Simplicial fibrations
D. Fernández-Ternero J. M. García Calcines E. Macías-Virgós J. A. Vilches

Evolution of the early Universe in the scale invariant theory
    [
    December 30, 2023
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We undertake a systematic study of the notion of fibration in the
setting of abstract simplicial complexes, where the concept of
“homotopy” has been replaced by that of “contiguity”. Then a
fibration will be a simplicial map satisfying the “contiguity
lifting property”. This definition turns out to be equivalent to
a known notion introduced by G. Minian, established in terms of a
cylinder construction K × I_m. This allows us to prove
several properties of simplicial fibrations which are analogous to
the classical ones in the topological setting, for instance: all
the fibers of a fibration have the same strong homotopy type, a
notion that has been recently introduced by Barmak and Minian; any
fibration with a strongly collapsible base is fibrewise trivial;
and some other ones. We introduce the concept of “simplicial
finite-fibration”, that is, a map which has the contiguity
lifting property only for finite complexes. Then, we prove that
the path fibration PK → K× K is a finite-fibration, where
PK is the space of Moore paths introduced by M. Grandis. This
important result allows us to prove that any simplicial map
factors through a finite-fibration, up to a P-homotopy
equivalence. Moreover, we introduce a definition of “Švarc
genus” of a simplicial map and, and using the properties stated
before, we are able to compare the Švarc genus of path
fibrations with the notions of simplicial LS-category and
simplicial topological complexity introduced by the authors in
several previous papers. Finally, another key result is a
simplicial version of a Varadarajan result for fibrations,
relating the LS-category of the total space, the base and the
generic fiber.






 

 

§ INTRODUCTION


In recent years there has been a renovated interest in abstract
simplicial complexes, as a setting which is well suited for
discretizing topological invariants and for designing computer
algorithms. Under this point of view and boosted by the
increasing computer capacities, several classical theories have
been developed, thus providing new powerful tools
like persistent homology <cit.> or discrete Morse
theory <cit.>, which are being applied in robotics,
neural networks or big data mining.

However, there is a lack of development of other ideas in
this new field of “applied algebraic topology”, like
Lusternik-Schnirelmann category or topological complexity, which
classically needed the use of notions such as homotopy, fibrations
or cofibrations.


In the framework of abstract simplicial complexes, the classical
notion of “contiguity” between simplicial maps <cit.>  plays
the role that “homotopy” plays in the context of topological
spaces. This notion has received new attention in the last years
after the work of J. Barmak and G. Minian <cit.>. They showed
that the equivalence under contiguity classes is the same as the
equivalence by “strong collapses”, a highly interesting idea
which is related on one hand with the classical Whitehead
collapses, and on the other hand with the theory of posets and
finite topological spaces <cit.>.

Using the ideas above, several of the authors have recently
introduced a notion of LS-category in the simplicial setting,
which generalizes the well known notion of “arboricity” in graph
theory <cit.>. Moreover,  we also introduced a
notion of topological complexity,  defined in purely combinatorial
terms <cit.>. Both invariants have similar properties to
the classical ones and also new results arise.

As a collateral result, cofibrations were studied in
<cit.>, but a systematic study of the notion of
simplicial fibration was lacking. This study is the aim of the
present paper.

The contents are as follows:

In Section <ref> we recall the basic notions of simplicial
complexes  and contiguity classes.

In Section <ref> we introduce two possible definitions of simplicial fibration in terms of a contiguity lifting
property and we show that in fact they are equivalent to a third
one introduced by Minian in <cit.>: a simplicial map
p E → B is a simplicial fibration if for any simplicial
map H K × I_m → B and any simplicial map f:K×{0}→ E, there is a simplicial map H̃ K ×
I_m → E such that p∘H̃=H and H̃∘
i_0^m= f (see Definition <ref>). There is a more general
notion of simplicial finite-fibration if we limit the
lifting property to finite complexes K in the definition
above. It will be necessary to obtain several key results along the paper.




In Section <ref> we give several basic examples and
constructions, including products and pullbacks of simplicial
fibrations. Then, we introduce (Section <ref>) the notion of
Moore path and the space K of Moore paths on a
simplicial complex K. This notion has been developed in <cit.>
by M. Grandis. The main result of this section is that the path map K →
K× K is a simplicial finite-fibration (Theorem
<ref>).

In Section <ref> we prove the important result that all
the fibers of a simplicial fibration have the same strong homotopy
type. In the same line, we adapt another classical result by
showing (Section <ref>) that a simplicial fibration with a
collapsible base is trivial. Here, “collapsible” means that
there is a finite sequence of strong collapses and expansions
transforming the base onto a point. Our next result (Section
<ref>) is a simplicial version of Varadarajan's theorem (see
Theorem <ref>) relating the LS-category of the total
space, the base and the generic fiber of a fibration.

In the last sections of the paper we introduce several new ideas.
The first one is based on a notion of “P-homotopy” modelled on
Moore paths (see Definition <ref>), which allows us to
prove the P-equivalence of the complexes K and K and to
give a general result  about the factorization of any map into a
P-equivalence and a finite-fibration (Section <ref>).

On the other hand, the P-homotopy equals
the usual contiguity property for finite complexes. This allows us to define in Section
<ref> a general notion of Švarc genus for
simplicial maps and to discuss its relationship with the
simplicial LS-category and the discrete topological complexity
introduced in our previous papers <cit.>.






§ SIMPLICIAL COMPLEXES AND CONTIGUITY


We start by briefly recalling the notions of simplicial complex
and contiguity. We are assuming that the reader
is familiarized with these notions as well as others that will be
appearing throughout the paper (see, for instance, <cit.> for
more details on this topic).



An (abstract) simplicial complex is a set V together with
a collection K of finite subsets of V such that if σ∈
K and τ⊆σ then τ∈ K.

Notice that K is not necessarily finite in the above definition.
As usual, K will denote the simplicial complex and V(K) the
corresponding vertex set.


Given two simplicial complexes K and L, a simplicial map
from K to L is a set map φ V(K)→ V(L) such
that if σ∈ K then φ(σ)∈ L.



Let K, L be two simplicial complexes.  Two simplicial maps
φ,ψ K→ L are contiguous <cit.>
if, for any simplex σ∈ K, the set
φ(σ)∪ψ(σ) is a simplex of L; that is,
if v_0,…,v_k are the vertices of σ then the vertices
f(v_0),…,f(v_k),g(v_0),…,g(v_k) span a simplex of L.


This relation, denoted by φψ, is reflexive and
symmetric, but in general it is not transitive. In order to
overcome this fact we use the notion of contiguity class.


Two simplicial maps φ,ψ K → L are in the same
contiguity class with m steps, denoted by φψ, if there is a finite sequence 
    φ=φ_0 ⋯φ_m=ψ
 of contiguous simplicial  maps
φ_i K → L, 0≤ i≤ m.


It is straightforward to prove the following:


Let f  K → L be a simplicial map between the simplicial
complexes K and L. If g V(K) → V(L) is a map such
that f g then g is a simplicial map.








Now we recall a formal notion of combinatorial homotopy
introduced by Minian in <cit.>. First of all we need a
triangulated version of the real interval [0,n].


For n≥ 1, let I_n be the one-dimensional simplicial complex
whose vertices are the integers {0,…,n} and the edges are
the pairs {j,j+1}, for 0≤ j<n.


<cit.>
Let K and L be two simplicial complexes. The categorical
product K× L is the simplicial complex whose set of
vertices is V(K× L)=V(K)× V(L), and  whose simplices
are given by the rule: σ∈ K× L if and only if
_1(σ)∈ K and _2(σ)∈ L, where _1,_2
are the canonical projections.


Given two simplicial maps, Minian proved that belonging to the
same contiguity class is equivalent to the existence of a
simplicial homotopy between them, modelled by a simplicial
cylinder.

<cit.>
Two simplicial maps f,g K → L are in the same contiguity
class, with m steps, f∼ g, if and only if there exists some m≥ 1 and
some simplicial map  H K × I_m→ L such that
H(v,0)=f(v) and H(v,m)=g(v), for all vertices v∈ K.



The preceding proposition holds when we consider K× I_m
to be the categorical product, but notice that the proof
does not work for the more usual notion of simplicial product,
namely the so-called simplicial cartesian product (see
<cit.>).


Barmak and Minian introduced <cit.> the so-called
strong homotopy type for simplicial complexes. Two simplicial
complexes have the same strong homotopy type, denoted by K∼
L, if they are related by a finite sequence of two kind of
simplicial moves, namely, strong collapses and expansions. An
elementary strong collapse consists of removing the open star
around a dominated vertex, where a vertex v is dominated by
another vertex v^' if every maximal simplex that contains
v also contains v^'. A complex is called strongly
collapsible if it has the same strong homotopy type of a point.

