacing X
with the space of triples (x, l, y) where x and y are points in X and Y respectively, and l
is a path from f (x) to y.
Similarly, if f : X → Y is a functor, one defines the comma category to be a category with
objects (x, l, y) where l is a morphism from f (x) to y. A morphism in the comma category
is a pair of morphisms sx from x to x0 and sy from y to y 0 such that sy ◦ l = l0 ◦ f (sx ).
The comma category is weakly equivalent to the original category X; the projection of the
comma category to X is right adjoint to the inclusion of X that takes x to (x, 1, f (x)).
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RUSTAM SADYKOV
9. Symmetric monoidal categories vs Segal Γ-categories
We start with a symmetric monoidal category S, and we aim to construct an
op-lax functor Γop → Cat. By the above construction, this op-lax functor
defines a genuine functor Γop → Cat, i.e., a Segal Γ-category.
The mentioned op-lax functor ΓS is defined by ΓS(n) = S × · · · × S, and
for any morphism ϕ : k → n, the morphism
S(ϕ) : S × · · · × S −→ S × · · · × S
is defined as above in an obvious way.
10. Γ-spaces
10.1. Simplicial category. The category ∆ is the category of finite ordered
sets and non-decreasing maps. The skeletal category of ∆ is isomorphic to
the full subcategory of finite sets n = {0, 1, ..., n}, which we also denote by
∆. Note that every morphism in ∆ is a composition of face maps δi and
degeneracy maps σi ,
δi : n → n + 1,
σi : n → n − 1.
These maps are unique order preserving maps such that δi is injective and
misses the value i and the map σi is surjective and assumes the value i
twice. In particular, there are n + 1 morphisms δi with target n + 1 and
n − 1 morphisms σi with target n − 1.
Example 6. The terminology comes from topology. Identify n with the
n-simplex ∆n spanned by vertices labbeled by 0, 1, ..., n. What are the n
faces ∆n−1 → ∆n and n − 1 degeneracy maps ∆n → ∆n−1 .
10.2. The category Γ. The category Γop is the category of finite pointed
sets and pointed maps. Its skeletal category is isomorphic to the full subcategory of pointed sets n+ = {0, 1..., n}.
A finite ordered set is regarded pointed with the base given by the minimal
element. For i = 0, ..., n, the map δi in ∆ corresponds to the order preserving
map (n + 1)+ → n+ in the opposite direction in Γop that is surjective and
assumes the value i twice. Thus, for example,
δ0∗ : (0, 1, ..., n + 1) 7→ (0, 0, 1, ..., n),
The category ∆op has objects n and morphisms δi∗ and σi∗ :
δi∗ : (0, 1, ..., n + 1) 7→ (0, ..., i − 1, ∅, i, ..., n),
where δi∗ (j) is a subset of n that consists of δi−1 (j). Similarly,
σi∗ : (0, 1, ..., n − 1) 7→ (0, ..., i − 1, {i, i + 1}, ..., n).
REVIEW OF MONOIDAL CONSTRUCTIONS
9
···
δn∗ : (0, 1, ..., n + 1) 7→ (0, 1, ..., n, n).
∗
The map δn+1
is defined slightly differently:
∗
δn+1
: (0, 1, ..., n + 1) 7→ (0, 1, ..., n, 0).
The map σi corresponds to the injective order preserving map (n−1)+ → n+
that misses the value i + 1. Note that in Γop there is a morphism σij missing
in ∆op that simply permutes two elements i and j in a set. In fact, any
morphism in Γop is a composition of permuting morphisms and morphisms
in the category ∆op .
10.3. Simplicial objects. Given a category C, a simplicial object is a functor ∆op → C. For example, a simplicial set K consists of sets K0 , K1 , ...
together with maps of sets. Each map is a composition of maps di =
K(δi ) : Kn → Kn−1 and si = K(σi ) = Kn → Kn+1 .
10.3.1. From topological spaces to simplicial sets. Given a topological space
X, we can associated a simplicial set of singular simplices ∆n → X. Let us
spell out this construction in the case of a circle X = S 1 ; though, we will
consider only linear singular simplices to simplify the construction.
Example 7. We regard S 1 to be the quotient of [0, 1] identifying the boundary points. Then the set Kn consists of n+1 simplices uni which is a sequence
of i + 1 ≥ 1 zeros and n − i copies of 1. Simplices in the image of the degeneracy operator are called degenerate. Note that in this case there are only
two non-degenerate simplicies; one in dimension zero, and one in dimension
1.
10.3.2. From categories to simplicial sets. Every category D defines a simplicial set, called the nerve N D of D. The 0-simplices are in bijective
correspondence with objects of D, the 1-simplices are in bijective correspondence with morphisms, and the n-simplices are the compositions of n
composable morphisms. Here the face map dj : Nn D → Nn−1 D replaces
morphisms aj and aj+1 with their composition; while the degeneracy map
sj : Nn D → Nn+1 D adds the identity morphism at the j-th place. For example, if D is the category with one object and one non-identity morphism,
then N D is the simplicial set of a circle.
