Table Of ContentYONEDA LEMMA FOR COMPLETE SEGAL SPACES
DAVIDKAZHDANANDYAKOVVARSHAVSKY
4
1
0 Abstract. In this note we formulate and give a self-contained proof of the
2 Yonedalemmafor∞-categoriesinthelanguageofcompleteSegalspaces.
b
e
F To the memory of I.M.Gelfand
7
Contents
]
T Introduction 1
C 1. Preliminaries 2
. 1.1. Model categories 2
h
1.2. Simplicial sets 4
t
a 1.3. Simplicial Spaces 6
m
1.4. Segal spaces 8
[ 2. The Yoneda lemma 10
2 2.1. Left fibrations 10
v 2.2. The ∞-category of spaces 13
6 2.3. The Yoneda embedding 17
5
3. Quasifibrations of simplicial spaces 20
6
3.1. Definitions and basic properties 20
5
. 3.2. Extension of fibrations, and fibrant replacements 23
1
4. Complements 25
0
4.1. Discrete iterated cylinder 25
4
1 4.2. Proofs 26
: References 30
v
i
X
r
a Introduction
In recentyears∞-categoriesor,more formally,(∞,1)-categoriesappearin vari-
ous areas of mathematics. For example, they became a necessary ingredient in the
geometricLanglands problem. In his books [Lu1, Lu2] Lurie developed a theory of
∞-categories in the language of quasi-categoriesand extended many results of the
ordinary category theory to this setting.
In his work [Re1] Rezk introduced another model of ∞-categories, which he
called complete Segal spaces. This model has certain advantages. For example, it
has a generalizationto (∞,n)-categories (see [Re2]).
It is natural to extend results of the ordinary category theory to the setting of
complete Segal spaces. In this note we do this for the Yoneda lemma.
D.K.waspartiallysupportedbytheERCgrantNo. 247049-GLC,Y.V.waspartiallysupported
bytheISFgrantNo. 1017/13.
1
2 DAVIDKAZHDANANDYAKOVVARSHAVSKY
To formulate it, we need to construct a convenientmodel of the ”∞-categoryof
spaces”, which an ∞-analog of the category of sets. Motivated by Lurie’s results,
we define this ∞-category to be the simplicial space ”classifying left fibrations”.
After this is done, the construction of the Yoneda embedding and the proof of the
Yoneda lemma goes almost like in the case of ordinary categories.
In our next works [KV1, KV2] we study adjoint functors, limits and colimits,
show a stronger version of the Yoneda lemma, and generalize results of this paper
to the setting of (∞,n)-categories.
We thank Emmanuel Farjoun, Vladimir Hinich and Nick Rozenblyum for stim-
ulating conversations and valuable remarks.
This paper is organizedas follows. To make the work self-contained, in the first
section we introduce basic definitions and discuss properties of model categories,
simplicial sets, simplicial spaces and Segal spaces, assuming only basic category
theory. Inthesecondsectionweintroduceleftfibrations,constructthe∞-category
ofspacesS,andformulateandprovetheYonedalemma. Next,inthethirdsection,
westudyquasifibrationsofsimplicialspaces,whichareneededforourargumentand
are also very interesting objects for their own. Finally, the last section is devoted
to the proof of the properties of S, formulated in the second section.
1. Preliminaries
1.1. Model categories.
1.1.1. Notation. Let C be a category. (a) For an element Z ∈ C, we denote by
C/Z the overcategoryover Z.
(b) For a pair of morphisms i : A → B and p : X → Y in C, we denote by
Hom (i,p) the set of commutative diagrams in C
C
a
A −−−−→ X
(1.1) i p
y y
b
B −−−−→ Y.
We say that i is a retract of p, if there exist α ∈ Hom (i,p) and β ∈ Hom (p,i)
C C
such that β◦α=Id .
i
(c)Wesaythatphastheright lifting property (RLP)withrespecttoi(andthat
i has the left lifting property (LLP) with respect to p), if for every commutative
diagram(1.1) there exists a morphism c:B →X such that p◦c=b and c◦i=a.
Equivalently, this happens if and only if the natural map of sets
(i∗,p ):Hom(B,X)→Hom(A,X)× Hom(B,Y)
∗ Hom(A,Y)
is surjective.
(d) Assume that C has fiber products. Then for every morphism f : X →Y in
C/Z and morphism g : Z′ → Z in C, we write g∗(f) : g∗(X) → g∗(Y) instead of
f × Z′ :X × Z′ →Y × Z′ and call it the pullback of f.
Z Z Z
(e) We say that a category C is Cartesian, if C has finite products, and for
every X,Y ∈ C there exists an element XY ∈ C, representing a functor Z 7→
Hom(Z×Y,X), which is called the internal hom of X and Y.
