HIGHER COHERENCE AND A GENERALIZATION OFHIGHER CATEGORIFIED ALGEBRAIC STRUCTURES
MATSUOKA, TAKUO

These (non-)slides reflect (to some degree) a talk I gave in Hiroshima and another in Himeji inNovember 2016. I am grateful to the organizers of the symposia in which the talks took place.

Reference: arXiv:1601.00301.
1. Introduction
Discovery or recognitionof the right kind of algebraic structure
is often importantin the development of mathematical subjects.
Starting perhaps with “group” in Galois theory,a list of examples would easily get long.
Can we systematicallyfind and treat algebraic structures?

Question seems important
for the use of higher categorical ideas.
With high categorical dimensionality,
more variety of structures available.
Why higher category theory?
• Necessary for both analysis and construction of topological field
theories, as has become clear
from the solution and generalizationof Baez and Dolan’s cobordism hypothesis by Lurie and Hopkins.
Lurie argues that an instanceof CH proved earlier was a theorem
by Costello, which was for proposing a definition of the B-model in themirror symmetry.
Date: 2016/11.
1• Categorification has been a useful method
for finding important new structures (since Grothendieck).

We apply a concrete understanding of higher coherence, and findsomething systematic at a quite global level.
In order to proceed, we need a few reminders.
Coloured operad/Multicategory, a generalizationof symmetric monoidal category:
For A symmetric monoidal category,underlying multicategory ΘA, from which
A can be recovered.
A multicategory controls algebras over it...
For U a multicategory
U-algebra in a symmetric monoidal category A= functor U → ΘA of multicategories.
Multicategory is analogous to an algebraic theory in the sense ofLawvere.
Many kinds of algebraic structure (in a symmetric monoidal category)controlled by a multicategory.
Commutative: by “Com”, terminal.
Associative: by “E1 ”.
Bare object (no structure): by “Init”, initial uncoloured operad.
...and so on.
With colours more variety of structures can be controlled.

Example of categorified structure:
For U a multicategory“U-monoidal category” — U-algebra in categories.
Symmetric monoidal, associative monoidal, braided, ...
Back to the subject...
When the notion “X-algebra” (e.g., U-algebra, multicategory, etc.)
has a categorification,

“categorified X-algebra” sometimes hasa good generalization, “X-theory”.
The meaning of “good” later.

We call the notion of X-theory a theorization of the notion of Xalgebra.
“Multicategory” is a theorizationof “commutative algebra”, which
generalizes “symmetric monoidal category”.
There is a natural
theorization “U-graded multicategory” of “U-algebra”,generalizing “U-monoidal category”.
E1 -graded = planar
E2 -graded = braided
Init-graded multicategory = category
MulticatU (Set) = Multicat(Set)/U .
In the case U ∈ Multicat(Gpd), e.g., U = E2 ,MulticatU (Gpd) = Multicat(Gpd)/U .
Example: For C a category= multicategory with only unary multimaps
Algebra: Functor on C (“left C-module”).
Categorification: Functor C → Cat.
Theorization: In Set,
category X + functor X → C,among which,categorifications (and their lax morphisms) correspond to fibrations
X op
C op
(and not necessarily Cartesian functors over C op ).
“(U-graded) 2-theory” — theorization of “(U-graded) multicategory”(= “1-theory”).

Example: For M a planar multicategory,“Morita” Init-graded 2-theory Alg1 (M), where
Alg1 (ΘA) ' ΘAlg1 (A)for A a associative monoidal category with nice behaviour,
Alg1 (A) “Morita” 2-category (= categorified Init-graded 1-theory) due
to Bénabou.
Alg1 will appear again later.
2. Coherence and higher theories
Concretely, can theorize
by using an inductivity
embedded in the structure of the coherence for higher associativity:
Consider situation:
m: system of operations
wanting to be associative, operating
as maps in an symmeric monoidal (∞, 1)-category.
E.g., “m + coherent associativity” maydefine an “X-algebra”.
m0 : system of (2-)isomorphisms/homotopies
giving an associativity of m.
m0 wants to be coherent.
In this situation,
see m0 as operations themselves.
Invertibility of those operations not required for a (op)lax X-algebra.

ThenCoherence of the associativity m0 for m= Coherent associativity of m0 as operations
Idea for theorizing: If mgives an X-algebra structure,then “X-theory” is a kinddefined by operations m0...considered in the “categorical deloop” BA, so 2-morphisms m0 are maps in
A = EndBA (∗).

Conceptual significance of theorization to be discussed later.