Strong homotopy type is deeply related to the notion of contiguity
between simplicial maps. More precisely, the following result
holds:


<cit.>
Two simplicial complexes K and L have the same strong homotopy
type if and only if there are simplicial maps φ K→
L and ψ L → K such that ψ∘φ∼ 1_K
and φ∘ψ∼ 1_L.






§ DEFINITIONS OF SIMPLICIAL FIBRATION

The goal of this section is to establish a notion of fibration
in the simplicial context. As we shall see, there are several
options of doing this, depending on the particular kind of lifting
property we deal with.


Our first definition of fibration in the simplicial context
corresponds to simplicial maps with the contiguity lifting
property with respect to any simplicial complex.

A simplicial map p E → B is a type I simplicial
fibration if  for any simplicial complex K (finite or not),
given  any two  contiguous simplicial maps f,g  K→ B, f
 g, and  any map f̃ K → E such that p∘f̃= f, there exists a simplicial map g̃ K →
E such that f̃ and g̃ are contiguous, f̃g̃, and p∘g̃=g (see Figure
<ref>).




A second option to give a simplicial notion of fibration consists
in generalizing the first definition to contiguity classes with a
given number of steps (see Definition <ref>).


A simplicial map p E → B is a type II simplicial
fibration if for any simplicial complex K, for any two
simplicial maps f,g  K→ B in the same contiguity class,
f ∼ g, with m steps, and for any map f̃ K →
E such that p∘f̃= f, there exists a simplicial map
g̃ K → E such that f̃ and g̃ are
in the same contiguity class with m steps, f̃∼g̃, and p∘g̃=g.


At this point we are ready for the third definition of simplicial
fibration in terms of the notion of homotopy, introduced by Minian
(see Prop. <ref>).


The map p  E → B is a type III simplicial
fibration if given simplicial maps H K× I_m→ B and
φ K×{0}→ E as in the following
commutative diagram:

    K×{ 0} @^(->[d]_i_0^m[r]^φ    E [d]^p 
    
    K× I_m @–>@<+0ex>[ru]^H [r]^ H     B


there exists a simplicial map H K
× I_m → E such that H∘ i_0^m=φ and
p∘H=H.









The three definitions of simplicial fibration are equivalent.


Type I ⇒   Type II: Assuming that the
simplicial map p  E → B satisfies Definition <ref>,
let us consider a simplicial complex K and two simplicial maps
f,g  K → B in the same contiguity class with m steps,
that is, there exists a finite sequence of direct contiguities

    f =φ_0 φ_1 ⋯φ_m = g.

Taking a simplicial map f such that p∘f̃= f, by Definition <ref>, there is a simplicial map
φ_1 K → E such that fφ_1 and p∘φ_1 =
φ_1. Iterating the same argument for every φ_i with
i=1,…, m, we obtain a finite sequence of direct
contiguities, f =φ_0 φ_1 ⋯φ_m,
where p∘φ_i = φ_i, i=1,…,m.
In particular, for i=m we get a simplicial map g=φ_m satisfying f̃∼g̃ with
m steps and p∘g̃=g since φ_m = g. So, we conclude that P is a type II simplicial fibration.













Type II ⇒   Type III: Let us assume that the
simplicial map p  E → B satisfies Definition
 <ref>. Consider a simplicial complex K and simplicial maps
φ K×{0}→ E and H  K× I_m →
B such that Diagram (<ref>) is commutative.

Now, we define φ_i  K → B by
φ_i(v)=H(v,i), where v∈ V(K) and i∈ V(I_m). By
means of Lemma <ref>, we only need to prove the
contiguity condition. These maps are contiguous because given
τ∈ K the following fact holds true:

    φ_i(τ)∪φ_i+1(τ)=H(τ×{i,i+1})∈ B

since H is a simplicial map.
Then, by hypothesis, there exists a finite chain of  simplicial maps φ_i K → E and
direct contiguities, with m steps,
φ_0 φ_1 ⋯φ_m,  such that p∘φ_i=φ_i, for all i=0,…,m.


Hence, the map H K× I_m → E given by
H(v, i)=φ_i(v) is simplicial, by an argument analogous to (<ref>), and satisfies
H∘ i_0^m= φ and p∘H=H. So, H satisfies Definition <ref>.






























Type III ⇒  Type I: Let us assume that the
simplicial map p  E → B satisfies Definition <ref>.
Consider a simplicial complex K and two contiguous simplicial
maps f,g  K → B. Now, by Proposition <ref>,
with m=1, there exist a homotopy H  K× I_1 → B
such that H(v,0)=f(v) and H(v,1)=g(v) for all v∈ V(K).

Consider a simplicial map f̃ K → E such that
p∘f̃= f. By Definition <ref>, there is a
simplicial map H K × I_1 → E such that
H∘ i_0^1=φ, where p∘φ (v,0)=
f(v) for all v∈ V(K), and p∘H=H. Let g K → E given by
g(v)=H(v,1), where v∈ V(K). By
Proposition <ref>, we conclude that
fg.



Notice that the complex K that we considered in the definitions
above may not be finite.


 Observe that it is possible to
restrict these definitions to the cases where K is finite. This
allows us to introduce the corresponding notions of
simplicial finite-fibration of type I, II and III, which
are equivalent by the finite version of Theorem <ref>.






§ EXAMPLES AND PROPERTIES


In this section we will introduce some important examples of
simplicial fibrations.  The following proposition will give us the
first basic ones. Notice that, unless otherwise specified, we will
use the notion of type III simplicial fibration given in
Definition <ref>.

 



  (i) Any simplicial isomorphism is a simplicial fibration.


  (ii) If * denotes the one-vertex simplicial complex,
then the constant simplicial map E→ * is a simplicial fibration,
for any  simplicial complex E.


  (iii) The composition of simplicial fibrations is a simplicial
fibration.




(i) is easily checked; indeed, if p E≅⟶B is a simplicial isomorphism, then, given any
m≥ 0 and a commutative diagram p∘φ =H∘
i^m_0, we have that the composition H
p^-1∘ H satisfies the required conditions. Item (ii) is
also straightforward, since for any simplicial map φ
K×{0}→ E the vertex map K× I_m→ E,
given by (v,i)↦φ (v,0), is simplicial.

Now, in order to prove (iii), consider p E→ B
and q B→ C simplicial fibrations, m≥ 1, and
the following commutative diagram of simplicial maps:

    K×{0}[rr]^φ@^(->[d]_i_0^m      E[d]^q∘ p
    K× I_m[rr]_G       C

Since q is a simplicial fibration we can consider a simplicial
map G K× I_m→ B satisfying
G∘ i_0^m=p∘φ and q∘G=G. Finally, as p is a simplicial fibration we can
also consider a simplicial map G K×
I_m→ E such that G∘ i_0^m=φ
and p∘G=G. From these conditions we
have that (q∘ p)∘G=q∘G=G.


For the next result we need to recall the pullback construction
for simplicial complexes. Given any pair of simplicial maps
f K→ M and g L→ M their
pullback is given as the following diagram:

    K× _ML[rr]^f'[d]_g'      L[d]^g 
    K[rr]_f       M
 where K× _M L is the full
simplicial subcomplex of K× L whose underlying vertex set
is given by those pairs of vertices (v,w)∈ K× L
satisfying f(v)=g(w). The induced simplicial maps f' and g'
are given by f'(v,w)=w and g'(v,w)=v. It is plain to check
that this construction is the pullback of f and g in the
category of simplicial complexes.


Let p E→ B be a simplicial fibration and
f K→ B any simplicial map. Then the simplicial
map p^' K× _B E→ K induced by p in
the pullback

    K× _BE[rr]^f'[d]_p'      E[d]^p 
    K[rr]_f       B
 is also a simplicial fibration.



Take any commutative diagram where L is a simplicial
complex:

    L×{0}[r]^g@^(->[d]_i_0^m   K× _BE[d]^p'
    L× I_m[r]_(.6)G   K
 Considering the
composition of this diagram with the pullback square and using the
fact that p is a simplicial fibration, one can take a simplicial
map H L× I_m→ E satisfying p∘H=f∘ G and H∘ i_0^m=f'∘ g.
By the pullback property there is an induced simplicial map
G L× I_m→ K× _B E making
commutative the following diagram:

    L× I_m@–>[dr]^G@/^1pc/[drrr]^H@/_1pc/[ddr]_G
       K× _BE[rr]^f'[d]_p'      E[d]^p
      
       K[rr]_f      B
 By the universal property of the pullback we have that G∘ i_0^m=g.



Let K and L be simplicial complexes and K× L be their
categorical product. Then the canonical projections _1
K× L→ K and _2 K× L→ L
are simplicial fibrations.