1.1.2. Example. Let C be the category of functors C =Fun(Γop,Set), where Γ is
a small category, and Set is the category of sets.
YONEDA LEMMA FOR COMPLETE SEGAL SPACES 3
(a) Since category Set has all limits and colimits, category C also has these
properties. Explicitly, every functor α : I → C defines a functor α(γ) : I → Set
for each γ ∈Γ, and we have lim (α)(γ)=lim (α(γ)) and similarly for colimits. In
I I
particular, category C has products.
(b) For every γ ∈ Γ, denote by F ∈ C the representable functor Hom (·,γ).
γ Γ
Then for every X,Y ∈ C there exists their internal hom XY ∈ C, defined by the
rule XY(γ) = Hom(Y × F ,X) with obvious transition maps. In other words,
γ
category C is Cartesian.
The following lemma is straightforward.
Lemma 1.1.3. Let C be a Cartesian category, and let i:A→B, j :A′ →B′ and
p:X →Y be morphisms in C. Then j has the LLP with respect to (i∗,p ):XB →
∗
XA×YA YB if and only if (i∗,j∗) : (A×B′)⊔(A×A′)(B ×A′) → B ×B′ has the
LLP with respect to p.
Definition1.1.4. (compare[GJ,II,1]). AmodelcategoryisacategoryC,equipped
with three collections ofmorphisms, calledcofibrations,fibrations andweak equiv-
alences, which satisfy the following axioms:
CM1: The category C has all finite limits and colimits.
f g
CM2 (2-out-of-3): In a diagram X → Y → Z if any two of the morphisms f,g
and g◦f are weak equivalences, then so is the third.
CM3(retract): Iff isaretractofg,andgisaweakequivalence/fibration/cofibration
then so is f.
CM4 (lifting property): Let i be a cofibration and p be a fibration. Then p has
the RLP with respect to i, if either i or p is a weak equivalence.
CM5 (decomposition property): any morphism f has a decomposition
(a) f =p◦i, where p is fibration, and i is a cofibration and weak equivalence;
(b) f =q◦j, where q is fibration and weak equivalence, and j is a cofibration.
1.1.5. Notation. (a) A map in a model category C is called a trivial cofibration
(resp. trivial fibration) if it is cofibration (resp. fibration) and a weak equivalence.
(b) By CM5, every morphism f : X → Y can be written as a composition
X →i X′ →p Y, where i is a trivial cofibration,and p a fibration. In such a case, we
say that p is a fibrant replacement of f.
(c) An element X ∈ C is called fibrant (resp. cofibrant), if the canonical map
X →pt (resp. ∅→X), where pt (resp ∅) is the final (resp. initial) object of C, is
a fibration (resp. cofibration).
For the following basic fact see, for example, [GJ, II, Lem. 1.1].
Lemma 1.1.6. A map f : X → Y in a model category C is a cofibration (resp.
trivial cofibration) if and only if it has the LLP with respect to all trivial fibrations
(resp. fibrations).
(b) A map f : X → Y in a model category C is a fibration (resp. trivial fi-
bration) if and only if it has the RLP with respect to all trivial cofibrations (resp.
cofibrations).
1.1.7. Remarks. (a) Lemma 1.1.6 implies in particular that (trivial) cofibrations
and (trivial) fibrations are closed under compositions, and that all isomorphisms
are trivial cofibrations and trivial fibrations.
4 DAVIDKAZHDANANDYAKOVVARSHAVSKY
(b)ItalsofollowsimmediatelyfromLemma1.1.6that(trivial)fibrationsarepre-
servedbyallpullbacks,andthat(trivial)cofibrationsarepreservedbyallpushouts.
(c) It follows from CM5 (a) and CM2 that every weak equivalence f has a
decomposition f =p◦i, where p is trivial fibration, and i is a trivial cofibration.
Definition 1.1.8. We call a model category C Cartesian, if C is a Cartesian cat-
egory, the final object of C is cofibrant, and for every cofibration i : A → B and
fibration p : X → Y, the induced map q : XB → XA× YB is a fibration and,
YA
additionally, q is a weak equivalence if either i or p is.
1.1.9. Remark. Taking A = ∅ or Y = pt in the definition of Cartesian model
category,we get the following particular cases.
(a) If B is cofibrant, then for every (trivial) fibration X →Y, the induced map
XB →YB is a (trivial) fibration.
(b) If X is fibrant, then for every (trivial) cofibration A→B, the induced map
XB →XA is a (trivial) fibration.
