Weak complicial sets [lecture notes]

19 Pages·2016·0.346 MB·English

by  Emily Riehl

WEAK COMPLICIAL SETS
EMILYRIEHL
Abstract. The aim of these talks will be to give a “hands on” introduction
to weak complicial sets, which are simplicial sets with certain marked “thin”
simplices satisfying a few axioms. By declaring all simplices above a certain
dimension to be “thin,” weak complicial sets can model (∞,n)-categories for
each n, including n = ∞. For this reason they present a fertile setting for
thinkingaboutweakinfinitedimensionalcategoriesinvaryingdimensions.
Contents
Lecture 1: Introducing Weak Complicial Sets 1
Quasi-categories as (∞,1)-categories 2
A simplicial model of (∞,2)-categories? 4
Stratified simplicial sets 4
Weak complicial sets 5
n-trivialization and the n-core 6
Lecture 2: the Street Nerve of an ω-category 8
ω-categories 8
Orientals 10
The Street nerve 12
Other markings 13
Lecture 3: Saturation and Model Structures 14
Saturation in low dimensions 14
Saturation in any dimension 15
Model categories of weak complicial sets 18
References 19
Lecture 1: Introducing Weak Complicial Sets
A weak complicial set is a stratified simplicial set, with a designed subset of
simplices marked “thin,” admitting extensions along a designated class of of maps.
Weak complicial sets model (∞,∞)-categories. By requiring all simplices above a
fixeddimensiontobethin,theycanalsomodel(∞,n)-categoriesforalln∈[0,∞].
Date:HigherStructuresinGeometryandPhysics,MATRIX,Melbourne,6-7June,2016.
Acknowledgments: The author wishes to thank the organizers of the Higher Structures in
GeometryandPhysicsworkshopandtheMATRIXInstituteforprovidingherwiththeopportunity
tospeakaboutthistopicandtheNSFforfinancialsupportthroughthegrantDMS-1509016.
1
2 EMILYRIEHL
Strict complicial sets were first defined by John Roberts [R] with the intention
of constructing a simplicial model of strict ω-categories. He conjectured that it
should be possible to extend the classical nerve to define an equivalence from the
category of strict ω-categories to the category of strict complicial sets. Ross Street
defined this nerve [S], providing a fully precise statement of what is known as the
Street–Roberts conjecture. Dominic Verity proved the Street–Roberts conjecture
[V1] and then subsequently defined and developed the theory of weak complicial
sets [V2, V3] that will be the focus here.
Webeginthislecturebyrevisitinghowquasi-categories(anunmarkedsimplicial
set) model (∞,1)-categories and then explore what would be needed to model an
(∞,2)-category as a simplicial set. These excursions will motivate the definition of
weak complicial sets. We will conclude by defining n-trivial weak complicial sets,
which model (∞,n)-categories.
Quasi-categories as (∞,1)-categories
Definition. Aquasi-categoryisasimplicialsetAsothateveryinnerhornadmits
a filler
Λk[n] A ∀n≥2, 0<k <n.
∆[n]
This presents an (∞,1)-category with:
• A as the set of objects;
0
• A as the set of 1-cells with sources and targets determined by the face
1
maps
d1=source
A A
1 0
d0=target
and degenerate 1-simplices serving as identities;
• A as the set of 2-cells;
2
• A as the set of 3-cells, and so on.
3
The weak 1-category structure arises as follows. A 2-simplex
1
(1) f g
α
0 2
h
provides a witness that h(cid:39)gf.
Notation. Throughout let us adopt the convention of always labeling the vertices
of an n-simplex by 0,...,n. This notation does not assert that the vertices are
necessarily distinct, but is used to help orient each picture.
WEAK COMPLICIAL SETS 3
A 3-simplex then
1
f k
g
0 (cid:96) 3
j h
2
provides witnesses that h(gf)(cid:39)hj (cid:39)(cid:96)(cid:39)kf (cid:39)(hg)f.
The homotopy category hA of a quasi-category A is then the category whose
objectsareverticesandwhosemorphismsareaquotientofA modulotherelation
1
f (cid:39)g that identifies a pair of parallel edges if and only if there exists a 2-simplex
1
(2) f
α
0 2
g
Notation. Here and elsewhere the notation “=” is used for degenerate simplices.
The composition operation witnessed by 2-simplices is not unique on the nose
but it is unique up to this notion of homotopy, witnessed by 2-cells. Hence the
homotopy category is a strict 1-category.