One has just to take into account  part (ii) of Proposition
<ref> because the following square is a pullback:

    K× L[r]^_2[d]__1   L[d] 
    K[r]    *



Another interesting example of simplicial fibration is given by
the product of simplicial fibrations. Recall that, if f_1
K_1→ L_1 and f_2 K_2→ L_2 are simplicial maps, then
one can construct their product simplicial map:

    f_1× f_2 K_1× K_2→ L_1×
    L_2

defined as

    (f_1× f_2)(v_1,v_2) (f_1(v_1),v_2(v_2)),

for any vertex (v_1,v_2)∈ K_1× K_2.



Let p_1 E_1→ B_1 and p_2 E_2→ B_2 be
simplicial fibrations. Then their product p_1× p_2 is also
a simplicial fibration.



Let K be a  simplicial complex, consider

    φ
    =(φ_1,φ _2) K×{0}→ E_1× E_2
 and

    H=(H_1,H_2) K× I_m→ B_1× B_2
 simplicial maps
such that (p_1× p_2)∘φ =H∘ i_0^m. As
p_i∘φ _i=H_i∘ i_0^m and p_i is a simplicial
fibration, there is a simplicial map H_i
K× I_m→ E_i such that p_i∘H_i=H_i and
H_i∘ i_0^m=φ _i for i=1,2. Hence, the
simplicial map

    H =(H_1,H_2) K×
    I_m→ E_1× E_2
 verifies the expected conditions.










§ THE PATH COMPLEX K




 §.§ Moore paths


Consider the one-dimensional simplicial complex Z, whose
vertices are all the integers i∈ℤ and whose
1-simplices are all the consecutive pairs {i,i+1}, that is,
ℤ is a triangulation of the real line.


Let K be a simplicial complex. A Moore path in K is a
simplicial map γ Z→ K which is eventually
constant on the left and eventually constant on the right, i.e.,
there exist integers i^-,i^+∈ℤ satisfying the two
following conditions:


  (i) γ (i)=γ (i^-), for all i≤ i^-,

  (ii)  γ (i)=γ (i^+), for all i≥ i^+.


Obviously, if i^-=i^+ we have the constant map.
For a non constant Moore path γ Z→ K we can
consider the integers

    γ^-   max{i^-γ (i)=γ
    (i^-),for
    all  i≤ i^-}
    γ ^+   min{i^+γ (i)=γ
    (i^+),for all i≥ i^+}.

Observe that γ ^-<γ ^+.



The images α(γ)γ (γ ^-) and
ω(γ)γ (γ ^+) are called the
initial vertex and final vertex of γ,
respectively. When γ is constant we set γ
^-=0=γ ^+.


If a,b∈ℤ with a≤ b, [a,b] will denote the
full subcomplex of ℤ generated by all vertices i with a≤
i≤ b. Considering this notation, any Moore path γ in
K may be identified with the restricted simplicial map γ [γ^-,γ ^+]→ K. The interval
[γ ^-,γ ^+] will be called the support of
γ.

If γ is a Moore path in K with support [γ
^-,γ ^+], then one can take the reverse Moore path
γ as 
    γ(i)=γ (-i),

whose support is [-γ ^+,-γ ^-]. Notice that this
reparametrization describes γ in the opposite direction.

If γ is a Moore path in K with support [γ
^-,γ ^+] such that γ ^+ - γ ^- = m, then we
define one normalized Moore path |γ | I_m→ K
as

    |γ |(i)=γ (i+γ^-) .

The advantage of this reparametrization is that the support of
|γ | is [0,m], and therefore it will be more manageable
when dealing with simplicial fibrations.












Given γ ,δ Moore paths in K such that
ω(γ)=α(δ), the product path γ
*δ it is defined as

    (γ *δ)(i)=γ (i-δ ^-),     i≤γ ^++δ ^-,
    δ (i-γ ^+),     i≥γ ^++δ ^-.



It is not difficult to see that the support of γ * δ
is [γ ^-+δ ^-,γ ^+ +δ ^+]. The product
of Moore paths is strictly associative, that is, given Moore paths
γ, δ, ε  such that
ω(γ)=α(δ) and
ω(δ)=α(ε), then

    γ *(δ *ε )=(γ *δ )*ε.


Moreover, if c_v denotes the constant path in a vertex v∈ K,
then it is immediate to check that γ *c_w=γ
=c_v*γ where v=α(γ) and w=ω(γ).




 §.§ The path complex


Next we will consider a suitable notion of Moore path complex
associated to a simplicial complex K. In order to do so we need
to recall some categorical properties in the category SC
of simplicial complexes and simplicial maps.

Indeed, if K and L are simplicial complexes, we define the
simplicial complex L^K, whose vertices are all simplicial maps
f K→ L and where we consider as simplices the
finite sets {f_0,…,f_p} of simplicial maps K→
L such that

    ⋃ _i=0^pf_i(σ )∈ L,   for any simplex σ∈ K.


It is not difficult to check that this definition induces a structure of
simplicial complex in L^K. Moreover, denoting by × the
categorical product in SC, we have that the evaluation map 
    L^K× K→ K,  
    (f,v)↦ f(v),
is simplicial. This fact allows us
to establish a natural bijection

    𝐒𝐂(M× K,L)≡𝐒𝐂(M,L^K).


Observe that for a simplicial map
h L→ L', there is a well defined map
h^K L^K→ (L')^K, which preserves the identities and
the compositions, that is, we have a functor
(-)^K𝐒𝐂→𝐒𝐂. More is true, the
functor (-)× K𝐒𝐂→𝐒𝐂 is left
adjoint to the functor (-)^K𝐒𝐂→𝐒𝐂.


Let K be a simplicial complex. We define the Moore path complex
of K, denoted by K, as the full subcomplex of K^Z
generated by all the Moore paths γ Z → K.


Then, {γ _0,…,γ _p}⊂ K defines
a simplex in K if and only if 
    {γ _0(i),…,γ
    _p(i),γ _0(i+1),…,γ _p(i+1)}
 is a simplex
in K, for any integer i∈ℤ.

An interesting property of K is that, for any bounded interval
[a,b]⊂ℤ, the complex K^[a,b] is, in fact, a full
subcomplex of K:

    K^[a,b]⊂ K.

Moreover, given a simplicial map f  K → L, since the
composite f∘γ is a Moore path in L for any Moore
path γ in K, we obtain the Moore path complex functor
𝐒𝐂→𝐒𝐂. One can check that
this functor preserves binary products and equalizers. Therefore
 preserves finite limits and, in particular, pullbacks. In
general  does not preserve limits; for instance,  does
not preserve infinite products.


The initial and final vertices of any given Moore path γ
define simplicial maps α K→ K and ω K → K.






 §.§ The path fibration

 The aim of this subsection  is to establish the following
important example of simplicial finite-fibration (see Remark
<ref>).



If K is any simplicial complex, then the following simplicial
map

    π=(α ,ω ) K→ K× K
 is a simplicial finite-fibration where α and ω are
the maps given in Definition <ref>.



Let L be a finite simplicial complex, m≥ 1, and a
commutative diagram of simplicial maps

    L[rr]^φ@^(->[d]_i_0^m       K[d]^π=(α,ω) 
    L× I_m[rr]_(G,H)      K× K
 We
recall that, as L is finite, there exists a factorization
of φ of the form

    L[rr]^φ[dr]        K 
        K^[a,b]@^(->[ur]

Indeed, if [a(v),b(v)] denotes the support of φ (v) for
any vertex v∈ L, then we may take a=min{a(v) v∈
L} and b=max{b(v) v∈ L} due to the fact that L is finite. We therefore obtain a
commutative diagram

    L[rr]^φ@^(->[d]_i_0^m      K^[a,b][d]_(_a,_b)@^(->[rr]        K[d]^π = (α,ω)
    L× I_m[rr]_(G,H)      K× K@=[rr]       K× K
 where _a and _b
K^[a,b]→ K denote the evaluation simplicial maps at
a and b, respectively. Our aim is to construct a simplicial
map

    Ω L× I_m→ K^[a-m,b+m]⊂ K

satisfying π∘Ω =(G,H) and Ω∘
i_0^m=φ. Notice that there is natural inclusion K^[a,b]↪ K^[a-m,b+m].


    L@^(->[d]_i_0^m[rr]^φ      K^[a,b]@^(->[rr]       K^[a-m,b+m][d]^(_a-m,_b+m)@^(->[rr]        K[d]_π
    L× I_m@–>[urrrr]^Ω[rrrr]_(G,H)            K× K@=[rr]       K× K


For this task, consider the simplicial maps
G,H L→ K^I_m respectively
associated to G,H L× I_m → K from the natural
adjunction (<ref>). Then, for any fixed vertex v∈ L, we
have three Moore paths

    φ(v)    [a,b]→ K, 
    G(v)    I_m→ K, 
    H(v)    I_m→ K,

satisfying
φ(v)(a)=G(v)(0) and φ(v)(b)=H(v)(0).

