What is an NCategory

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What is an N-Category?

• A talk by Chris Schommer-Pries

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History
1942-45
Eilenberg Maclane Categories

• Things we wont discuss
• Double categories
• Poly categories
• Internal categories
• Weak versions of the these

Ehresmann strict n-categories
1950s-65
Bénebou bicategories (weak 2-categories)
1967
Monoidal, weak monoidal, braided,
multi-categories, operads
1987
Street weak n-category
1995
Many more definitions of weak n-category
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Warm-Up

• What structure does the category of categories
  have?
  
• Given two categories C, D we have a new
  category
  
• Fun(C,D) objects functors from C to D
• mor nat. transformations
• Fun(C,D) ? Fun(D, E) ? Fun(C,E)

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Enriched Categories

• Definition See board
• Examples
• Top
• Abelian categories
• Category of categories
• Vect

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Strict n-Categories

• Definition A strict (n1)-category is a category
  enriched over the category of n-categories.

• Non-Examples
• (Algebras, Bimodules, Intertwiners)
• The Fundamental 2-groupoid

• Examples
• 0-Cats Set
• 1-Cats ordinary categories
• 2-Cats ?

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Bicategories

• Sets of objects B0, 1-morphisms B1, and
  2-morphisms B2
  
• Vertical and Horizontal compositions

• Compositions are compatible?
• Have identities?

Isomorphisms aL,M,N AL ?B (M ?C N)D ? A(L ?B
M) ?C ND rM AM ?B 1B ? AMB, lM A1 ?A MB ? AM
B
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Coherence
Isomorphisms (2-morphisms) aL,M,N AL ?B (M ?C
N)D ? A(L ?B M) ?C ND rM AM ?B IB ? AMB,
lM AI ?A MB ? AMB
Axioms
Maclane All diagrams formed with these commute.
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Approaches to Weak n-Cats

• View strict n-category as a collection of sets
  with structure
  
• Weaken the structure
• Multicategories
• Operads
• Monads
• Higher Operads

• View strict n-category as a presheaf with certain
  properties
  
• Weaken the properties
• Simplicial Sets variants
• Presheaves
• Dense subcategories

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Two Algebraic Approaches

• Mimic inductive definition.
• Weaken notion of enriched category
• Weak n-cat Weakly enriched categories

• Study Strict n-cats as sets with ways to combine
  things
  
• Weaken the ways to combine things
• More Later

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Operads

• A sequence of sets P(n) for n ? N
• Maps
• P(n) ? P(k1) ? ? P(kn) ? P( k1 kn)
• An identity 1? P(1)
• Satisfying associativity and identity axioms

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Examples of Operads
Binary Trees
Planar Diagrams
P(3) ? P(2) ? P(3) ? P(1) ? P(6)
P(2) ? P(3) ? P(1) ? P(4)
Reparametrizations
Linear Graphs
P(3) ? P(2) ? P(3) ? P(3) ? P(8)
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Weakly Enriched Categories

• Idea weaken notion of composition
• Use an operad to parametrize this
• E an operad, Fn(E(k)) is an n-cat
• A weakly enriched n-category has
• objects a1, a2,
• Morphism categories A(a1, a2) ( weak (n-1)-cats
  )
  
• Compositions
• Fn-1(E(3)) ? A(a0, a1) ? A(a1, a2) ? A(a2, a3)
  ?A(a0, a3)

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Weak n-category family tree
Weak n-cat
Algebraic
Non-algebraic
Weakly Enriched
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Second Algebraic Approach
forget
T forget(free(-)) T is a monad
Cat
Basic Data
free
Category C ? Algebra for monad T

• Idea
• Define n-category as an algebra for a certain
  monad on the basic data.

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Globular Sets
N-morphisms

• A sequence of sets Bi
• Maps satisfying
• ss st
• tt ts
• Presheaf on the globular category G.