Lemma 1.1.10. Let C be a model category, which is Cartesian as a category, and
such that the final object of C is cofibrant. Then C is a Cartesian model category
if and only if for every two cofibrations i : A → B and i′ : A′ → B′, the induced
morphism j :(A×B′)⊔(A×A′)(B×A′)→B×B′ is a cofibration and, additionally,
j is a weak equivalence, if either i or i′ is.
Proof. This follows from a combination of Lemma 1.1.6 and Lemma 1.1.3. (cid:3)
Definition 1.1.11. A model category C is called:
(a)right proper,ifweakequivalencesarepreservedbypullbacksalongfibrations;
(b) left proper, ifweakequivalence arepreservedby pushoutsalongcofibrations;
(c) proper, if it is both left and right proper.
1.2. Simplicial sets.
1.2.1. Category ∆. (a) For n≥0, we denote by [n] the category, corresponding
to a partially ordered set {0< 1< ...<n}. Let ∆ be the full subcategory of the
category of small categories Cat, consisting of objects [n],n≥0.
(b) For each (m+1)-tuple of integers 0 ≤ k ≤ k ≤ ... ≤ k ≤ n, we denote
0 1 m
by δk0,...,km the map δ :[m]→[n] such that δ(i)=ki for all i.
(c) For 0 ≤ i ≤ n we define an inclusion di : [n−1] ֒→ [n] such that i ∈/ Imdi;
for 0≤i<j ≤n, we define aninclusiondi,j :[n−2]֒→[n] suchthat i,j ∈/ Imdi,j;
for 0≤i≤n−m we define an inclusion ei :[m]→[n] defined by ei(k):=k+i.
1.2.2. Spaces. (a) By the category of spaces or, what is the same, the category
simplicial sets we mean the category of functors Sp:=Fun(∆op,Set).
(b) For X ∈ Sp, we set X := X([n]). For every τ : [n] → [m] in ∆, we denote
n
by τ∗ : X →X the induced map of sets. For every morphism f : X →Y in Sp
m n
we denote by f the corresponding map X →Y .
n n n
(c) By 1.1.2, category Sp is Cartesian and has all limits and colimits.
1.2.3. The standard n-simplex. (a) For every n ≥ 0, we denote by ∆[n] ∈ Sp
the functor Hom (·,[n]):∆op →Set. Then pt:=∆[0] is a final object of Sp.
∆
(b) The Yoneda lemma defines identifications Hom (∆[n],X)=X and
Sp n
Hom (∆[n],∆[m])=Hom ([n],[m]) for all X ∈Sp and all n,m≥0.
Sp ∆
(c) We denote by ∆i[n] the image of the inclusion di :∆[n−1]→∆[n], and set
∂∆[n]:=∪n ∆i[n]⊂∆[n] and Λk[n]:=∪ ∆k[n]⊂∆[n] for all k =0,...,n.
i=0 i6=k
YONEDA LEMMA FOR COMPLETE SEGAL SPACES 5
1.2.4. Fibers. (a) For X ∈ Sp, we say x ∈ X instead of x ∈ X . By 1.2.3 (b),
0
each x∈X corresponds to a map x:pt→X.
(b)Foreverymorphismf :Y →X,wedenotebyf−1(x)orY thefiberproduct
x
{x}× Y :=pt× Y and call it the fiber of f at x.
X x,X
(c) For every Z ∈ Sp and X,Y ∈ Sp/Z, we denote by Map (X,Y) the fiber of
Z
YX →ZX over the projection (X →Z)∈ZX.
Definition 1.2.5. (a) Amapf :X →Y inSp is calleda (Kan) fibration, if ithas
the RLP with respect to inclusions Λk[n]֒→∆[n] for all n>0,k =0,...,n.
(b) A map f : X → Y in Sp is called a weak equivalence, if it induces a weak
equivalence |f|:|X|→|Y| between geometric realisations (see [GJ, p. 60]).
(c)Amapf :X →Y inSpiscalledacofibration,iff :X →Y isaninclusion
n n n
for all n.
Theorem1.2.6. Category Sphas astructureofaproper Cartesian model category
such that cofibrations, fibrations and weak equivalences are defined in Definition
1.2.5. In particular, all X ∈ Sp are cofibrant, and trivial fibrations are precisely
the maps which have the RLP with respect to inclusions ∂∆[n]֒→∆[n],n≥0.
Proof. See [GJ, I, Thm 11.3, Prop 11.5 and II, Cor 8.6] and note that in the case
of model category Sp, ”Cartesian” means the same as ”simplicial”. (cid:3)
Definition 1.2.7. We say that X ∈ Sp is a (contractible) Kan complex, if the
projection X →pt is a (trivial) fibration.
1.2.8. Connected components. (a) We say that X ∈ Sp is connected, if it can
not be written as X = X′ ⊔X′′, where X′,X′′ 6= ∅. We say that Y ⊂ X is a
connectedcomponent of X,ifitisamaximalconnectedsubspaceofX. Noticethat
X is a disjoint union of its connected components.