For any fixed i∈ I_m we may also consider the i-truncated Moore paths, G(v)_i and
H(v)_i, whose supports are contained in [0,i] (and
therefore in [0,m]), given by:

    (G(v)_i)(j)=G(v)(j),    0≤ j≤ i 
    G(v)(i),    i≤ j≤ m  (H(v)_i)(j)=H(v)(j),    0≤ j≤ i 
    H(v)(i),    i≤ j≤ m

Observe that
G(v)_i starts at G(v)(0)=φ(v)(a) and ends at
G(v)(i)=G(v,i); moreover, when i=0 we obtain the
constant path at G(v,0). Similarly, H(v)_i starts at
H(v)(0)=φ(v)(b) and ends at H(v)(i)=H(v,i); we
also have that, when i=0, it is the constant path at H(v,0).

Let us denote by -G(v)_i the reverse of the normalized
Moore path of G(v)_i, that is,

    -G(v)_i|G(v)_i|


In this way, -G(v)_i, φ(v) and
H(v)_i can be multiplied and its multiplication
(-G(v)_i)*φ(v)*H(v)_i is a Moore path
starting at G(v,i) and ending at H(v,i), so that for i=0 it
equals  φ(v). Such multiplication gives rise to the
desired simplicial map Ω by establishing the identity

    Ω
    (v,i)(j) ((-G(v)_i)*φ(v)*H(v)_i)(j).
 Its explicit
expression is as follows:

    Ω (v,i)(j)=
    
    G(v,i),     a-m≤ j≤ a-i 
    
    G(v,a-j),     a-i≤ j≤ a 
    φ(v)(j),     a≤ j≤ b 
    
    H(v,j-b),     b≤ j≤ b+i 
    
    H(v,i),     b+i≤ j≤ b+m.

At this point we will prove that Ω is a simplicial map.
Taking into account the exponential law, this is equivalent to
prove that the following map is simplicial:

    Ω L× I_m × [a-m,b+m] → K,    (σ, i, j)↦Ω(σ,i)(j).

Now, given σ∈ L, i∈ I_m and j∈ [a-m,b+m], we will prove that

    μΩ(σ×{i,i+1}×{j,j+1})

is a simplex in K. Taking into account that Ω
is piecewise defined, we have to consider the following cases:


    
  * If i=0 and j=a-i, then
    
    μ=Ω(σ×{0,1}×{a,a+1})=φ(σ)({a,a+1}).


    
  * If i=0 and j=b+i, then
    
    μ=Ω(σ×{0,1}×{b,b+1})=H(σ×{0,1}.


    
  * If i> 0 and j=a-i, then
    μ=G(σ×{i-1,i}).

    
  * If i> 0 and j=a-1, then
    μ=G(σ×{0,1}).

     
  * If i> 0 and j=b, then
     μ=H(σ×{0,1}).

     
  * If i> 0 and j=b+i, then μ=H(σ×{i,i+1}).



Since φ, G and H are simplicial maps, in all cases we
obtain a simplex in K and hence Ω is a
simplicial map.

Finally, it is not difficult to check that Ω satisfies the
commutativity in Diagram (<ref>).





 §.§ P-homotopy


The maps α and ω of Definition <ref> allow us
to introduce the following notion of homotopy:



Given f,g K→ L simplicial maps, we will say that f
is P-homotopic to g, denoted by f≃ g, when there
exists a simplicial map

    H K→ L

such that α∘ H=f and ω∘ H=g.




This relation is certainly reflexive and symmetric but presumably
non transitive (see <cit.>). More is true, it is compatible with left and right compositions.














Let f K→ L be a simplicial map. Then f is said
to be a P-homotopy equivalence if there exists a
simplicial map g L→ K such that g∘ f≃
1_K and f∘ g≃ 1_L.


Taking into account the links between strong homotopy type and contiguity classes established by Barmak and Minian in <cit.> and <cit.>, notice that if P-homotopies of the above definition are switched by finite sequences of contiguous maps, then we conclude that f is a strong equivalence. 

An alternative equivalent form for the notion of  P-homotopy is
given the following result:


Let f,g K→ L be simplicial maps. Then f≃
g if and only if there exists a sequence of simplicial maps
{f_i K→ L}_i∈ℤ indexed by the
integer numbers, such that:


  (i) f_i∼ _c f_i+1 are contiguous maps;

  (ii) For all vertex v∈ K there exist integers n_v and m_v such
that f_i(v)=f(v), for all i≤ n_v and f_i(v)=g(v) for all
i≥ m_v.




If H K → L is a homotopy between f and g then we
define the simplicial map f_i H(-)(i) K→ L.
Obviously, as H is simplicial and {i,i+1} is a simplex in
ℤ, we have that f_i and f_i+1 are contiguous, so condition
(i) holds. Moreover, since for any v∈ K we have that H(v)∈ L is a Moore path with support [n_v,m_v], condition (ii) is
also fulfilled as α∘ H=f and ω∘ H=g.

Conversely, given a sequence of simplicial maps
{f_i K→ L}_i∈ℤ satisfying (i) and (ii)
we define H K → L as

    H(v)(i) f_i(v)

It is straightforward to check from (i) that H is simplicial
and from (ii) that α∘ H=f and ω∘ H=g.







Now we will see the relationship between the homotopy ≃ and the
class of contiguity ∼. It is clear that Proposition
<ref> can be rewritten as follows: f and g are in
the same class of contiguity if and only if there exist integers
a≤ b and a simplicial map H K× [a,b]→
L such that H(v,a)=f(v) and H(v,b)=g(v), for all vertex v∈
K.


Let f,g K→ L be simplicial maps. Then


  (i) If f∼ g,
then f≃ g.


  (ii) If K is finite and f≃ g, then f∼ g.

In particular, if
K is finite, then f and g are in the
same contiguity class, f∼ g, if and only if they are
-homotopic, f≃ g.



First consider H K× [a,b]→ L such that
H(v,a)=f(v) and H(v,b)=g(v). Taking the composition of the
adjoint simplicial map K→ L^[a,b], given by the bijection
(<ref>), with the inclusion L^[a,b]⊂ L, we
obtain a simplicial map H' K→ L satisfying
α∘ H'=f and ω∘ H'=g.

Conversely, suppose that K is finite and consider a simplicial
map G K→ L satisfying α∘ G=f and ω∘ G=g. For any v∈ K we have that G(v) is a Moore path in
L with support [a_v,b_v]. Taking a=min{a_v v∈ K}
and b=max{b_v v∈ K} we obtain a factorization

    K[rr]^G [rd]        L
       L^[a,b]@^(->[ur]
 Considering the adjoint of K→ L^[a,b] we obtain a
simplicial map G' K× [a,b]→ L such that
G'(v,a)=f(v) and G'(v,b)=g(v), for all v∈ K. Hence, this
means that f∼ g.


We have the following immediate result:


Let f K→ L be a simplicial map between finite
simplicial complexes. Then f is a P-homotopy equivalence if
and only if f is a strong equivalence.






§ HOMOTOPY FIBER THEOREM



Recall that for any natural number n, I_n is the full subcomplex of ℤ generated by
all vertices 0≤ i≤ n. If n^'≥ n, then a
subdivision map t I_n^'→ I_n is any simplicial map satisfying t(0)=0 and t(n^')=n.
The proof of the following Lemma can be found in <cit.>.


Given natural numbers n, m, there exist n^', m^' with
n^'≥ n and m^'≥ m and a simplicial map ϕ I_n^'× I_m^'→ I_n× I_m
with the following sketches on the boundaries:




    < g r a p h i c s >
  





where a^'→ a, b^'→
b, etc. are subdivision maps. Moreover, there exist n^''≥ n^', m^''≥ m^' and a
simplicial map ϕ^' I_n^''×
I_m^''→ I_n^'× I_m^'
with the opposite sketches on the boundaries such that the
composition

    ϕ∘ϕ^' I_n^''×
    I_m^''→ I_n× I_m
 has the form
ϕ∘ϕ^' =t_1× t_2, where t_1
I_n^''→ I_n and t_2
I_m^''→ I_m are subdivision maps.


Another technical lemma is the so-called Simplicial Pasting
Lemma, whose proof can be found in <cit.>.


Let U,V be subcomplexes of a simplicial complex K and let f
 U → L, g V → L be simplicial maps such that
f(v)=g(v), for all vertex v∈ U∩ V.  Then, the vertex
function h U∪ V→ L, defined as

    h(v)=
    f(v),     if v∈ U, 
    
    g(v),     if v∈ V,

is a simplicial map.



Given v,v^' vertices in a simplicial complex K and a Moore path γ
joining v and v^', that is, α(γ)=v and
ω(γ)=v^', let us consider the Moore path
γ |γ|, that is the normalized
of the reverse Moore path of γ. It is plain to check that
the simplicial map H I_2n× I_n→ B defined as

    H(i,j)γ (max{n-i,j}),     0≤ i≤ n, 
    γ (max{i-n,j}),     n≤ i≤ 2n,

gives a homotopy H between γ*γ and
the constant Moore path c_b^' relative to {0,n}.
Similarly, one can also check that γ *γ∼ c_b
rel. {0,n}.


