Michael Robinson

Categorification and Chain Complexes

SIMPLEX Program

© 2015 Michael Robinson This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Michael Robinson2

Recap of yesterday● Sheaves can support faithful models information

integration problems – indeed, they're canonical● But they can become too complicated to be useful● The issue is that the stalks are sets, without any

algebraic structure

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Focus for today: computation● We want to derive actionable, relevant summaries

of sheaves● The summaries should be sheaf invariants● The summaries should be computationally tractible

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Mathematical dependency tree

Sheaves

Cellular sheaves

Linear algebra

Set theory

Calculus

Topology

Homology

SimplicialComplexes

CW complexes

Sheaf cohomology

AbstractSimplicialComplexes

de Rham cohomology(Stokes' theorem)

Manifolds

Lecture 2

Lectures 3, 4

Lectures 5, 6

Lectures 7, 8

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Session objectives● How do we encode data for convenient

computation?

● Build out homological algebra, which is really multiscale linear algebra

● What kind of homological invariants are there?

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Encoding data for computation

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Encoding data for computationFasten your seatbelts...

this is a bit abstract!

But the payoff is worth it

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What data types are useful?

Thanks to Cliff Joslyn for this graphic!

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Lifting functions into linear mapsConsider any function between sets f: AB● Let ℝ(A) be the vector space generated by A

– The basis of ℝ(A) is the set of elements of A● Then f lifts uniquely to a linear map Rf

ℝ(A) ℝ(B)

A Bf

Rf Notice that generally we cannot recover a unique element of B from ℝ(B).

But we can if we've used Rf ∘ (1×)

(1×) (1×)

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Categorification

Sets

Functions

Categories

Functors

Before we start in with precision, some notes:● There may be several possible categorifications for a given set. Choosing the best one is still an art

● This process allows us to normalize a sheaf with many different data types into a sheaf with just vector data

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Data types as categoriesA category C consists of● A class Ob(C) of objects● A class Mor(C) of morphisms

For which● Each morphism m has a source and target object, usually written

m: A → B● Morphisms can be composed: if p : A → B and q : B → C, then

there is unique morphism q ∘ p : A → C called their composition● Composition is associative: (p ∘ q) ∘ r = p ∘ (q ∘ r)

● There is an identity morphism 1A for every object A for which

p ∘ 1A = p and 1A ∘ q = q

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Isomorphisms● An isomorphism in a category C is a morphism

f: A → B

for which there is another morphism in Cf-1: B → A

satisfying

f ∘ f-1 = 1B and f-1 ∘ f = 1A.

● We say that A and B are isomorphic objects

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Changing types via functorsIf C and D are categories, a covariant functor F : C → D assigns:● An object F(A) in D for each object A in C● A morphism F(m) : F(A) → F(B) in D for each

morphism m : A → B in Cso that composition is preserved F(m ∘ n) = F(m) ∘ F(n).● Contravariant functors are the same, but reverse the

direction of morphisms and the order of composition.● A functor is faithful if it is injective on morphisms

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Categorification ● The decategorification Decat(C) of a category C is

the set of isomorphism classes● If S is a set, then C is a categorification of S if there

is a decategorification function

d : Decat(C) → S● Categorifications are not unique!● With good categorifications, morphisms of C turn

into appropriate functions or relations on S

See John Baezhttps://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html

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OK. That was probably altogether too much abstraction!

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Getting everything vectorified...

Start with a set S :● Categorify it into a category C● Decategorify C into a vector space V

Decategorification map Decategorification map

bijective injectiveS Decat(C) V

We make the stipulation here that morphisms of C turn into linear maps V → V

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Getting everything vectorified...

Start with a set S :● Categorify it into a category C● Decategorify C into a vector space V

Categorification map Decategorification map

bijective injectiveS Decat(C) V

This is the (1×) map we saw earlier

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Vectorification of BooleansStarting category: Bool● One object (the set {0,1}) and its Cartesian products

(sets of tuples)● Logic functions as morphisms

A

BC

A B C0 0 10 1 11 0 11 1 0

Not linear!

