Magnitude Homology

Tom Leinster Edinburgh A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. 1. apologize; 2. blame John Baez.

A theme of this conference so far When introducing a piece of category theory during a talk: Plan

1. The idea of magnitude

2. The magnitude of a metric space

3. The idea of magnitude homology

4. The magnitude homology of a metric space 1. The idea of magnitude Size For many types of mathematical object, there is a canonical notion of size. • Sets have cardinality. It satisﬁes |X ∪ Y | = |X | + |Y | − |X ∩ Y | |X × Y | = |X | × |Y | .

n • Subsets of R have volume. It satisﬁes vol(X ∪ Y ) = vol(X ) + vol(Y ) − vol(X ∩ Y ) vol(X × Y ) = vol(X ) × vol(Y ). • Topological spaces have Euler characteristic. It satisﬁes χ(X ∪ Y ) = χ(X ) + χ(Y ) − χ(X ∩ Y ) (under hypotheses) χ(X × Y ) = χ(X ) × χ(Y ).

Challenge Find a general deﬁnition of ‘size’, including these and other examples. One answer The magnitude of an enriched category. Enriched categories A monoidal category is a category V equipped with a product operation. A category X enriched in V is like an ordinary category, but each HomX(X , Y ) is now an object of V (instead of a set). iercategories linear ercspaces metric categories

enriched posets categories

monoidal • Vect • Set • ([0, ∞], ≥) categories • (0 → 1) The magnitude of an enriched category

There is a general definition of the magnitude |X| of an enriched category. (Definition and details omitted.) Examples This gives definitions of: • the magnitude of a poset • the magnitude of an ordinary category • the magnitude of a linear category • the magnitude of a metric space. The magnitude of a poset

The magnitude of a poset is better known as its Euler characteristic (1960s). Example Let M be a triangulated manifold. Write P for the poset of simplices in the triangulation, ordered by inclusion. Then |P| = χ(M). The magnitude of an ordinary category

Let X be a ﬁnite category.

‘Recall’: X gives rise to a topological space BX (its classifying space), built as follows:

• for each object of X, put a point • into BX; • for each map x → y in X, put an interval •——• into BX; • for each commutative triangle in X, put a 2-simplex N into BX; • ...

Theorem Let X be a ﬁnite category. Then

|X| = χ(BX), under hypotheses ensuring that χ(BX) is well-deﬁned. The magnitude of a linear category

. . . is related to the Euler form in commutative algebra. The magnitude of a metric space

. . . is something new! 2. The magnitude of a metric space

— done explicitly — The magnitude of a ﬁnite metric space

Let X be a ﬁnite metric space.

Write ZX for the X × X matrix with entries

−d(x,y) ZX (x, y) = e

(x, y ∈ X ). (Why e−distance? Because e−(u+v) = e−ue−v .)

n If ZX is invertible (which it is if X ⊆ R ), the magnitude of X is

X −1 |X | = ZX (x, y) ∈ R x,y∈X

—the sum of all the entries of the inverse matrix of ZX . First examples

• |∅| = 0. • |•| = 1. −0 −`−1 ← ` → e e 2 • • • = sum of entries of = e−` e−0 1 + e−`

2 1 0 `

• If d(x, y) = ∞ for all x 6= y then |X | = cardinality(X ).

Slogan: Magnitude is the ‘eﬀective number of points’. • When t is moderate, X looks like a 2-point space. • When t is large, X looks like a 3-point space.

Example: a 3-point space (Simon Willerton) Take the 3-point space

X =

• When t is small, X looks like a 1-point space.

• • When t is large, X looks like a 3-point space.

Example: a 3-point space (Simon Willerton) Take the 3-point space

X =

• When t is small, X looks like a 1-point space. • When t is moderate, X looks like a 2-point space.

•• Example: a 3-point space (Simon Willerton) Take the 3-point space

X =

• When t is small, X looks like a 1-point space. • When t is moderate, X looks like a 2-point space. • When t is large, X looks like a 3-point space.