Under the action of dj : Kn → Kn−1 , for example, we have
n−1
un
i 7→ ui−1
if
j > i + 1,
un
i
if
j ≤ i + 1.
7→
un−1
i
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RUSTAM SADYKOV
For example, let D be the category of an abelian group A. Then Nn D = An ,
the map dj replaces aj and aj+1 with a single element aj + aj+1 , while sj
adds zero at the j-th position.
10.3.3. From simplicial sets to topological spaces. We have seen how to
pass from topological spaces to simplicial sets. Conversely, there is a construction that allows us to pass from simplicial sets to topological spaces.
Under this construction, we take one copy (u, ∆n ) of the simplex ∆n for
each element u ∈ Kn . And then identify them by means of the equalities
(di u, ∆n−1 ) = (u, δi ∆n−1 ) and (si u, ∆n+1 ) = (u, σi ∆n+1 ). In particular,
when the geometric realization is taken one can forget about all degenerate
simplices.
10.4. Γ-objects. A pointed category is a category with a zero object, i.e.,
an object 0 such that for any object x in the category there is a unique
morphism 0 → x and a unique morphism x → 0.
For any category C, a simplicial object is a functor ∆op → C. If C is
pointed, then a Γ-object is a functor Γop → C with Γ(0) = ∗. A Γ-object
is completely detemined by the restriction of A to the full subcategory of
pointed sets n+ .
A special Γ-object is a Γ-object such that the natural map A(S ∨ T ) →
A(S) × A(T ) is an equivalence for every pointed sets S and T . Thus, for
a special Γ-object A, the object A(S) is equivalent to A × · · · × A where
the number of factors is |S|. Let us now spell out the definition. We know
that in Γop every morphism is a composition of basic morphisms, which
are permuting morphisms, coface and codegeneracy operators. Permuting
morphism in A(S) simply permutes the factors. Every face operator is a
product of the identity map and the face operator d∗1 : 2+ → 1+ ; the latter
induces an operation
A(1+ ) × A(1+ ) ' A(2+ ) −→ A(1+ ).
The associativity of the operation is clear, while the identity is the element in
the image of A(0+ ) → A(1+ ) of the map induced by the morphism 0+ → 1+ .
In fact, the monoidal operation is commutative, since the operator d∗1 is the
same as d∗1 ◦ σ12 . Finally, the codegeneracy operator s∗i omits the i-th factor
in A(1+ ) × · · · × A(1+ ).
Note that A(1+ ) is not a strict monoid, since we do not really have a strict
operation; the operation is defined to be a map not from the product of
objects A(1+ ) but from an equivalent object A(2+ ).
Thus, for example, a Γ-space consists of a family of simplicial sets A(n+ ), one for each
non-negative integer n, together with maps A(n+ ) → A(m+ ) for each map of pointed sets
n+ → m+ . In particular, each space A(n+ ) has n! symmetries. Furthermore, the maps
A(n+ ) → A(m+ ) are equivariant with respect to these symmetries. A Γ-space differes
from a simplicial space in that its maps are equivariant.
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11
A basic example of a Γ-object is a Γ-space associated with an abelian topological group A. In this case A(n+ ) is the product of n-copies of A and
all the maps are strict. Since every Γ-space is a simplicial space, there is
a well-defined notion of the geometric realization of A. In this particular
example, the geometric realization |A| of the Γ-space A coincides with the
classifying space BA of the abelian topological group A. Indeed, a simplex
in BA corresponds to a diagram
a
a
a
1
2
n
· −→
· −→
· · · −→
·
Similarly, a simplex in |A| corresponds to an element in An . The i-th face
of a simplex in BA corresponds to replacing two consequeting arrows ai and
ai+1 with one arrow ai + ai+1 . The same is true for a simplex in |A|. Finally,
the i-th degeneracy forgets the i-th component.
10.5. Extension of Γ-objects. The Γ-functor A admits various extensions.
We require that A preserves colimits, and thus define A over infinite sets. For
a simplicial set K, define AK to be the simplicial space with (AK)n = A(Kn )
for n ≥ 0.
For example, let K be the simplicial set of the two-point-set S 0 , and A the
Γ-space of an abelian group A. Then in each dimension n the simplicial set
has precisely two simplices, namely, the base simplex ∗ = (0, ..., 0) and the
simplex 1 = (1, ..., 1). Consequently, the simplicial set AK(n) = A(1+ )n
is precisely the nerve of the category associated with the abelian group
A.