A quasi-category is understood as presenting an (∞,1)-category because each
2-simplex is invertible up to a 3-simplex, and each 3-simplex is invertible up to a
4-simplex, and so on, in a sense we will now illustrate. First consider a 2-simplex
as in (2). This data can be used to define a horn Λ1[3]→A whose other two faces
are degenerate
1 2 1 2
⇑α
f f =
g β ⇑ (cid:86) =
0 3 0 3
f f
whichcanbefilledtodefinea“rightinverse”β inthesensethatthispairof2-cells
bound a 3-simplex with other faces degenerate. Similarly, there is a “special outer
horn”1 Λ3[3]→A
1 2 1 2
⇑γ
g g =
f α⇑ (cid:86) =
0 3 0 3
g g
which can be filled to define a “left inverse” γ. In this sense, α is an equivalence
up to 3-simplices.
1SpecialouterhornsΛ0[n]andΛn[n]havefirstorlastedgesmappingtoequivalences(such
asdegeneracies)inA,moreaboutwhichbelow.
4 EMILYRIEHL
What if α has the form (1). In this case, we can define a Λ1[3]→A horn
g g
1 2 1 2
(cid:39)
f f =
gf αˆ ⇑ (cid:86) ⇑α g
0 3 0 3
h h
whose 3rd face is constructed by filling a horn Λ1[2] → A. In this sense, any
2-simplex is equivalent to one with last (or dually first) edge degenerate.
A simplicial model of (∞,2)-categories?
How might a simplicial set model an (∞,2)-category? A natural idea would be
interpret the 2-simplices as inhabited by not necessarily invertible 2-cells pointing
in a consistent direction. The problem with this is that the 2-simplices need to
play a dual role: they must also witness composition of 1-simplices, in which case
it doesn’t make sense to think of them as inhabited by non-invertible cells. The
ideawillbetomarkasthinthewitnessesforcomposition. Themarked2-simplices
shouldbethoughtofas2-dimensionalequivalencesinasensethatwillbeexamined
more precisely later.
Then 3-simplices can be used to witness composition of not-necessarily thin 2-
simplices. For instance, given a pair of 2-simplices α and β with boundary as
displayed below, the idea is to build a Λ2[3]-horn
g g
1 2 1 2
⇑α
f k f (cid:39) k
β ⇑ (cid:86) ⇑α∗β
h kg
0 3 0 3
(cid:96) (cid:96)
whose 0th face is a thin filler of the Λ1[2]-horn formed by g and k. The 2nd
face, defined by filling the horn Λ2[3]-horn, defines a composite 2-simplex, which
is witnessed by the (thin) 3-simplex. Note that because the 0th face is thin, its
1st edge is interpreted as a composite kg of g and k, which is needed so that the
boundary of the new 2-cell agrees with the boundary of the pasted composite of β
and α.
A similar Λ1[3]-horn can be used to define composites where the domain of α
is the last, rather than the first, edge of the codomain of β. It is in this way
that simplicial sets with certain marked simplices will be used to model (∞,2)-
categories or indeed (∞,n)-categories for any n ∈ [0,∞]. We will now formally
introduce stratified simplicial sets before stating the axioms that define these
weak complicial sets.
Stratified simplicial sets
Definition. Astratifiedsimplicialsetisasimplicialsetwithadesignatedsubset
of marked or thin simplices in positive dimensions containing all degeneracies. A
map of stratified sets is a simplicial map that preserves thin simplices.
WEAK COMPLICIAL SETS 5
There are left and right adjoints
(−)(cid:91)
⊥
Strat U Simp
⊥
(−)(cid:93)
to the forgetful functor from stratified simplicial sets to ordinary simplicial sets,
both of which are full and faithful.
Definition. An inclusion U (cid:44)→V of stratified simplicial sets is:
• regular, denoted U (cid:44)→ V, if thin simplices in U are created in V (a
r
simplex is thin in U if and only if its image in V is thin); and
• entire, denoted U (cid:44)→ V, if it is an identity on underlying simplicial sets
e
(in which case the only difference between U and V is that V has more
thin simplices).
The monomorphisms in Strat are generated under pushout, transfinite composi-
tion, and coproduct by
{∂∆[n](cid:44)→ ∆[n]|n≥0}∪{∆[n](cid:44)→ ∆[n] |n≥1},
r e t
where the top-dimensional n-simplex in ∆[n] is thin.
t
Weak complicial sets
Definition. A weak complicial set is a stratified simplicial set that admits ex-
tensions along any of the elementary anodyne extensions. These include:
(i) the complicial horn extensions Λk[n] (cid:44)→ ∆k[n] for n ≥ 1, 0 ≤ k ≤ n,
r
i.e., the regular inclusions of k-admissible n-horns.