• • • Example: a 3-point space (Simon Willerton) Take the 3-point space

X =

• When t is small, X looks like a 1-point space. • When t is moderate, X looks like a 2-point space. • When t is large, X looks like a 3-point space. Indeed, the magnitude of X as a function of t is: Magnitude functions Magnitude assigns to each metric space not just a number, but a function. For t > 0, write tX for X scaled up by a factor of t. The magnitude function of a metric space X is the partially-deﬁned function

(0, ∞) → R t 7→ |tX | .

E.g.: the magnitude function of X = (•← ` →•) is |tX | 2 2/(1 + e−`t )

0 t n A magnitude function has only ﬁnitely many singularities (none if X ⊆ R ). It is increasing for t 0, and lim |tX | = cardinality(X ). t→∞ • When t is moderate, tX looks nearly 1-dimensional. • When t is large, tX looks 0-dimensional again.

Dimension at diﬀerent scales

2 Let X = , with subspace metric from R .

• When t is small, tX looks 0-dimensional.

• • When t is large, tX looks 0-dimensional again.

Dimension at diﬀerent scales

2 Let X = , with subspace metric from R .

• When t is small, tX looks 0-dimensional. • When t is moderate, tX looks nearly 1-dimensional. Dimension at diﬀerent scales •

2 Let X = , with subspace metric from R .

• When t is small, tX looks 0-dimensional. • When t is moderate, tX looks nearly 1-dimensional. • When t is large, tX looks 0-dimensional again.

•

• Dimension at diﬀerent scales

2 Let X = , with subspace metric from R .

• When t is small, tX looks 0-dimensional. • When t is moderate, tX looks nearly 1-dimensional. • When t is large, tX looks 0-dimensional again.

The magnitude function sees all this! Here’s how. . . Dimension at diﬀerent scales (Willerton) For a function f : (0, ∞) → R, the instantaneous growth of f at t ∈ (0, ∞) is d(log f (t)) growth(f , t)= = slope of the log-log graph of f at t. d(log t)

E.g.: If f (t) = Ctn then growth(f , t) = n for all t. For a space X , the magnitude dimension of X at scale t is dim(X , t) = growth(|tX | , t).

1.0 E.g.: Let X = . X dim(X , t) 0.5

0 0 10 20 30 40 50 t The magnitude of a compact metric space

A metric space M is positive definite if for every finite Y ⊆ M, the matrix ZY is positive definite. n E.g.: R with Euclidean or taxicab metric; sphere with geodesic metric; hyperbolic space; any ultrametric space.

Theorem (Mark Meckes) All sensible ways of extending the definition of magnitude from finite metric spaces to compact positive definite spaces are equivalent.

For a compact positive deﬁnite space X ,

|X | = sup{|Y | : finite Y ⊆ X }. Magnitude encodes geometric information Theorem (Juan-Antonio Barceló& Tony Carbery) n For compact X ⊆ R , |tX | voln(X ) = Cn lim t→∞ tn where Cn is a known constant. Theorem (Heiko Gimperlein & Magnus Goffeng) n Assume n is odd. For ‘nice’ compact X ⊆ R (meaning that ∂X is smooth and Cl(Int(X )) = X ),

n n−1 n−2 |tX | = cn voln(X )t + cn−1 voln−1(∂X )t + O(t )

as t → ∞, where cn and cn−1 are known constants.

The magnitude function knows the volume and the surface area. Magnitude encodes geometric information Magnitude satisﬁes an asymptotic inclusion-exclusion principle: Theorem (Gimperlein & Goﬀeng) n Assume n is odd. Let X , Y ⊆ R with X , Y and X ∩ Y nice. Then |t(X ∪ Y )| + |t(X ∩ Y )| − |tX | − |tY | → 0 as t → ∞.

n n But not all results are asymptotic! Let B denote the unit ball in R . Theorem (Barcel´o& Carbery; Willerton) n Assume n is odd. Then |tB | is a known rational function of t over Z. Examples

• tB1 = |[−t, t]| = 1 + t 3 2 1 3 • tB = 1 + 2t + t + 3! t 2 3 4 5 5 24 + 72t + 35t + 9t + t t • tB5 = + . 8(3 + t) 5! 3. The idea of magnitude homology Two perspectives on Euler characteristic

So far: Euler characteristic has been treated as an analogue of cardinality.