(b)WedenotethesetofconnectedcomponentsofX byπ (X). Theneverymap
0
f :X →Y inSp induces a map π (f):π (X)→π (Y). Note that X is connected
0 0 0
if and only if its geometric realization |X| is connected. In particular, we have an
equality π (X)=π (|X|). Therefore for every weak equivalence f :X →Y in Sp,
0 0
the map π (f) is a bijection.
0
(c) For x,y ∈ X, we say that x ∼ y, if x and y belong to the same connected
component of X. If X is a Kan complex, then x ∼ y if and only if there exists a
map α:∆[1]→X such that α(0)=x and α(1)=y (see [GJ, Lem 6.1]).
Lemma 1.2.9. (a) Let f :X →Y be a trivial fibration. Then the space of sections
Map (Y,X) of f is non-empty and connected.
Y
(b) Let f :X →Y be a fibration in Sp. Then f is trivial if and only if the Kan
complex f−1(y) is contractible for every y ∈Y.
(c) Let f : X → Y is a map of fibrations over Z in Sp. Then f is a weak
equivalence if and only if the map of fibers f : X →Y is a weak equivalence for
z z z
every z ∈Z.
Proof. (a) Since Y is cofibrant, the projection XY →YY is a trivial fibration (by
1.1.9(a)). HenceitsfiberMap (Y,X)isacontractibleKancomplex(by1.1.7(b)),
Y
therefore it is non-empty and connected by 1.2.8 (b).
(b)By the lastassertionofTheorem1.2.6, thefibrationf is trivialifandonlyif
its pullback τ∗(f) is a trivial fibration for all τ :∆[n] →Y. Thus we may assume
that Y = ∆[n]. Then for each y ∈ ∆[n], the inclusion y : ∆[0] → ∆[n] is a weak
6 DAVIDKAZHDANANDYAKOVVARSHAVSKY
equivalence. Thus X → X is a weak equivalence, because Sp is right proper.
y
Hence, by 2-out-of-3, f is a weak equivalence if and only if X →∆[0] is.
y
(c) will be proven in 1.3.13. (cid:3)
1.3. Simplicial Spaces.
1.3.1. Notation. (a) By the category of simplicial spaces, we mean the category
of functors sSp=Fun(∆op,Sp)=Fun(∆op×∆op,Set).
(b)ForX ∈sSpandn,m≥0,wesetX :=X([n])∈SpandX :=(X ) ∈
n n,m n m
Set. For every morphism f : X → Y in sSp, we denote by f : X → Y the
n n n
corresponding morphism in Sp.
(c)Foreveryτ :[n]→[m]in∆,wedenotebyτ∗ :X →X theinducedmapof
m n
spaces. We also set δk0,...,km :=(δk0,...,km)∗ :Xn →Xm, di :=(di)∗ : Xn →Xn−1,
and e :=(ei)∗ :X →X .
i n m
(d) By 1.1.2, category sSp is Cartesian and has all limits and colimits. For
X,Y ∈sSp, we define the mapping space Map(Y,X):=(XY) ∈Sp.
0
1.3.2. Two embeddings Sp ֒→ sSp. (a) Denote by diag : Sp → sSp (resp.
diag :Set→Sp)the mapwhichassociatestoeachX the constantsimplicialspace
(resp. set) [n]7→X,τ 7→Id . For each X ∈Sp, we denote the constant simplicial
X
space diag(X)∈sSp simply by X.
(b) The embedding diag : Set → Sp gives rise to an embedding disc : Sp =
Fun(∆op,Set)→sSp=Fun(∆op,Sp). Then the image of disc, consists of discrete
simplicial spaces, thatis, X ∈sSpsuchthatX ∈Spis discrete(thatis,eachmap
n
∆[1]→X is constant) for all n.
n
(c) We set F[n]:=disc(∆[n]), ∂F[n]:=disc(∂∆[n]) and Fi[n]:=disc(Λi[n]).
1.3.3. Standard bisimplex. (a) For n,m ≥ 0, we set [n,m] := ([n],[m]) ∈ ∆2
and✷[n,m]:=F[n]×∆[m]∈sSp. Inparticular,wehaveequalitiesF[n]=✷[n,0],
∆[m]=✷[0,m] and pt=F[0]=∆[0].
(b) Note that ✷[n,m] is the functor Hom (·,[n,m]). Then, by the Yoneda
∆×∆
lemma, we get identifications Hom(✷[n,m],✷[n′,m′]) = Hom([n,m],[n′,m′]) and
Hom(✷[n,m],X)=X . In particular, we have identifications
n,m
Map(F[n],X)=X and Hom(F[n],F[m])=Hom([n],[m]).
n
(c) We also set ∂✷[n,m]:=(∂F[n]×∆[m])⊔ (F[n]×∂∆[m]) and
(∂F[n]×∂∆[m])
X :=Map(∂F[n],X).