Using this language, a simplicial complex B is connected
if and only for any pair of vertices b,b^'∈ B there
exists a normalized Moore path γ [0,n]→ B such
that γ (0)=b and γ (n)=b^'.

Given a simplicial fibration f E→ B and a vertex b_0∈
B, the simplicial fiber of p over b_0, denoted by
F_b_0, is the full subcomplex of E generated by all the
vertices e∈ E such that p(e)=b_0. In other words,
F_b_0=p^-1(b_0).

The following theorem is one of the main results of this paper,
since it shows that our notion of simplicial fibration has nice
properties like the homotopy invariance of the fiber.

Let p E→ B be a simplicial fibration where B is a
connected simplicial complex. Then any two simplicial fibers of
p have the same strong homotopy type.



Let γ I_n→ B be a Moore path such that γ
(0)=b and γ (n)=b^'. Let us first check that there is
a simplicial map F_b→ F_b^'.  Indeed, if
i_b denotes the inclusion F_b⊂ E and _2
F_b× I_n→ I_n the natural projection, then take a
lift in the diagram:

    F_b×{0}[rr]^i_b@^(->[d]       E[d]^p 
    F_b× I_n[rr]_γ∘pr_2@–>[urr]^γ      B
 Clearly, we have
that γ(v,n)∈ F_b^', for all v∈
F_b. Therefore there is an induced simplicial map γ
^♯:F_b→ F_b^' given by γ
^♯(v)γ(v,n).

Suppose δ I_n→ B is another Moore path
satisfying that δ (0)=b, δ (n)=b^' and there
is H I_n× I_m→ B such that Hγ∼δ rel. {0,n}.  We shall prove that γ^♯
and δ^♯ are in the same contiguity class. In
particular we will prove that γ ^♯ is independent,
up to contiguity class, of the chosen lift γ.

First, let us consider the commutative diagram:

    F_b× J_nm[rr]^(.6)f @^(->[d]       E[d]^p 
    F_b× I_n× I_m[rr]_(.6)H∘      B

where:


  * J_nm=(I_n×{0})∪ ({0}×
I_m)∪ (I_n×{m}),

  * pr F_b× I_n×
I_m→ F_b× I_n is the projection on the two first
complexes,

  * f is the
simplicial map (see Lemma <ref> above) defined as

    f(v,i,0)   =γ(v,i), 
    
    f(v,0,j)   =v, 
    
    f(v,i,m)   =δ(v,i).



Now, take the simplicial maps ϕ I_n^'×
I_m^'→ I_n× I_m and ϕ^'
I_n^''× I_m^''→
I_n^'× I_m^' given by Lemma <ref>.
As ϕ(I_n^'×{0})⊂ J_nm, we have a lift
H F_b× I_n^'× I_m^'→ E, represented by the dotted arrow in the composite
diagram:

    F_b× I_n^'×{0}[rr]^1_F_b×ϕ@^(->[d]       F_b× J_nm[rr]^f @^(->[d]       E[d]^p 
    F_b× I_n^'× I_m^'[rr]_1_F_b×ϕ@–>[urrrr]^H      F_b× I_n× I_m[rr]_H∘pr      B


Taking into account that ϕ^'
(J_n^''m^'')⊂ I_n^'×{0} and that ϕ∘ϕ^'  =t_1× t_2, we have
that the simplicial map 
    H^'H∘
    (1_F_b×ϕ ^'  ) F_b×
    I_n^''× I_m^''→ E

satisfies that H^'  (v,n^'',j)∈
F_b^', for all v∈ F_b and j∈
I_m^''. Therefore, there is an induced simplicial
map

    H H^'  (-,n^'',-)^'  F_b× I_m^''→ F_b^'.

It is not difficult to see that H(v,0)=γ ^♯(v) and H(v,m^'')= δ ^♯(v); that is, γ ^♯∼δ ^♯ are in
the same contiguity class.

It is immediate to check that, if c_b  I_n→ B
denotes the constant Moore path in b∈ B, then c_b^♯
is (up to contiguity class) the identity 1_F_b
F_b→ F_b. Moreover, if γ^'
I_n→ B and δ ^' I_m→ B are
Moore path spaces such that γ (0)=b, γ (n)=b^'
=δ (0) and δ (m)=b^'', then we can prove
that

    (γ
    *δ )^♯ F_b→ F_b^''
 belongs to
the same contiguity class of the composition δ
^♯∘γ ^♯. Indeed, the simplicial map
γ *δ F_b× I_n+m → E
defined as

    (γ *δ)(v,i)=γ(v,i),     0≤ i≤ n 
    δ(γ(v,n),i-n),     n≤ i≤
    n+m
 gives a lift:

    F_b×{0}[rr]^i_b@^(->[d]       E[d]^p 
    F_b× I_n+m[rr]_(γ *δ )∘_2@–>[urr]^γ *δ      B


This proves that (γ *δ )^♯ and δ
^♯∘γ ^♯ are in the same contiguity class.

Using the above reasonings and the fact that γ*γ∼ c_b^' rel. {0,n} and γ
*γ∼ c_b rel. {0,n}, we conclude the proof
of the result.






§ COLLAPSIBLE BASE



Minian's Lemma (Lemma <ref>) and the Simplicial Pasting Lemma
(Lemma <ref>) will be crucial for the next results. We
start with this fairly general proposition.



Let p E→ B be a simplicial fibration and let
F_0,F_1 K× I_n→ E be simplicial maps such
that p∘ F_0 and p∘ F_1 are in the same contiguity
class with m steps. Let H p∘ F_0∼ p∘ F_1
be the map given by Proposition <ref> and analogously,
let G F_0|_K×{0}∼ F_1|_K×{0} be the
map given by Proposition <ref>. Consider the following
commutative diagram:

    K×{0}× I_m[rr]^(.6)G @^(->[d]       E[d]^p 
    K× I_n× I_m[rr]_(.6)H       B
 

Then, for suitable q≥ n and p≥ m, there exist
subdivision maps t_1 I_q→ I_n and t_2 I_p→
I_m and a simplicial map

    H' K× I_q× I_p→ E

such that H^' defines a homotopy
F_0 ∘ (1_K× t_1)∼ F_1 ∘ (1_K× t_1), which is an extension of
G∘ (1_K×{0}× t_2).



Consider the commutative diagram

    K× J_nm[rr]^(.6)f @^(->[d]       E[d]^p 
    K× I_n× I_m[rr]_(.6)H       B
 where
J_nm is the full subcomplex of I_n× I_m given in the
proof of Theorem <ref> and f is the
simplicial map (see Lemma <ref>) defined as

    f(v,i,0)   =F_0(v,i), 
    
    f(v,0,j)   =G(v,0,j), 
    
    f(v,i,m)   =F_1(v,i).

Now we take the simplicial maps ϕ I_n'×
I_m'→ I_n× I_m and ϕ'
I_n^''× I_m^''→
I_n^'× I_m^' given by Lemma <ref>. As
ϕ∘ (I_n^'×{0})⊂ J_nm we have a
lift H K× I_n^'× I_m^'→ E, represented by the dotted arrow in the composite
diagram:

    K× I_n^'×{0}[rr]^1_K×ϕ@^(->[d]       K× J_nm[rr]^f @^(->[d]       E[d]^p 
    K× I_n^'× I_m^'[rr]_1_K×ϕ@–>[urrrr]^H      K× I_n× I_m[rr]_H       B

Renaming q=n^'', p=m^'', and taking into account that 
    ϕ
    ^' (J_n^''m^'')⊂ I_n^'×{0}
 and that ϕ∘ϕ' =t_1× t_2, we have that the simplicial map
H^'H∘ (1_K×ϕ' ) fulfills all the expected
requirements.


Now, for our next result, we need to establish the following natural
definition:


Let p E→ B be a simplicial fibration and consider
f,g K→ E simplicial maps. Then it is said that f and
g are in the same class of fibrewise contiguity, denoted
by f∼ _p g, if there exists a finite sequence of simplicial
maps

    f=f_0∼ _c  …∼ _c f_n=g
 such that
p ∘ f=p ∘ f_i, for all i=0,…,n.


In other words, f and g are in the same class of fibrewise
contiguity if there exists a simplicial map F K×
I_n→ X satisfying


  (i) F(v,0)=f(v) and F(v,n)=g(v), for all v∈ K;


  (ii) p(F(v,i))=p(f(v)), for all v∈ K and i∈ I_n
(therefore p ∘ f=p ∘ g).


As a corollary of Proposition <ref>, we have:


Let p E→ B be a simplicial fibration and let
F_0,F_1 K× I_n→ E be two lifts of the same
map

    K×{0}[rr]^f @^(->[d]       E[d]^p 
    K× I_n[rr] @–>[urr]^F_0_F_1      B
 Then,
for a suitable q≥ n, there exists a subdivision map
t I_q→ I_n such that F_0 ∘ (1_K× t)∼
_p F_1∘ (1_K× t).