Alternatively: Given any homology theory H∗ of any kind of object X , can deﬁne ∞ X n χ(X )= (−1) rank Hn(X ). n=0

Note: • χ(X ) is a number

• H∗(X ) is an algebraic structure, and functorial in X .

We say that H∗ is a categoriﬁcation of χ. So, homology categoriﬁes Euler characteristic. Towards magnitude homology of enriched categories

Any ordinary category X gives rise to a chain complex C∗(X): a Cn(X)= Z · X(x0, x1) × · · · × X(xn−1, xn) x0,...,xn∈X where Z · −: Set → Ab is the free abelian group functor.

The homology H∗(X) of X is the homology of C∗(X).

Theorem H∗(X) = H∗(BX).

Generalizing this, Michael Shulman gave a deﬁnition of the homology of an enriched category (omitted here).

It should categorify magnitude. In particular, it gives a homology theory of metric spaces. . . 4. The magnitude homology of a metric space

Special case of graphs: Hepworth and Willerton (2015) General case of enriched categories: Shulman (n-Category Caf´e,Aug 2016) The shape of the deﬁnition

In this talk, a persistence module is a functor

A: ([0, ∞], ≥) → Ab.

That is: it’s a family (A(`))`∈[0,∞] of abelian groups, together with a homomorphism α`,k : A(`) → A(k) whenever ` ≥ k, such that α`,k ◦ αm,` = αm,k and α`,` = id.

We will define the homology H∗(X , A) of a metric space X with coefficients in a persistence module A. Each Hn(X , A) is an abelian group. This is a special case of the general definition for enriched categories. The magnitude homology H∗(X , A) is the homology of C∗(X , A).

The deﬁnition Let X be a metric space and let A be a persistence module. There is a chain complex C(X , A) with a Cn(X , A) = A d(x0, x1) + ··· + d(xn−1, xn) .

x0,...,xn∈X

The diﬀerential is

n X i ∂ = (−1) ∂i : Cn(X , A) → Cn−1(X , A) i=0 where (e.g.) in the case n = 2, the maps ∂0, ∂1, ∂2 are given as follows:

the inequality d(x0, x1) + d(x1, x2) ≥ d(x1, x2) induces a homomorphism ∂0 : A d(x0, x1) + d(x1, x2) → A d(x1, x2) . The magnitude homology H∗(X , A) is the homology of C∗(X , A).

The deﬁnition Let X be a metric space and let A be a persistence module. There is a chain complex C(X , A) with a Cn(X , A) = A d(x0, x1) + ··· + d(xn−1, xn) .

x0,...,xn∈X

The diﬀerential is

n X i ∂ = (−1) ∂i : Cn(X , A) → Cn−1(X , A) i=0 where (e.g.) in the case n = 2, the maps ∂0, ∂1, ∂2 are given as follows:

the inequality d(x0, x1) + d(x1, x2) ≥ d(x0, x2) induces a homomorphism ∂1 : A d(x0, x1) + d(x1, x2) → A d(x0, x2) . The deﬁnition Let X be a metric space and let A be a persistence module. There is a chain complex C(X , A) with a Cn(X , A) = A d(x0, x1) + ··· + d(xn−1, xn) .

x0,...,xn∈X

The diﬀerential is

n X i ∂ = (−1) ∂i : Cn(X , A) → Cn−1(X , A) i=0 where (e.g.) in the case n = 2, the maps ∂0, ∂1, ∂2 are given as follows:

the inequality d(x0, x1) + d(x1, x2) ≥ d(x0, x1) induces a homomorphism ∂2 : A d(x0, x1) + d(x1, x2) → A d(x0, x1) .

The magnitude homology H∗(X , A) is the homology of C∗(X , A). Magnitude homology with coeﬃcients in a point module For each ` ∈ [0, ∞], deﬁne a persistence module A` by ( Z if k = ` A`(k) = 0 otherwise.

Then Cn(X , A`) = Z · (x0,..., xn): d(x0, x1) + ··· + d(xn−1, xn) = ` .