∂n
1.3.4. Fibers. (a)ForX ∈sSp,wesayx∈X insteadofx∈X ,andx∼y ∈X
0,0
insteadofx∼y ∈X . By 1.3.3(b), eachx∈X correspondsto a map x:pt→X.
0
(b) As in 1.2.4, for every morphism f :Y →X in sSp, we denote by f−1(x) or
Y the fiber product {x}× Y :=pt× Y and call it the fiber of f at x.
x X x,X
(c) For every Z ∈sSp and X,Y ∈sSp/Z we denote by Map (X,Y)∈sSp the
Z
fiber of YX →ZX over the projection (X →Z)∈ZX. We also set
Map (X,Y):=Map (X,Y) ∈Sp.
Z Z 0
Definition 1.3.5. We say that a map f : X → Y in sSp is a (Reedy) fibration if
for every n≥0 the induced map f¯ :X →Y × X is a Kan fibration in Sp.
n n n Y∂n ∂n
Theorem 1.3.6. Category sSphas Cartesian proper model category suchthat cofi-
brations and weak equivalences are degree-wise and fibrations are Reedy fibrations.
Proof. All the assertions, except that the model category is Cartesian, are proven
in [GJ, IV,Thm 3.9]. For the remaining assertion, we use Lemma 1.1.10. Now
YONEDA LEMMA FOR COMPLETE SEGAL SPACES 7
the assertion follows from the fact that pushouts, products, cofibrations and weak
equivalences are defined degree-wise, and the model category Sp is Cartesian. (cid:3)
1.3.7. Remarks. (a) It follows from Lemma 1.1.3 that a map f : X →Y in sSp
is a fibration if and only if it has the RLP with respect to all inclusions
(∂F[n]×∆[m])⊔ (F[n]×Λi[m])֒→✷[n,m].
(∂F[n]×Λi[m])
(b) If f :X →Y is a Reedy fibration, then the map f :X →Y is a fibration
n n n
for all n. Indeed, f = f¯ is a fibration by definition, fF[n] : XF[n] → YF[n] is a
0 0
fibration by 1.1.9 (a), hence f =(fF[n]) is a fibration.
n 0
(c) Let X → Y be a fibration in sSp, and let i : A → B be a cofibration
over Y. Then the map XB → YB × XA is fibration by Theorem 1.3.6. Hence
YA
taking fibers at (B → Y) ∈ YB and passing to zero spaces, we get that the map
i∗ :Map (B,X)→Map (A,X)isafibration,thusMap (B,X)isaKancomplex.
Y Y Y
1.3.8. Homotopy equivalence. Let Z ∈sSp (resp. Z ∈Sp).
(a) We say that maps f : X → Y and g : X → Y in sSp/Z (resp. Sp/Z) are
homotopic over Z and write f ∼ g, if f ∼g as elements of Map (X,Y).
Z Z
NoticethatifY →Z isafibration,thenMap (X,Y)∈SpisaKancomplex(by
Z
1.3.7(c)),thusby1.2.8(c)f ∼ g meansthatthereexistsamaph:X×∆[1]→Y
Z
over Z such that h| =f and h| =f.
0 1
(b)Wesaythatamapf :X →Y isahomotopyequivalenceoverZ,ifthereexists
a map g : Y → X over Z, called a homotopy inverse of f, such that f ◦g ∼ Id
Z Y
and g◦f ∼ Id .
Z X
1.3.9. Remarks. (a) Let f : X → Y be a homotopy equivalence over Z with
homotopy inverse g. Then for every τ :Z′ →Z, the pullback τ∗(f) is a homotopy
equivalenceoverZ′ withhomotopyinverseτ∗(g). Similarly,foreveryK ∈sSp,the
mapfK :XK →YK isahomotopyequivalencewithhomotopyinversegK :YK →
XK. Also, a composition of homotopy equivalences is a homotopy equivalence.
(b) Any homotopy equivalence is a weak equivalence. Indeed, the assertion for
Sp follows from the fact that if f ∼ g, then the geometric realizations satisfy
Z
|f|∼ |g|, and the assertion for sSp follows from that for Sp.
|Z|
1.3.10. Strong deformation retract. Let Z ∈sSp (resp. Z ∈Sp).
(a) We say that an inclusion i : Y ֒→ X over Z is a strong deformation retract
over Z, if there exists a map h : X ×∆[1] → X over Z such that h| = Id , and
0 X
pr
h| (X)⊂Y, h| is Y ×∆[1]−→2 Y ֒→X.