Just apply Proposition <ref> to the simplicial maps G,H
defined as G(v,0,j)=f(v,0) and H(v,i,j)=(p ∘ f)(v,0), for
all v∈ K and j∈ I_n.



Observe that the previous corollary also holds true when we
consider K×{n} in the diagram instead of K×{0}.
In this case one just have apply the corollary by carefully using
the simplicial isomorphism I_n≅→I_n
given by i↦ n-i. Using this trick one can also check that
a fibration p E→ B may be also characterized by
the existence of a lift in any diagram of the form

    K×{n}[rr] @^(->[d]       E[d]^p 
    K× I_n[rr] @–>[urr]       B


For our next result we will consider the notion of having the same
type of fibrewise contiguity.


We say that two simplicial fibrations p_1 E_1→
B, p_2 E_2→ B have the same type of
fibrewise contiguity when there exist simplicial maps f
E_1→ E_2 and g E_2→ E_1 such that
p_2 ∘ f=g, p_1 ∘ g=f and g ∘ f∼ _p_11_E_1
and f ∘ g∼ _p_21_E_2.


The following important result asserts that simplicial maps in the
same class of contiguity induce simplicial fibrations having the
same type of fibrewise contiguity.


Let p E→ B be a simplicial fibration and let
f_0,f_1 K→ B simplicial maps. Consider, for each
i=1,2, the pullback of f_i along p:

    E_i[d]_p_i[r]^f^' _i   E[d]^p 
    K[r]_f_i   B
 If f_0 and f_1 are in the same class of
contiguity, then the simplicial fibrations p_0 and p_1 have
the same type of fibrewise contiguity.



Consider a simplicial map F K× I_n→ B such
that F(v,0)=f_0(v) and F(v,n)=f_1(v), for all v∈ K. Take
F^' _0, F^' _1 lifts of the following diagrams (for
the second diagram see Remark <ref>):

    E_0×{0}[rr]^f^' _0@^(->[d]       E[d]^p
          E_1×{n}[rr]^f^' _1@^(->[d]       E[d]^p   
    E_0× I_n@–>[urr]^F^' _0[rr]_F∘(p_0× 1)      B      E_1× I_n@–>[urr]^F^' _1[rr]_F∘(p_1×
    1)      B

By the pullback property there are well defined
simplicial maps g_0 E_0→ E_1 and g_1 E_1→
E_0 satisfying the commutativities given in the following
diagrams:

    E_0@–>[dr]^g_0@/^1pc/[drr]^F^' _0(-,n)@/_1pc/[ddr]_p_0            E_1@–>[dr]^g_1@/^1pc/[drr]^F^' _1(-,0)@/_1pc/[ddr]_p_1      
       E_1[r]^f^' _1[d]_p_1   E[d]^p          E_0[r]^f^' _0[d]_p_0   E[d]^p 
       K[r]_f_1   B         K[r]_f_0   B
 Let us first check that g_0 ∘ g_1∼ _p_11_E_1.
Indeed, observe that F^' _0 ∘ (g_1× 1) and
F^' _1 are lifts of the same map

    E_1×{0}[rr]^f^' _0∘ g_1@^(->[d]       E[d]^p 
    E_1× I_n[rr]_F∘ (p_1× 1)@–>[urr]       B
 Corollary <ref> assures the existence of a
suitable subdivision map t I_n^'→ I_n
and a simplicial map G E_1× I_n^'×
I_m→ E satisfying that

    G F^' _0 ∘ (g_1× 1)∘(1× t)∼ _p F^' _1∘ (1× t)

Again, using the pullback property, there is an induced simplicial
map G E_1× I_m→ E_1 satisfying

    E_1× I_m@–>[dr]^G@/^1pc/[drr]^G(-,n^' ,-)@/_1pc/[ddr]_p_1∘
       E_1[r]^f^' _1[d]_p_1   E[d]^p
      
       K[r]_f_1   B
 A simple inspection proves that G g_0 ∘
g_1∼ _p_11_E_1.

Analogously, by applying Corollary <ref> to the diagram

    p∘ f^' _1∘ g_0=F∘ (p_0× 1)|_E_0×{n}
 for the common lifts
F^' _1∘(g_0× 1) and F^' _0, and taking into
account Remark <ref>, one can find a simplicial map
H satisfying 
    H F^' _1∘ (g_0× 1)∘
    (1× s)∼ _p F^' _0∘ (1× s)
 for a suitable
subdivision map s I_n^''→ I_n.
Taking H the simplicial map characterized by the
equalities f^' _0 ∘H=G(-,0,-) and p_0
∘H=p_0∘ we obtain H
g_1∘ g_0∼ _p_01_E_0.




Let p E→ B be a simplicial fibration where B
is strongly collapsible. Then p has the same class of fibrewise
contiguity of the trivial fibration B×
p^-1(b_0)→ B, for any vertex b_0∈ B.



Just take into account that 1_B B→ B and the
constant path c_b_0 B→ B are in the same class
of contiguity and use Theorem <ref>.


We finish this section with an interesting property whose proof
relies on the proof of Theorem <ref>. Indeed, remember
that, in the statement of such theorem, we have pullbacks
(i=0,1):

    E_i[d]_p_i[r]^f^' _i   E[d]^p 
    K[r]_f_i   B
 where f_0∼ f_1 (i.e., f_0
and f_1 are in the same class of contiguity). From the proof one
obtains simplicial maps g_0:E_0→ E_1 and
g_1:E_1→ E_0 (with p_1∘ g_0=p_0 and p_0∘
g_1=p_1) and Minian simplicial homotopies

    F'_0:E_0× I_n→ E;F'_1:E_1×
    I_n→ E
 satisfying F'_0:f'_0∼ f'_1∘
g_0 and F'_1:f'_0∘ g_1∼ f'_1.  Moreover, g_0 and g_1
satisfy g_1∘ g_0∼ _p_01_E_0 and g_0∘ g_1∼
_p_11_E_1. In particular g_1∘ g_0∼ 1_E_0 and
g_0∘ g_1∼ 1_E_1.

We easily have the following lemma:


    Consider the pullback of a simplicial fibration p:E→ B
    along a simplicial map f:B→ B such that f∼ 1_B:
    
    E'[d]_p'[r]^f'   E[d]^p 
    B[r]_f   B
 Then f'∼ g_0 and f'∘ g_1∼ 1_E. In
    particular, f' is a strong equivalence with g_1:E→
    E' as a homotopy inverse.



    Just observe that the following square is a pullback
    
    E[d]_p[r]^1_E   E[d]^p 
    B[r]_1_B   B
 and apply the argument above.



And finally our result. Observe that such result completes
Proposition <ref>.


    Consider the pullback of a simplicial fibration p:E→ B
    along a simplicial map f:K→ B:
    
    E'[d]_p'[r]^f'   E[d]^p 
    K[r]_f   B
 If f is a strong equivalence, then f' is also a strong
    equivalence.



Suppose g:B→ K a simplicial map such that g∘
f∼ 1_K and f∘ g∼ 1_B. Consider first the following
diagram, where E” is the pullback of p' along g:

    E”[r]^g'[d]_p”   E'[d]^p'[r]^f'   E[d]^p 
    B[r]_g    K[r]_f    B
 As f∘ g∼ 1_B, by
the lemma above (note that the composite of pullbacks is a
pullback) one can find simplicial maps g_0:E”→ E and
g_1:E→ E” such that g_1∘ g_0∼ 1_E”,
g_0∘ g_1∼ 1_E, f'∘ g'∼ g_0 and f'∘ g'∘
g_1∼ 1_E. Therefore, g' has a left homotopy inverse and f'
has a right homotopy inverse:

    (g_1∘ f')∘ g'∼ 1_E”f'∘ (g'∘ g_1)∼ 1_E

Similarly, from the diagram of pullbacks

    E”'[r]^f”'[d]_p”'   E”[d]^p”[r]^g'   E'[d]^p'
    K[r]_f    B[r]_g    K
 we have that g'
has a right homotopy inverse. It straightforwardly follows that
g' is a strong equivalence. Since f'∘ g'∼ g_0 we
conclude that f' is a strong equivalence.







§ VARADARAJAN'S THEOREM

As an application of Theorem <ref> about the strong
homotopy type of the fibers of a fibration, we prove in this
section a simplicial version of a well-known result from
Varadarajan for topological fibrations <cit.>,
establishing a relationship between  the LS-categories of the
total space, the base and the homotopic fiber. A more general
result for smooth foliations was proved by Colman and Macías
in  <cit.>.


