Categorification and Chain Complexesdrmichaelrobinson.net/sheaftutorial/20150826_tutorial_1.pdf ·...
Transcript of Categorification and Chain Complexesdrmichaelrobinson.net/sheaftutorial/20150826_tutorial_1.pdf ·...
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.
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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