1 Y×∆[1]
(b) If i : Y ֒→ X is a strong deformation retract over Z, then i is a homotopy
equivalence over Z, and h| : X → Y is its homotopy inverse. Also in this case,
1
Y →Z is a retract of X →Z. In particular, if X →Z is a fibration, then Y →Z
is a fibration as well (by CM3).
(c) Conversely, a trivial cofibration i : Y → X between fibrations over Z is a
strong deformation retract over Z.
Proof. Since i : Y → X is trivial cofibration, while Y → Z is a fibration, there
exists a map p : X → Y over Y such that p◦i = Id . Next, the induced map
Y
(∂∆[1]×X)⊔ (∆[1]×Y)֒→∆[1]×X is a trivial cofibration (see Lemma
(∂∆[1]×Y)
1.1.10). Since X → Z is a fibration, there exists a map h :∆[1]×X → X over Z
such that h| =Id , h| =p and h| =pr . (cid:3)
0 X 1 ∆[1]×Y 2
8 DAVIDKAZHDANANDYAKOVVARSHAVSKY
Lemma 1.3.11. (a) A morphism f : X → Y over Z is a homotopy equiva-
lence over Z if and only if for every map α : K → Z in sSp, the induced map
π (Map (K,X))→π (Map (K,Y)) is a bijection.
0 Z 0 Z
(b) Every weak equivalence between fibrations is a homotopy equivalence. In
particular, apullback of aweakequivalence betweenfibrations is a weakequivalence.
(c) For each fibration X →Y ×∆[1], there exists a weak equivalence X| →X|
0 1
of fibrations over Y.
(d) Let f : Y → A be a fibration and let A ֒→ B be trivial cofibration. Then
A
there exists a fibration g :Y →B, whose restriction to A is f.
B
Proof. (a) If f is a homotopy equivalence, then the induced map Map (K,X) →
Z
Map (K,Y) is a homotopy equivalence (by 1.3.9 (a)), thus the assertion follows
Z
from1.3.9(b)and1.2.8(b). Conversely,applyingtheassumptionfortheprojection
Y → Z, we find a morphism g : Y → X over Z such that f ◦g ∼ Id . Next
Z Y
applying it to the projection X →Z, we find that g◦f ∼ Id .
Z X
(b)By1.1.7(c)and1.3.9(a),itisenoughtoconsiderseparatelycasesofatrivial
cofibration and a trivial fibration. When f is a trivial cofibration, the assertion
follows from 1.3.10 (c) and (b). When f is a trivial fibration, the assertion follows
from(a). Indeed,eachmapMap (K,X)→Map (K,Y)isatrivialfibration,thus
Z Z
the map on π is a bijection by 1.2.8 (b). The last assertionfollows from 1.3.9 (a).
0
(c) Since each map δi : ∆[0] ֒→ ∆[1] is a trivial cofibration, the induced map
(δi)∗ :X∆[1] →X× (Y ×∆[1])∆[1] is atrivialfibration. Takingthe pullback
∆[1]×Y
with respect to the inclusion Y ֒→ (Y ×∆[1])∆[1], corresponding to Id , we
Y×∆[1]
get a trivial fibration X := X∆[1]× Y → X| over Y. Thus both X|
e (∆[1]×Y)∆[1] i 0
and X| are homotopy equivalent to X over Y (by (b)), hence they are homotopy
1 e
equivalent.
(d) will be proven in 3.2.3. (cid:3)
1.3.12. Remark. It follows from Lemma 1.3.11 (b) and 1.2.8 (b) that a Kan
complexX ∈Sp is contractibleifandonly ifthe projectionX →pt is a homotopy
equivalence. ThusbydefinitionthishappensifandonlyifX isnon-emptyandId
X
is homotopic to a constant map X →{x}⊂X.
1.3.13. Proof of Lemma 1.2.9 (c). Iff isaweakequivalence,theneachf :X →
z z
Y is a weak equivalence by Lemma 1.3.11 (b).
z
Conversely, write f as p ◦ i, where i : X → X′ is a trivial cofibration, and
p : X′ → Y is fibration. By the ”only if” assertion, each i is a weak equivalence.
z
Since each f is a trivial fibration by assumption, each p is a weak equivalence
z z
by 2-out-of-3. Since p : X′ → Y is a fibration, it is a trivial fibration. Hence all
z z z
fibers of each p are contractible. Thus all fibers of p are contractible, hence p is a
z
trivial fibration by Lemma 1.2.9 (b). Therefore f is a weak equivalence. (cid:3)
1.4. Segal spaces. We follow closely [Re1].
1.4.1. Notation. (a) We say that X ∈sSp is a Segal space, if X is fibrant, and
ϕ =:δ × ...× δ :X →X × ...× X
n 01 δ1 δn−1 n−1,n n 1 X0 X0 1
is a weak equivalence for each n≥2.