Here a non-degenerate m-simplex in ∆k[n] is thin if and only if it con-
tains the vertices {k−1,k,k+1}∩[n]. Thin faces include
• the top dimensional n-simplex
• all codimension-one faces except for the (k−1)th, kth, and k+1th
• the 2-simplex spanned by [k−1,k,k+1] in the case of inner horns
or the edge [k−1,k,k+1]∩[n] in the case of outer horns.
Thek-admissiblen-hornistheregularsubsetofthek-admissiblen-simplex.
For the inner horns, it parametrizes “admissible composition” of a pair of
(n−1)-simplices.
(ii) the complicial thinness extensions ∆k[n](cid:48) (cid:44)→ ∆k[n](cid:48)(cid:48) for n ≥ 1, 0 ≤
e
k ≤ n, an entire inclusion of two entire supersets of ∆k[n]. The stratified
simplicial set ∆k[n](cid:48) is obtained from ∆k[n] by also marking the (k−1)th
and (k+1)th faces, while ∆k[n](cid:48)(cid:48) marks all of the codimension one faces.
This extension problem demands that whenever the composable pair of
simplices in an admissible horn are thin, then so is any composite.
Example (complicial horn extensions). To gain familiarity, let us draw the com-
plicial horn extensions in low dimensions, using red to depict simplices present in
the codomain but not the domain and “(cid:39)” to decorate thin simplices. The labels
onthesimplicesareusedtosuggesttheinterpretationofcertaindataascomposites
6 EMILYRIEHL
of other data.
1
f g
• Λ1[2](cid:44)→r ∆1[2] (cid:39)
0 2
gf
1
• Λ0[2](cid:44)→r ∆0[2] e(cid:39) (cid:39) fe−1
0 2
f
• Λ2[3](cid:44)→ ∆2[3]
r
g g
1 2 1 2
⇑α
f k f (cid:39) k
β ⇑ (cid:39) ⇑α∗β
h kg
0 3 0 3
(cid:96) (cid:96)
• Λ0[3](cid:44)→ ∆0[3]
r
fe−1 fe−1
1 2 1 2
(cid:39) αe−1
e(cid:39) g (cid:39) ⇑ g
f α⇑ (cid:39) (cid:39) he−1
0 3 0 3
h h
• For Λ2[4] (cid:44)→ ∆2[4] the non-thin codimension-one faces in the horn look like
r
the picture on the left, while their composite looks like the picture on the right.
1 1
2
0 4
0 4
3 3
It makes sense to interpret the 2nd face as a composite of the 3rd and 1st faces
because the 2-simplex
2
(cid:39)
1 3
is thin.
n-trivialization and the n-core
Definition. A stratified simplicial set X is n-trivial if all r-simplices are marked
for r >n.
WEAK COMPLICIAL SETS 7
The full subcategory of n-trivial stratified simplicial sets is reflective and core-
flective
trn
Strat ⊥ Strat
n-tr
⊥
coren
in the category of stratified simplicial sets. That is n-trivialization defines an
idempotent monad on Strat with unit the entire inclusion
X (cid:44)→ tr X
e n
of a stratified simplicial set X into the stratified simplicial set tr X with the same
n
markedsimplicesindimensions1,...,n,andwithallhighersimplices“madethin.”
The n-core core X, defined by restricting to those simplices whose faces above di-
n
mensionnareallthin,isanidempotentcomonadwithcounittheregularinclusions
core X (cid:44)→ X.
n r
As is always the case for a monad-comonad pair arising in this way, these functors
are adjoints: tr (cid:97)core .
n n
These assemble to define a string of inclusions with adjoints
tr1 trn−1
Simp (−)(cid:93) Strat ⊥ Strat ··· Strat ⊥ Strat ··· Strat
∼= 0-tr ⊥ 1-tr (n−1)-tr ⊥ n-tr
core1 coren−1
Remark. Thetworightadjointsrestricttoweakcomplicialsetstogivetheinclusion
of (∞,n−1)-categories into (∞,n)-categories and its right adjoint, which takes an
(∞,n)-category to the “groupoid core.” The left adjoint, which just marks things
arbitrarily, does not preserve weak complicial sets being too naive to define the
“freely invert n-arrows” functor from (∞,n)-categories to (∞,n−1)-categories.2
Example.
• A0-trivialweakcomplicialsetisexactlyaKancomplexwiththemaximal
“(−)(cid:93)” marking.