2-morphisms
morphisms
B0
objects
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Second Algebraic Approach
forget
T forget(free(-)) T is strict n-cat monad
Strict n-Cat
Globular Sets
free
Strict n-Category C ? Algebra for monad T

• Replace T by similar monad
• Use globular operads to construct new monad

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Pasting Diagrams

• T(1) pasting diagrams

• They have a compositions

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Globular Operads
WA

• Operad sets indexed by numbers n
  compositions
  
• Globular Operad sets indexed by pasting
  diagrams compositions

W
TA
T(1)

• W operad
• ? A ? WA a monad

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Second Algebraic Approach
WA
Additional Weakened Ways to Compose
Labeled Diagrams of Strict Compositions
W
TA
T(1)
pasting diagrams with parenthetheses
How to choose W?

• W has enough cells to have all the compositions
• W is close to T(1)

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Weak n-category family tree
Weak n-cat
Algebraic
Non-algebraic
Weakly Enriched
Algebra for Operad/Monad
Trimble topological operad
May Any contractible operad in a model

category
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Categories as Simplicial Nerves

• ? Cat, full subcategory
• n ?

N Cat ? ?op, Set C ? Hom(-, C)
Simplicial sets
N arrows

• N is full and faithfull
• Can define categories as simplicial sets with
  certain properties

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Why Does this work?

• ? is dense in the category of categories
• FD ? C is dense in C if either
• The functor C ? Dop, Set is fully faithfull
• For each object c in C,
• Is an isomorphism

Colim F(d) ? c F(d) ? c
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Non-algebraic Approach

• A strict n-cat is a presheaf on the globlar cat
  G.
  
• No hope of identifying properties
• Replace G with a better category

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Street

• ? strict N-Cat, subcategory
• n ?

N-Cat ? ?op, Set C ? Hom(-, C)
Simplicial sets
Free cat generated by n-simplex

• This subcategory is not dense.
• Can fix by adding hollowness or thinness

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Simpson Tamsamani

• Cat ? ?op, Set properties
• Strict 2-cat cat enriched in cat
• Strict 2-cat ?op, cat properties
• Strict 2-cat ?op, ?op, set properties
• Strict 2-cat (?op)2, set prop.
• Strict n-cat (?op)n, set prop.

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Graphical interpretation
C(4, 2, 0, )

• Grids of composible morphisms
• Strict n-cat certain multisimplicial sets are
  equal
  
• Weak n-cat certain multisimplicial sets are
  equivalent.
  

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Joyal
T(1) ? strict N-Cat, subcategory ?
Free cat generated by pasting diagram

• T(1) is dense
• Strict n-categories are presheaves on T(1)
  unique inner horn fillers
  
• Weak n-cat drop uniqueness.

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Weak n-category family tree
Weak n-cat
Non-algebraic
Algebraic
Weakly Enriched
Algebra for Operad/Monad
Leinster Globular Operad contraction
Trimble topological operad
Penon ?-magma monad
May Any contractible operad in a model

category
Batanin Globular Operad coherence contract
ion
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References

• Higher-Dimensional Categories an illustrated
  guide book
  
• Eugenia Cheng and Aaron Lauda
• Higher Operads, Higher Categories
• Leinster

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Opetopic
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Quasicategories
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Variations on a theme

• Opetopic sets

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Globular Sets
N-morphisms

• A sequence of sets Bi
• Maps satisfying
• ss st
• tt ts
• Presheaf on the globlar category G.

2-morphisms
B1
morphisms
s
t
B0
objects
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Second Algebraic Approach

• Data Globular Set (objects, morphisms, etc)
• Characterize ways of putting things together
• Define operads based on this structure
• Identify operads with compositions and coherence
  relations
  
• Weak n-cat algebra for this operad

Same?
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Ways of Putting Things Together
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Composition Coherence
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Operadic view of Categories
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The Categorical Family Tree
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The Plan

• Strict n-categories
• Operads
• Weak n-categories?

• Test

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Test
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Why are Categories Good?

• replaced with isom.
• Study how things are related.
• How are categories related?

A pcture of a category
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Coherence Axioms