The simplicial LS-category (K) of the simplicial complex
K is the least integer n≥ 0 such that K can be covered by
n+1 subcomplexes K_j such that each inclusion ι_j
K_j⊂ K belongs to the contiguity class of some constant map
c_v_j K_j → K.









This notion is the simplicial version of the well known homotopic
invariant (X), the so-called Lusternik-Schnirelmann category
of the topological space X <cit.>. It has been introduced
by the authors in <cit.>, its most important
property being the invariance by strong homotopy equivalences.

Accordingly to Theorem <ref>, all the fibers
p(v), v∈ B, of a fibration with connected base p E → B have the
same strong homotopy type, so we call generic fiber F of
the fibration any simplicial complex into that equivalence class,
and its simplicial category (F) is well defined.


Let p E → B be a simplicial fibration with connected
base B and generic fiber F.Then

    (E)+1≤ ( (B) +1)( (F)+1).









Let B=m, and take a categorical covering U_0,…,U_m
of B.  From Theorem <ref> we know that all the
fibers have the same strong homotopy type. We identify F to the
fiber p(v) over some base point v∈ B. Let F=n,
with V_0,…,V_n a categorical covering of F.

For each i∈{0,…,m} let I_i U_i⊂ B be the
inclusion. By definition of simplicial category, the map I_i is
in the contiguity class of a constant map, say c_i U_i →
B. Since B   is path connected we can assume that c_i is the
constant map  corresponding to the base point v. Consider the
map

    p_i=I_i∘ p p(U_i)→ B.

If  ϵ_i p(U_i)⊂ E is the inclusion, then
p∘ϵ_i=p_i. Now, from I_i∼ c_v it follows that
p_i∼ c_v, the latter being the constant map with domain
p(U_i). By the contiguity lifting property, there exists

    G_i p(U_i)→ E
 such that ϵ_i ∼ G_i and p∘ G_i=c_v.
The latter means that the  map G_i takes its values in F. We
denote

    g_i p(U_i)→ F
 the map given by g_i(v)=G_i(v).
In this way G_i=ι_F∘ g_i, where ι_F F ⊂ E is the inclusion.

For each i∈{0,…,m}, j∈{0,…,n}, we take the subcomplex

    W_ij=g_i(V_j)⊂ p(U_i)⊂ E.

Since B=U_0∪⋯∪ U_m implies
E=p(U_0)∪⋯∪ p(U_m), and since
F=V_0∪⋯∪ V_n implies p(U_i)=W_i0∪⋯∪
W_i n, it follows that {W_ij} is a   covering of E.

It only remains to prove that each W_ij is categorical in E.
Let g_ij W_ij→ V_j  be the restriction of g_i to
W_ij⊂ p(U_i). We know that each V_j⊂ F is
categorical, so the inclusion J_j V_j⊂ F is in the
contiguity class of some constant map c_j V_j → F, whose
image is some vertex f_j of F. Then the composition

    W_ijg_ij→V_iJ_j⊂F,

belongs to the contiguity class of the constant map f_j
W_ij→ F since

    J_j∘ g_ij∼ c_j∘ g_ij=f_j.

Let ϵ_ij W_ij⊂ E be the inclusion, that
is, the restriction of ϵ_i p(V_j)⊂ E to
W_ij. Since ϵ_i∼ G_i it follows that
ϵ_ij∼ G_ij, the latter being the restriction of
G_i to W_ij. Finally we have

    ϵ_ij∼ G_ij=ι_F∘  J_j ∘ g_ij = ι_F∘ f_j ,

so the inclusion W_ij⊂ E is in the contiguity class of a constant map.






§ FACTORIZATION


In this section we will see that any simplicial map may be
considered, in a homotopical sense, as a simplicial fibration.









For our main result in this section we will use the fact that
(α,ω ) K → K× K is a simplicial
finite-fibration, for any simplicial complex K (see Theorem
<ref>).


Let f K → L be a simplicial map. Then there is a
factorization

    K[dr]_j [rr]^f       L
        P_f [ur]_p
 where j is a P-homotopy
equivalence and p is a simplicial finite-fibration.



We shall denote by P_f=K× _L  L the pullback of f
K→ L along α L → L, where α is the
initial vertex map. In this way we have a commutative square:

    P_f[d]_α '[r]^f'    L[d]^α
    K[r]_f    L


As we have previously observed, the simplicial complex P_f is
defined as the full subcomplex of K× L whose set of
vertices is given as

    P_f={(v,γ )∈ K× L f(v)=α (γ )}
 and α^',f^'
are the restrictions of the obvious projections. We define the
simplicial map j K→ P_f as j(v) (v,c_f(v)),
where c_f(v)ℤ→ L denotes the constant Moore path at
f(v). On the other hand, the simplicial map p P_f→ L
is defined as p(v,γ )ω (γ ), where
ω is the final vertex map. Obviously, we have that p∘
j=f.

Let us first check that p is a simplicial fibration. Indeed,
consider the following commutative diagram:

    P_f[rr]^f^'[d]_(α^', p)       L[d]^(α,ω)
    K× L[rr]^f× 1_L[d]__1      L× L[d]^_1
    K[rr]^f       L


It is not difficult to check that the bottom diagram is a
pullback. As, by construction, the composition diagram is also a
pullback, we obtain that the top diagram must be a pullback. But
then, being (α ,ω ) a simplicial finite-fibration, the
simplicial map (α^',p) P_f→ K× L must be
also a simplicial finite-fibration. Therefore, the composition of
the fibrations p=_2∘ (α^',p) P_f→ L is
a finite-fibration.

Next we check that j K→ P_f is a P-homotopy
equivalence. As α^'∘ j=1_K it only remains to see
that j∘α^'≃ 1_P_f. We directly define the
homotopy H P_f→ P_f as H(v,γ )(i)
(v,γ _i) where γ _i∈ L is defined as

    γ _i(j)=γ (j),     j≤ i 
    γ (i),     j≥
    i,

for any γ∈ L and i∈ℤ. Observe that
γ _i is the i-truncated Moore path of γ and hence,
α (γ _i)=α (γ ) and ω (γ
_i)=γ (i), so (v,γ _i) is a well defined element in
P_f. Moreover, if (v,γ )∈ P_f is fixed, we are going to
check that H(v,γ )∈ P_f; equivalently, we have to
prove that H(v,γ )ℤ→ P_f is a simplicial map and a
eventually constant Moore path, that is, for any integer i∈ℤ, the image

    H(v,γ )({i,i+1})={(v,γ _i),(v,γ
    _i+1)}={v}×{γ _i,γ _i+1}
 is a
simplex in P_f, or equivalently, that {γ _i,γ
_i+1} is a simplex in L. But this is true, as, for any
integer j∈ℤ, the set of vertices 
    {γ
    _i(j),γ _i(j+1),γ _i+1(j),γ _i+1(j+1)}

is precisely γ ({j,j+1}) or γ
({i,i+1}), depending on whether j≤ i or j≥ i. Thus
H is a simplicial map. Now, taking into account that

    H(v,γ )(i)=(v,c_f(v)),     i≤γ ^- 
    
    (v,γ ),     i≥γ ^+
 we have
that H(v,γ )∈ P_f. Moreover, α∘ H=j∘α ' and ω∘ H=1_P_f. Therefore, to conclude
that H is a P-homotopy equivalence between j∘α^' and 1_P_f, we just need to justify that it is a
simplicial map. For this final task, considering the exponential
law and the fact that P_f is a full subcomplex of K×
L^ℤ, it suffices to check that the following map is simplicial:

    Λ K× L^ℤ×ℤ→
    K× L^ℤ,(σ,γ,i)↦ (σ,γ
    _i).
 Now if consider σ a simplex in K, {γ
_0,…,γ _p} a simplex in L^ℤ and {i,i+1} a
simplex in ℤ, then we have to prove that the image

    Λ (σ×{γ _0,...,γ _p}×{i,i+1}),

that is,

    σ×{(γ _0)_i,...,(γ _p)_i,(γ
    _0)_i+1,…,(γ _p)_i+1}
 is a simplex in K×
L^ℤ, or equivalently

    {(γ _0)_i,…,(γ
    _p)_i,(γ _0)_i+1,…,(γ _p)_i+1}
 is a simplex
in L^ℤ. So take any integer j∈ℤ and consider

    {(γ _0)_i(j),…,(γ
    _p)_i(j),(γ _0)_i+1(j),…,(γ _p)_i+1(j), 
          (γ
    _0)_i(j+1),…,(γ _p)_i(j+1),(γ
    _0)_i+1(j+1),…,(γ _p)_i+1(j+1)}

Then, it
is not difficult to check that this set of vertices is precisely

    {γ _0(j),…,γ _p(j),γ
    _0(j+1),…,γ _p(j+1)}
 or 
    {γ _0(i),…,γ
    _p(i),γ _0(i+1),…,γ _p(i+1)},
 depending on
whether j≤ i or j≥ i. But whatever the case is, we have
a simplex in L because {γ _0,…,γ _p} is a
simplex in L^ℤ. We conclude the proof of the theorem.