Equivalently, we can replace C∗(X , A) by a normalized version:

] Cn(X , A`) = Z· (x0,..., xn): d(x0, x1)+···+d(xn−1, xn) = `, x0 6= · · · 6= xn .

Pn−1 i The diﬀerential is ∂ = i=1 (−1) ∂i , where ( (x0,..., xi−1, xi+1,..., xn) if xi is between xi−1 and xi+1 ∂i (x0,..., xn) = 0 otherwise.

‘Between’ means that d(xi−1, xi ) + d(xi , xi+1) = d(xi−1, xi+1). H1 detects convexity A metric space X is Menger convex if for all distinct x, y ∈ X , there exists z ∈ X between x and y with x 6= z 6= y.

Theorem Let X be a metric space. Then

X is Menger convex ⇐⇒ H1(X , A`) = 0 for all ` > 0.

n Corollary Let X be a closed subset of R . Then

X is convex ⇐⇒ H1(X , A`) = 0 for all ` > 0.

And, for instance, if X = ••••••• ⊆ R with all gaps of length < ε, then H1(X , A`) = 0 for all ` ≥ ε. Back to Euler characteristic Let X be a metric space. For any persistence module A, put ∞ X n χ(X , A)= (−1) rank Hn(X , A) n=0 (if deﬁned). In particular, we have an Euler characteristic ∞ X n χ(X , A`) = (−1) rank Hn(X , A`) n=0 for each ` ∈ [0, ∞). Not just one Euler characteristic: many! Make these Euler characteristics into the coeﬃcients of a formal expression:

X ` χ(X )= χ(X , A`) q . `∈[0,∞) Claim: χ(X ) is formally equal to |tX |, where q = e−t . So I’m claiming:

Magnitude homology categoriﬁes magnitude Magnitude homology categoriﬁes magnitude: ‘proof’

X X n ` X n ] ` χ(X ) = (−1) rank Hn(X , A`) q = (−1) rank Cn (X , A`) q `∈[0,∞) n∈N `,n X n ` = (−1) (x0,..., xn): d(x0, x1) + ··· + d(xn−1, xn) = `, x0 6= · · · 6= xn q n,` X X = (−1)n qd(x0,x1)+···+d(xn−1,xn)

n x0,...,xn: x06=···6=xn

X n X d(x0,x1) d(xn−1,xn) = (−1) q − δx0,x1 ··· q − δxn−1,xn

n x0,...,xn∈X X n X = (−1) (ZtX − I )x0,x1 ··· (ZtX − I )xn−1,xn

n x0,...,xn∈X X n n = (−1) sum (ZtX − I ) where sum means the sum of all entries n X n −1 −1 = sum (I − ZtX ) = sum I − (I − ZtX ) = sum ZtX n∈N = |tX | . X . . . formally, at least! Open questions

1. What information does the magnitude homology H∗(X , A) capture when we use other persistence modules A as our coeﬃcients (e.g. interval modules)? 2. What is the relationship between magnitude homology and persistent homology? 3. Which theorems about magnitude of metric spaces can be categoriﬁed to give theorems about magnitude homology? Compare:

I Many theorems about topological Euler characteristic are shadows of theorems about homology.

I For the special case of graphs, Hepworth and Willerton already proved a Künneththeorem (categorifying the formula for |X × Y |) and a Mayer–Vietoris theorem (categorifying formula for |X ∪ Y |). References (titles are clickable links) General references on magnitude: • Leinster, The magnitude of metric spaces • Leinster and Meckes, The magnitude of a metric space: from category theory to geometric measure theory Magnitude of finite metric spaces (in direction of data): • Willerton, Spread: a measure of the size of metric spaces • Willerton, Instantaneous dimension of finite metric spaces via magnitude and spread Analytic aspects of magnitude: • Meckes, Positive definite metric spaces • Meckes, Magnitude, diversity, capacities, and dimensions of metric spaces • Barcelóand Carbery, On the magnitudes of compact sets in Euclidean spaces • Willerton, The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials • Gimperlein and Goffeng, On the magnitude function of domains in Euclidean space Magnitude homology: • Hepworth and Willerton, Categorifying the magnitude of a graph • Shulman, Leinster, et al., Magnitude homology