A

B

C

Logic circuit Connection graph

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Vectorification of Posets

Start with a poset P :● Categorify it into a category P in which

A→B whenever A ≤ B● Decategorify P into a vector space V

Decategorification map Decategorification map

bijective injectiveP Decat(P) V

Morphisms of P turn into linear maps V → V, which in this case are various permutation maps or projection maps

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Returning to types...

First, think of this hierarchy in terms of sets, as we usually do

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Categorify: step 1

Categorify: reinterpret each type as a category

NB: There are many ways to do this...Note: The arrows become functors

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Categorify: step 2

Decategorify back into Vec, the category of vector spaces

NB: There are even more ways to do this...

Vec

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Payoff: preserving structure of heterogeneous data

● If the data we started with was heterogeneous, we probably had to work with a sheaf of sets and forgot much of the data's internal structure

● After categorifying, we (temporarily) made a sheaf of categories (each stalk is a category), but we're able to capture all of the data's internal structure

● When decategorifying back into vector spaces, we preserve that structure through the presence of linear self-maps!

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Hybrid datatypes● Categorification respects the constraints inherent in

data. This can result in coefficients being implied...

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What to do with vectorified data?

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The dimension theoremTheorem: Linear maps between vector spaces are characterized by four fundamental subspaces

A Bf

ker fcoimage f image f

coker f

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Exactness of a sequenceExactness of a sequence of maps,

means that image f = ker gA → B → C

f g

f g

A B C

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Example: exact sequence

00

1 00 00 10 0

0 1 0 00 0 0 10 1 0 1 (1 1 -1) (0)

0 ℝ2 ℝ4 ℝ3 ℝ 0

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Properties of exact sequencesExactness encodes useful properties of maps● Injectivity

0 → A → B ● Surjectivity

A → B → 0● Isomorphism

0 → A → B → 0● Quotient

0 → A → B → B / A → 0

f

f

f

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Chain complexesExactness is special and delicate. Usually our sequences satisfy a weaker condition:

A chain complex

satisfies image f ⊆ ker g or equivalently g ∘ f = 0

Exact sequences are chain complexes, but not conversely

Homology measures the difference

A → B → Cf g

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The category Kom● Objects of Kom are chain complexes● Morphisms of Kom are diagrams like this…

● Homology is a collection of covariant functors

Hk : Kom → Vec

Vk Vk-1

dk Vk-2

dk-1Vk+1

dk+1

WkWk-1

ek Wk-2

ek-1Wk+1

ek+1

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Homology of a chain complexStarting with a chain complex

Homology is defined as

Vk Vk-1

dk Vk-2

dk-1Vk+1

dk+1

Hk = ker dk / image dk+1

All the vectors that are annihilated in stage k ... … that weren't already present in

stage k + 1Homology is trivial if and only if the chain complex is exact

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A summary: Euler characteristicStarting with a chain complex

The Euler characteristic is the alternating sum of dimensions of the Vk,, which happens to be a homological property

χ(V) = Σ (-1)k dim Vkk

= Σ (-1)k dim Hkk

Vk Vk-1

dk Vk-2

dk-1Vk+1

dk+1

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Next up...● Interactive session: Constructing and Categorifying● Next lecture: Computing topological features

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Further reading...● John Baez and James Dolan, “Categorification”, in Ezra Getzler,

Mikhail Kapranov, Higher Category Theory, Contemp. Math. 230, Providence, Rhode Island: American Mathematical Society, pp. 1–36, 1998.

● Robert Ghrist, Justin Curry, and Michael Robinson “Euler calculus and its applications to signals and sensing,” in Proceedings of Symposia in Applied Mathematics: Advances in Applied and Computational Topology, Afra Zomorodian (ed.), 2012.

● Allen Hatcher, Algebraic Topology, Cambridge, 2002.

● Michael Robinson, “Asynchronous logic circuits and sheaf obstructions,” Electronic Notes in Theoretical Computer Science (2012), pp. 159-177.

● Gilbert Strang, “The Fundamental Theorem of Linear Algebra,” The American Mathematical Monthly, Vol. 100, No. 9 (Nov., 1993), pp. 848-855