As a consequence of this theorem new finite-fibrations appear.


For instance, for any simplicial complex K and a based vertex
v_0∈ K one can consider the full subcomplex of K

    _0 K={γ∈ Kα (γ )=v_0}.

Observe that this simplicial complex is nothing else than the
construction P_f given in the theorem above, where f{v_0}↪ K is the inclusion map. Then, the
simplicial finite-fibration associated to f is precisely the map

    p_0 K → K,  γ↦ω (γ).




There is also a special factorization that we are specially
interested in.  Although it does not come from the general
construction of the above theorem, one can check that the
following diagram gives a factorization of the diagonal map
Δ K→ K× K through a P-homotopy
equivalence followed by a simplicial finite-fibration

    K[dr]_c [rr]^Δ      K× K
        K [ur]_(α ,ω )
 We already know that
(α ,ω ) is a simplicial finite-fibration. On the other
hand, c K→ K is defined as the simplicial map
sending each v∈ K to the constant Moore path at v,
c(v) c_v. This map is, indeed, a P-homotopy equivalence:
the simplicial map α K → K satisfies α∘ c=1_K. Moreover, the homotopy H K→^2 K
defined as H(γ )(i)=γ _i, where γ _i is the
i-truncated Moore path of γ, satisfies α∘
H=c∘α and ω∘ H=1_ K; that is,
H c∘α≃ 1_ K.


 




§ ŠVARC GENUS






In the classical topological setting, the Lusternik-Schnirelmann
category can be seen as a particular case of the so-called
“Švarc genus” or sectional category of a continuous
map.

In this Section we adapt this definition to the simplicial
setting.


The simplicial Švarc genus of a simplicial map
φ K → L is the minimun integer n≥ 0 such that
L  is the union L_0∪…∪ L_n of  n+1 subcomplexes,
and for each j there exists a section σ_j of φ,
that is, a simplicial map σ_j L_j → K such that
φ∘σ_j is the inclusion ι_j L_j⊂
L.


We denote this genus by (φ). A slight modification in the above definition is
to change the equality by “being in the same contiguity class”.


The homotopy simplicial Švarc genus of a simplicial map
φ K → L, denoted by (φ), is the
minimum n≥ 0 such that L=L_0∪…∪ L_n , and for
each j∈{0,…,n} there exists an “up to contiguity
class” simplicial section σ_j of φ, that is, a
simplicial map σ_j L_j→ K such that φ∘σ_j∼ι_j.



Note that when the complex L is finite, the condition
φ∘σ_j ∼ι_j (same contiguity class) can
be changed to φ∘σ_j ≃ι_j (P-homotopy),
by Proposition <ref>.


Obviously, (φ)≤(φ). The equality holds
for some particular classes of maps.


Let p E → B be a simplicial fibration. Then
(p)=(p).



We only have to prove that (φ)≤(φ), so
let (φ)=n and L_0,…, L_n be a covering of B by
subcomplexes such that there exist simplicial maps σ_j
L_j → E with p∘σ_j ∼ι_j, as in Definition
<ref>. By Theorem <ref>, for each j we have simplicial
maps

    H_j L_j × [0,m_j]→ B
 with H_j(-,0)=p∘σ_j
and H_j(-,m_j)=ι_j the inclusion L_j⊂ B. Take a lift
H̃_j L_j× [0,m_j]→ E in the following
diagram,


L_j×{0} @^(->[d] [r]^    σ_j     E[d]^p

L_j× [0,m_j][r]^   H_j@–>[ru]^H̃_j   B


in such a way that p∘H̃=H and H̃(v,0)=σ_j(v). Then the map ξ_j L_j → E given by
ξ_j(v)=H̃(v,m_j) is simplicial and verifies

    p∘ξ_j(v)=p∘H̃(v,m_j)=H(v,m_j)=ι_j(v)=v,

so it is a true section of p. Then (φ)≤ n.


 Note that if p E → B is
a simplicial finite-fibration then Theorem <ref> above
is still true when the base B is finite.
























 §.§ Simplicial L-S category


Taking into account the notion of simplicial LS-category (Definition <ref>), what we want
to see is that it equals (as in the classical case) the Švarc
genus of a certain fibration.









Assume that the complex K is connected. Then every two
constant maps c_v_i, c_v_j, belong to the same contiguity
class, as can be easily seen by considering a path γ
[0,m] → K connecting v_i and v_j and the sequence of
contiguous maps given by the truncated paths γ_k 
[0,k] → K. So we can choose a base point v_0 and assume that
all the constant maps in Definition <ref> equal c_v_0.


Let P_0K be the full subcomplex of the path complex PK
consisting on all Moore paths whose initial point is the base
point,

    P_0K={γ∈ PKα(γ)=v_0}.


The map ω P_0K → K, sending each path to its final
point, is then the pullback of the simplicial finite-fibration
(α,ω) PK → K× K by the map f, where
f(v)=(v_0,v):



P_0K [d]_ω@^(->[r]   PK[d]^(α,ω)

K[r]^f   K× K




Let K be a connected finite complex. Then the simplicial
LS-category (K) equals the Švarc genus of the
simplicial finite-fibration ω P_0K → K.



We have to prove that (K)=(ω), but we know that the
latter equals (ω).

So, first, assume that (K)=n and let K=K_0∪…∪
K_n such that each inclusion ι_j K_j⊂ K belongs
to the contiguity class of the constant map v_0,  that is, there
is a Minian simplicial homotopy H_j K_j× [0,m_j] → K with
H_j(v,0)=v_0 and H_j(v,m_j)=v.

Take a lift H̃_j of H_j in the following diagram:



K_j×{0} @^(->[d] [r]^    c_j     P_0K [d]^ω

K_j× [0,m_j][r]^   H_j@–>[ru]^H̃_j   K


where c_j(v) is the constant Moore path at v_0, for  all  v. Define
σ_j K_j→ P_0K as σ_j(v)=H̃_j(v,m_j). Then
ω∘σ_j(v)=H_j(v,m_j)=v, so σ _j is a
section of the fibration. The map σ _j is simplicial
since 


    v∈ K_j ↦ (v,m_j)∈ K_j× [0,m_j]


is a simplicial map. We have then proved that (ω)≤ n.

Conversely, if s L → P_0K is a section of ω, each
path s(v) has initial vertex v_0 and final vertex v. Denote by
[v^-,v^+] the support of s(v). Then, since K is finite,
there is an interval [m,n] containing all the supports, so we
can define H L× [m,n] → K as H(v,i)= s(v)(i).
Then H(v,m)=s(v)(m)=s(v)(v^-)=α(s(v))=v_0, while
H(v,m)=s(v)(n)=s(v)(v^+)=ω(s(v))=v, showing that the
inclusion L⊂ K belongs to the contiguity class of the
constant map v_0.

We must check that H is simplicial: if {v_0,…,v_p} is a
simplex of L and {i,i+1} is a simplex in [m,n], the image
by H of σ×{i,i+1} is the set

    {s(v_0)(i),…,s(v_p)(i),s(v_0)(i+1),…,s(v_p)(i+1)},

which is a simplex of K because s(σ) is a simplex of P_0K.

So each covering of K by subcomplexes L verifying the
Definition <ref> gives the same covering verifying
Definition <ref>. Then (K)≤(ω).




 §.§ Discrete topological complexity

In <cit.>, a subset of the authors introduced a notion of
discrete topological complexity  which is a version of
Farber's topological complexity <cit.>, adapted to the
simplicial setting.

By using well known equivalences in the topological setting, the
simplicial definition avoids the use of any path complex.


<cit.>
The discrete  topological complexity (K) of the
simplicial complex K is the least integer n≥ 0 such that
K× K can be covered by n+1 “Farber subcomplexes”
Ω_j, where the latter means that there exist simplicial
maps σ_jΩ_j → K such that Δ∘σ_j is in the contiguity class of the inclusion
ι_jΩ_j⊂ K× K. Here Δ :K→ K× K denotes the diagonal map v↦ (v,v).


In other words:


The discrete topological complexity of the abstract simplicial
complex K is the homotopic Švarc genus of the diagonal map
Δ K → K × K, i.e., (K)=(Δ).



Our main result in this ection is the following one.
 Let K be a finite complex. The discrete topological complexity of K
equals the Švarc genus of the finite-fibration (α,ω) PK → K× K.


As was stated in Example <ref>, there is a
P-equivalence c between the complexes K and PK, in such a
way that the diagonal factors through the finite-fibration
(α,ω) PK → K × K. Since K × K is
finite, Remark <ref> applies and the homotopic Švarc
genus of Δ and (α,ω) can be computed by means
of P-homotopies. Then it is clear that

    (K)=(Δ)=((α,ω))= ((α,ω)),

where the latter equality follows from Proposition <ref>.


 


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