(b) Notice that since Reedy model category is Cartesian,when X is fibrant, the
map ϕ is a fibration. Thus a fibrant object X ∈ sSp is a Segal space if and only
n
if each map ϕ is a trivial fibration.
n
YONEDA LEMMA FOR COMPLETE SEGAL SPACES 9
1.4.2. ”Objects” and ”Mapping spaces”. Let X be a Segal space.
(a) By a space of objects of X we mean spaceX . We set ObX :=X andcall
0 0,0
it the set of objects of X. As in 1.3.4, we say x∈X instead of x∈ObX.
(b) For each x,y ∈ X, we denote by map(x,y) = map (x,y) ∈ Sp the fiber of
X
(δ ,δ ):X →X ×X over(x,y). Notice that sinceX isfibrant,the map(δ ,δ )
0 1 1 0 0 0 1
is a fibration, thus each space map(x,y) is a Kan complex. For each x,y,z ∈ X,
we denote by map(x,y,z) the fiber of (δ ,δ ,δ ):X →(X )3 over (x,y,z).
0 1 2 3 0
(c) For each x∈X, we set id :=δ (x)∈map (x,x).
x 0,0 X
(d) We call a map between Segal spaces f : X →Y is fully faithful, if for every
x,y ∈X the induced map map (x,y)→map (f(x),f(y)) is a weak equivalence.
X Y
1.4.3. The homotopy category. (a) Let X be a Segal space, and x,y,z ∈
X. Then the trivial fibration ϕ : X → X × X induces a trivial fibration
2 2 1 X0 1
map(x,y,z)→map(x,y)×map(y,z)(by 1.1.7(b)), whichbyLemma 1.2.9(a)has
a section s, unique up to homotopy. Thus we have a well-defined map
[s]:=π (s):π (map(x,y))×π (map(y,z))→π (map(x,y,z)).
0 0 0 0
(b) The map δ : X → X induces a map δ : map(x,y,z) → map(x,z).
02 2 1 02
Therefore for every [α]∈π (map(x,y)) and [β]∈π (map(y,z)) we can define
0 0
(1.2) [β]◦[α]:=π (δ )([s]([α],[β]))∈π (map(x,z)).
0 02 0
Itisnotdifficulttoprove(see[Re1,Prop. 5.4])thatthiscompositionisassociative
and satisfies [α]◦[id ]=[α]=[id ]◦[α] for all α∈map(x,y).
x y
(c)Using(b),onecanassociatetoX itshomotopy categoryHoX,whoseobjects
are ObX, morphisms defined by Hom (x,y) := π (map (x,y)), the composi-
HoX 0 X
tion is defined by (1.2), and the identity map is [id ]∈Hom (x,x).
x HoX
1.4.4. Complete Segal spaces. Let X be a Segal space.
(a) We say that α ∈ map (x,y) ⊂ X is a homotopy equivalence, if the corre-
X 1
sponding morphism[α]∈MorHoX is anisomorphism. Explicitly, this means that
there exist β ∈map(y,x) such that [β]◦[α]=[id ] and [α]◦[β]=[id ].
x y
(b) Let X ⊂ X be the maximal subspace such that each α ∈ X is a
heq 1 heq
homotopyequivalence. Itisnotdifficulttoprove(see[Re1,Lem. 5.8])thatX ⊂
heq
X is a union of connected components.
1
(c) Notice that since each [id ] is an isomorphism, we have id ∈X for every
x x heq
x∈X. Thereforethemaps :=δ :X →X factorsthroughX . We saythat
0 0,0 0 1 heq
X iscalledacompleteSegalspace,ifthemaps :X →X isaweakequivalence.
0 0 heq
Lemma 1.4.5. Let X ∈Sp be a Segal space.
(a) LetX′ ⊂X be theunion of connectedcomponents, intersectings (X ), and
1 1 0 0
set X′ :=δ−1(X′)∩δ−1(X′)⊂X . Then X′ ⊂X and δ (X′)=X .
3 02 1 13 1 3 1 heq 12 3 heq
(b) X is complete if and only if δ :X ֒→X →X is a trivial fibration.
0 heq 1 0
Proof. (a)SinceX ⊂X isaunionofconnectedcomponents,inclusions (X )⊂
heq 1 0 0
X implies that X′ ⊂ X . Next, for each α ∈ X′ we have δ (α),δ (α) ∈
heq 1 heq 3 02 13
X′ ⊂X , hence [δ (α)]=[δ (α)]◦[δ (α)] and [δ (α)]=[δ (α)]◦[δ (α)] are
1 heq 02 12 01 13 23 12
isomorphisms in HoX. Therefore [δ (α)] is isomorphism, thus δ (α)∈X .