For another example, let K be the simplicial set of the circle S 1 , and A the
Γ-space associated with an abelian topological group A. We consider S 1 to
be the quotient of [0, 1] that identifies the two boundary points. Then in
K = S 1 there are precisely n + 1 simplices in Kn , namely, simplices ui that
consist of i + 1 copies of 0 and n − i copies of 1. In other words,
uni = (00 , ..., 0i−1 , 1i , ..., 1n )
with the first occurence of 1 at the position i. The simplex un0 is a basepoint;
then each Kn is isomorphic to n+ .
Thus (AK)n = A(n+ ). For example, the 0-space is ∗, and the n-th space is
An . Consider, now, a morphism δi : n → n + 1 in the category ∆. It defines
the map dj : Kn+1 → Kn on the sets of simplices, namely,
(un0 , ..., unn+1 ) 7→ (u0n−1 , un−1
, ..., un−1
, ujn−1 , · · · , un−1
n )
1
j
In other words, it is a surjective map that assumes the j-st value twice:
dj
uj = (00 , ..., 0j−1 , 1j , ..., 1n+1 ) −→ (00 , ..., 0j−1 , 1j , ..., 1n ) = uj ,
dj
uj = (00 , ..., 0j , 1j+1 , ..., 1n+1 ) −→ (00 , ..., 0j−1 , 1j , ..., 1n ) = uj .
12
RUSTAM SADYKOV
Thus the morphism dj : An+1 → An replaces aj and aj+1 with their sum,
while the morhpism sj simply forgets the j-th factor. In other words A(δj ) : An+1 →
An and A(σj ) are the expected maps. To summarize, the geometric realization of the simplicial set A(S 1 ) is the classifying space of the nerve of the
categroy A.
Next, let us consider the case of the simplicias set S n . We regard S n to be
the smash product S 1 ∧ · · · ∧ S 1 . In other words, the set Nk S n consists of
n-tupple of k-simplices x1 , ..., xn in S 1 such that each of the simplicies is
distinct from the basesimplex. All the tupples containing the basesimplex
represent the basesimplex in Nk S n . Therefore, the set Nk S n is isomorphic
to (kn)+ . The desired simplicial set is k 7→ A(kn+ ), and the construction
repeats the one we have already seen; the realization of the simplicial set
A(S n ) is B n A.
10.6. From Γ-spaces to spectra. Let A be an arbitrary Γ-space. We
have seen in the previous section that if A is associated with an abelian
group, then the classical spectrum consists of the spaces A(S n ) for n ≥ 0.
In general, we define the spectrum associated with A to be the spectrum
A:
A(S 0 ), A(S 1 ), A(S 2 ), · · · .
We do not require that A is associated with an abelian group. To complete
the definition we need to define the maps
S 1 ∧ A(S n ) −→ A(S n+1 );
recall that all spaces under the consideration are simplicial spaces.
In order to define the suspension maps, let us construct a map
K ∧ A(L) −→ A(K ∧ L)
where K and L are arbitrary simplicial sets. On the level m, we need to
construct a map
Km ∧ A(Lm ) −→ A(Km ∧ Lm ).
Note that for each k ∈ Km , we have a map Lm = k × Lm → Km ∧ Lm and
the induced map A(Lm ) → A(Km ∧ Lm ). These maps define the desired
map.
10.7. Simplicial Γ-spaces. In practice it is convenient to consider not only
Γ-spaces, but also simplicial Γ-sets. In other words, functors Γop → Sset∗
with values in pointed simplicial sets. Recall that a pointed simplicial set
differs from a simplicial set in that it is a functor K : ∆op → Set∗ with values
in the category of pointed sets. In fact, the base point in all sets K(n) is
determined by the distinguished point in K(0). The basepoints in K(n) are
obtained from that in K(0) by the degeneracy maps.
REVIEW OF MONOIDAL CONSTRUCTIONS
13
Let A be a simplicial Γ-set and let K be an arbitrary simplicial set. Then
composing K with A we obtain a simplicial object ∆op → Sset∗ , i.e., a
bisimplicial set AK. Its realization is defined to be the diagonal realization
of the bisimiplicial set. However, we may consequently take the composition
of the two realizations (in two different ways) and the resulting space is
always the same.
Thus we can first take the geometric realization in the direction of the differentals in A. Thus, if |A| denote the Γ-space obtained from A by applying
the geometric realization functor, then after taking A-directional geometric
realization of AK we obtain the simplicial space |A|K.
Now, let us take first the K-directional differential. This case is more complicated, therefore let us first consider the example where K is the simplicial
set of S 1 and A is the simplicial Γ-space associated with a simplicial abelian
group A. In this case we have a bisimplicial set
k 7→ A(k + )
where A(k + ) = A × · · · × A is the product of k-copies of a simplicial set A.
In particular, for each n, we have a simplicial set
k 7→ An (k + ),
where An is the Γ-set that associates to each k + the the set of n-simplicies
in Ak . The geometric realization of this simplicial set is |An |. Thus it is a
Γ-space n 7→ (BA)n . Its geometric realization is the classifying space of the
simplicial group A.