We can iterate theorization,
get to “n-theory”, generalizing n-categories in particular.
Definition of an n-theory can be written explicitly.
There are general constructionsof higher theories, which play various roles.
• “Delooping” construction B, of an (n + 1)-theory from an ntheory.
For A a symmetric monoidal category (= categorified 0-theory)Θn+1 B n A ' Bn ΘA,equivalence of (n + 1)-theories.
• Change of “grading”: pull-back,push-forward on left and on right.
• Day convolution.
Many of these raises theoretic order.
If at least interested
in multicategories, thenwould want to know allhigher theories.
Also concrete constructions
in more specific situations.
3. Meaning of theorization
Relevance of categorification for us:“X-algebra” makes sense
in a categorified formof X-algebra.
E.g.,
U-algebra in a U-monoidal category A= lax U-monoidal functor 10U → A,
10U unit U-monoidal.
...an enriched notionof U-algebra.

“U-graded multicategory”
was a common generalizationof “U-monoidal category” and “Multicategory”.
Notion of U-algebra can be enriched in a U-graded multicategory.
In general, by theorizing “X-algebra”,
we want...
“For A a categorified form of X-algebra
X-algebra in A
→ ΘA of X-theories”= “coloured” functor 1TheoryX
terminal X-theory,• 1TheoryX• ΘA denotes A as an X-theory.
• Left hand side = “coloured” lax functor 1Algebra→ A of XXalgebras.
3X-theory

U1TheoryX“virtualize”

Categorified
X-algebra

control

control

U-algebra

X-algebra.

theorize“grade” by U(see below)

categorifyX-algebra
Enriched notions
FunctorU →Vforbidcolours

U-algebra
in V

U=1TheoryXX-algebra
in V

V=ΘA

V=ΘA

U-algebra
in A

X-algebra
in A

U=1Theory
X

whereU, V: X-theories
A: categorified X-algebra
4. Graded higher theories
Are graded multicategories controlled? By what?
— By a very simple 2-theory.
In comparison,classical answer: “By a multicategory”.

forbidcolours

Lax functor
1Algebra→ A,X

E.g., “slice” multicategory U + of Baez and Dolan
AlgU + (Set) = MulticatSame colours as U’s (Set)/U .
But a simpler answer above by:expressing “1-dimensional” structure (multicategory)as algebra over a 2-dimensional structure.
Purpose of Baez and Dolan was to define n-category.
Iterated theorization is a more direct route to n-categories
compared to Baez and Dolan.
Key: an n-category is naturally n-dimensional as is an n-theory,
rather than 1-dimensional like a multicategory.

Also helps with enriching the notion...
Where to enrich “U-graded multicategory”?
— In a (U ⊗E1 )-monoidal category (folklore? ...if U-graded multicategories
in a symmetric monoidal category had been known
as seems possible.)(E1 ⊗ E1 = E2 etc.),

and more generally, ...
Let us see the details.
For U an n-theory, “U-graded n-theory”, theorizing “U-algebra”.
Theorem: Following notions are equivalent:• U-graded n-theory.
• ΘU-algebra.
n = 1: U-graded multicategories are controlled
by the 2-theory ΘUMulticatU = AlgΘU .
ΘU is simple and direct...compared to U + , which was the main construction
for Baez and Dolan.
Construction of ΘU is in one step:
Replace composition operation in U
with the corepresented “bimodule”.

...essentially as simple as the underlying multicategory of a symmetric monoidal category.

“U-graded n-theory” has a natural theorization, andTheorem: Following notions are equivalent:• U-graded (n + 1)-theory.
• ΘU-graded (n + 1)-theory.
Leads to iterative theorization.
For U a multicategory = 1-theory,a U-graded 2-theory
is a general place where
the notion of U-graded multicategory
can be enriched,specializing to enrichment
in a (U ⊗ E1 )-monoidal category.
Example: U = Init,
M a planar mutlicategory.
There is forgetful functor
Alg1 (M) −→ BMof Init-graded 2-theories,generalizing the forgetful lax functor
Alg1 (A) −→ BAfor A a nice associative monoidal category.
A category = Init-graded 1-theory, enrichedin M or “along” BM,
has a canonical lift
to a category enrichedalong Alg1 (M).
5. Graded lower theories
U an n-theory.
For 0 ≤ m ≤ n − 1, there is a notion of U-graded m-theory so that
U-algebra = U-graded (n − 1)-theory.

Moreover, a notion of “`-theory” graded by a U-graded m-theory.
E.g.,
1mU -graded `-theory = U-graded `-theory
6. More general higher theories
Can also theorize some structures
involving operations with multiple inputs and multiple outputs,
e.g.,“coloured properad” of Vallette, which turns out
to be “Cocorr(Fin)-graded” 1-theory,Cocorr(Fin) cocorrespondence category on finite sets.
We get iterative theorizations.
For C a category,a “Bordoriented-graded” 1-theory ZC such that1every 1-dimensional “TFT” in ZC
11Bord1 −→ ZC
is of the formZx
11Bord1 = Z1 −→ZC

for an unique object
x : 1 −→ Cof C.
Very different from field theories in untheorized context,classified by dualizable objects of a symmetric monoidal category.