12 12 heq
Conversely,let α∈map(x,y)⊂X andletβ ∈map(y,x) suchthat [β]◦[α]=
heq
[id ] and [α]◦[β]=[id ]. Since ϕ is a trivial fibration, it is surjective. Thus there
x y 3
existsγ ∈X suchthatδ (γ)=δ (γ)=β andδ (γ)=α. Then, byassumption,
3 01 23 12
δ (γ)∼id and δ (γ)∼id , thus γ ∈X′.
02 y 13 x 3
10 DAVIDKAZHDANANDYAKOVVARSHAVSKY
(c) By 1.4.4 (b), the composition X ֒→ X → X ×X is a fibration, thus a
heq 1 0 0
projection δ : X → X is a fibration. Since δ ◦s = Id , we conclude that
0 heq 0 0 0 X0
s : X → X is a weak equivalence if and only if δ : X → X is a trivial
0 0 heq 0 heq 0
fibration. (cid:3)
1.4.6. Cartesian structure. Rezk showed (see [Re1, Cor 7.3]) that if X is a
(complete) Segal space, then XK is a (complete) Segal space for every K ∈sSp.
2. The Yoneda lemma
2.1. Left fibrations.
Definition 2.1.1. We calla fibrationf :X →Y insSp aleft fibration,ifthe map
(f ,(δ0)∗):XF[1] →X× YF[1],inducedbyδ0 :F[0]֒→F[1],isatrivialfibration.
∗ Y
Lemma 2.1.2. (a) A pullback of a left fibration is a left fibration.
(b) If f : X → Y is a left fibration, then fZ : XZ → YZ is a left fibration for
every Z ∈sSp.
Proof. (a) follows from the fact that a pullback of a (trivial) fibration is a (trivial)
fibration (see 1.1.7 (b)).
(b) By definition, the map XF[1] → X × YF[1] is a trivial fibration. Since
Y
Reedy model structure is Cartesian, we conclude that fZ is a fibration, while the
map(XZ)F[1] =(XF[1])Z →(X× YF[1])Z =XZ× (YZ)F[1] isatrivialfibration
Y YZ
(use 1.1.9). Thus fZ :XZ →YZ is a left fibration. (cid:3)
Lemma 2.1.3. Let f :X →Y be a fibration in sSp. The following conditions are
equivalent:
(a) f is a left fibration.
(b) For every n ≥ 1, the map (f ,(δ0)∗) : XF[n] → X × YF[n], induced by
∗ Y
δ0 :[0]֒→[n], is a trivial fibration.
(c) For every n≥1, the map p :X →X × Y , induced by δ0 :[0]֒→[n], is
n n 0 Y0 n
a trivial fibration.
Proof. (a)=⇒ (b) By (a) and 1.1.9 (a), the map
p:XF[1]×F[n] =(XF[1])F[n] →(X × YF[1])F[n] =XF[n]× YF[1]×F[n]
Y YF[n]
isatrivialfibration. Sincetrivialfibrationsarestableunderretracts(axiomCM3),
it remains to show that the map XF[n+1] → X × YF[n+1] is a retract of p. It is
Y
enoughtoshowthatδ0 :[0]֒→[n+1]isaretractofδ0 :[n]×[0]֒→[n]×[1]. Consider
α β
maps[n+1]−→[n]×[1]−→[n+1],where α(0)=(0,0), α(i)=(i−1,1)for i≥1
andβ(i,j)=(i+1)j. Then β◦α=Id, α(0)∈[n]×{0}andβ([n]×{0})=0,thus
α and β realize δ0 :[0]֒→[n+1] as a retract of δ0 :[n]×[0]֒→[n]×[1].
(b)=⇒(c) Pass to the zero spaces.
(c)=⇒(a)FirstweassumethatY isfibrant. Sincef isafibration,X isfibrant,
and the induced map XF[1] → X × YF[1] is a fibration. It remains to show
Y
that each map (XF[1]) →X × (YF[1]) is a weak equivalence. Since the map
n n Yn n
X →X × Y is a trivial fibration, its pullback
n 0 Y0 n
X × (YF[1]) →(X × Y )× (YF[1]) =X × (YF[1])
n Yn n 0 Y0 n Yn n 0 Y0 n
is a trivialfibration. It remains to show that the map (XF[1]) →X × (YF[1])
n 0 Y0 n
or, equivalently, q :(XF[1]×F[n]) →X × (YF[1]×F[n]) is a weak equivalence.
n 0 0 Y0 0