• A 1-trivial weak complicial set defines a quasi-category. Conversely, any
quasi-category admits a stratification (the best being the “natural” one,
about more which below) making it a weak complicial set. The markings
onthe1-simplicescan’tbearbitrary. Atminimum,certainautomorphisms
(endosimplicesthatarehomotopictoidentities)mustbemarked. Moreto
the point, each edge that is marked necessarily defines an equivalence in
thequasi-category. Butit’snotnecessarytomarkalloftheequivalences.3
• Strict n-categories or even ω-categories define strict complicial sets (with
unique fillers for the admissible horns) via the Street nerve, about more
which in lecture nerve.
2For instance, if A is a naturallymarked quasi-category, that’s 1-trivial, then its zerotrivial-
ization isn’t a Kan complex (because we haven’t changed the underlying simplicial set) but it’s
groupoidcoreis(byatheoremofJoyal).
3Laterwe’ll say astratified simplicial setis saturated if allequivalences are marked. Inthe
caseofann-trivialstratification,theequivalencesarecanonicallydeterminedbythestructureof
thesimplicialset. Withthis“natural”marking,quasi-categoriesareisomorphictothesaturated
1-trivialweakcomplicialsets.
8 EMILYRIEHL
Lecture 2: the Street Nerve of an ω-category
As is usual for nerve constructions, the Street nerve, a functor
N: ω-Cat→Simp
from strict ω-categories to simplicial sets is determined by a functor
O
∆−→ω-Cat.
The image of [n]∈∆ is an n-category O defined by Street, the nth oriental. The
n
nerve of an ω-category C is then defined to be the simplicial set whose n-simplices
NC :=hom(O ,C)
n n
are ω-functors O → C. There are various ways to define a stratification on the
n
nerve of an ω-category, defining a lift of the Street nerve to a functor valued in
stratifiedsimplicialsets. TheStreet–Robertsconjecturemotivatedthedefinitionof
strict complicial sets, i.e., stratified simplicial sets that admit unique extensions
along the complicial horn inclusions and complicial thinness extensions.
Theorem (conjecture of Street–Roberts, proof by Verity [V1]). The Street nerve
ω-Cat N Strat
embedsω-categoriesfullyfaithfullyintostratifiedsimplicialsets,whereann-simplex
O →C in NC is marked if and only if it carries the top dimensional n-cell on O
n n
to an identity in C. The essential image is the category of strict complicial sets.
In this talk, we will introduce (strict) ω-categories and the orientals. We then
define the Street nerve and revisit the Street–Roberts conjecture, though we’ll say
nothing about its proof. We will conclude by exploring other markings of strict
n-categories. In this way, the Street nerve will provide an important source of
examples of weak complicial sets. This will lead to a consideration of equivalences
in a weak complicial set and a discussion of saturation, which will be a main topic
for the third lecture.
ω-categories
Street’s “The algebra of oriented simplexes” [S] gives a single-sorted definition
of an n-category in all dimensions n=1,...,ω.
Definition. A category (C,s,t,∗) consists of
• a set C of cells
• functions s,t: C → C so that ss = ts = s and tt = st = t (a target or
source has itself as its target and its source).
• afunction∗fromthepullbackofsalongttoC sothats(a∗b)=s(b)and
t(a∗b)=t(a) (the source of a composite is the source of its first cell and
the target is the target of the second cell).
and so that
• s(a)=t(v)=v implies a∗v =a (right identity)
• u=s(u)=t(a) implies u∗a=a (left identity)
• s(a)=t(b) and s(b)=t(c) imply a∗(b∗c)=(a∗b)∗c (associativity).
Theobjectsor0-cellsarethefixedpointsforsandthenalsofortandconversely.
WEAK COMPLICIAL SETS 9
Definition. A 2-category (C,s ,t ,∗ ,s ,t ,∗ ) consists of two categories
0 0 0 1 1 1
(C,s ,t ,∗ ) and (C,s ,t ,∗ )
0 0 0 1 1 1
so that
• s s = s = s s = s t , t = t s = t t (globularity plus 1-sources and
1 0 0 0 1 0 1 0 0 1 0 1
1-targets of points are points)
• s (a)=t (b) implies s (a∗ b)=s a∗ s b and t (a∗ b)=t (a)∗ t (b)
0 0 1 0 1 0 1 1 0 1 0 1
(1-cell boundaries of horizontal composites are composites).
• s (a)=t (b) and s a(cid:48) =t b(cid:48) and s a=t a(cid:48) imply that
1 1 1 1 0 0
(a∗ b)∗ (a(cid:48)∗ b(cid:48))=(a∗ a(cid:48))∗ (b∗ b(cid:48))
1 0 1 0 1 0
(middle four interchange).
Identities for ∗ are 0-cells and identities for ∗ are 1-cells.
0 1
Definition. An ω+-category4 consists of categories (C,s ,t ,∗ ) for each n∈ω
n n n
so that (C,s ,t ,∗ ,s ,t ,∗ ) is a 2-category for each m<n. The identities for
m m m n n n
∗ are n-cells. A cell is an n-cell for some n.
n
An ω+-functor is a function that preserves sources, targets, and composition
for each n.
An ω-category is an ω+-category in which every element is a cell. Every ω+-
categoryhasamaximalsubω-categoryofcellsandalloftheconstructionsdescribed
here restrict to ω-categories.
An n-category is an ω-category comprised of only n-cells. This means that the
category structures for m>n are all discrete.
Example. The free ω+-category 2 on one generator5, which represents the
ω
underlying set functor ω+-Cat→Set, has its underlying set
(2×ω)∪{ω}.
Theelementω istheuniquenon-cell, whiletheobjects(0,n)and(1,n)aren-cells,
respectively its n-source and n-target:
s (ω)=(0,n) and t (ω)=(1,n).
n n
An m-cell should be its own n-source and n-target for m≤n; thus:
s ((cid:15),m)=t ((cid:15),m)=((cid:15),m) for m≤n,
n n
while:
s ((cid:15),m)=(0,n) and t ((cid:15),m)=(1,n) for n<m.
n n
Composition is by
ω∗ (0,n)=ω =(1,n)∗ ω
n n
and
(cid:40)
((cid:15),m) m≤n
((cid:15),m)∗ ((cid:15)(cid:48),m(cid:48))=
n ((cid:15)(cid:48),m(cid:48)) m(cid:48) ≤n.
Using 2 one can define the functor ω+-category [A,B] for two ω+-categories
ω
AandB: elementsareω+-functorsA×2 →B. Weleaveitasanexercisetowork
ω
out the rest of this definition and prove that ω+-Cat is cartesian closed.
4Streetcalledthese“ω-categories”butwe’llreservethistermforsomethingelse.
5In personal communication, Ross suggests that there may be something wrong with this
example,butIdon’tseewhatitis.
10 EMILYRIEHL
Theorem (Street). There is an equivalence of categories
(ω+-Cat)-Cat−(cid:39)→ω+-Cat
which restricts to define an equivalence
(cid:39)
(n-Cat)-Cat−→(1+n)-Cat
for each n∈[0,ω].6
Proof. The construction of this functor is an extension of the construction of a 2-
categoryfromaCat-enrichedcategory. LetCbeanω+-categoryenrichedcategory.
Define an ω+-category C whose underlying set is
(cid:97)
C := C(u,v).
u,v∈obC
The 0-source and 0-target of an element a∈C(u,v) are u and v, respectively, and
0-composition is defined using the enriched category composition. The n-source
and n-target of a ∈ C(u,v) is defined using the (n−1)-category structure of the
ω+-category C(u,v).
Conversely, the ω+-category enriched category of an ω+-category C can be de-
fined by taking its 0-cells as its objects, defining C(u,v) to be the collection of
elementswith0-sourceuand0-targetv,usingtheoperations∗ forn>0todefine
n
the ω+-category structure on C(u,v). (cid:3)
Orientals
The idea is that nth oriental O is a strict n-category with a single n-cell whose
n
source is the pasted composite of (n−1)-cells, one for each of the odd faces of
the simplex ∆[n], and whose target is a pasted composite of (n−1)-cells, one for
each of the even faces of the simplex ∆[n]. These can be recognized as sub ω-
categories (throwing away the top dimensional cells) of an ω-category O , which
ω
can be thought of as the free ω-category on the ω-simplex ∆[ω] (in a sense that
is made precise by the notion of a parity complex, which is a quite complicated
combinatorial gadget that we’ll avoid defining). This was all worked out by Street.
The orientals O ,O ,O ,... are a sequence of ω-categories, with O actually
0 1 2 n
an n-category. In low dimensions:
(n=0) O is the ω-category with a single 0-cell:
0
•
(n=1) O is the ω-category with two 0-cells 0,1 and a 1-cell:
1
0 1
(n=2) O is the ω-category with three 0-cells 0,1,2, four 1-cells as displayed:
2
1
01 12
{01,12}
0 2
02
6Here1+ω=ω.