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 jX [ Y j = jX j + jY j − jX \ Y j jX × Y j = jX j × jY j : 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). linear categories metric spaces categories enriched posets categories monoidal • Vect • Set • ([0; 1]; ≥) categories • (0 ! 1) The magnitude of an enriched category There is a general definition of the magnitude jXj 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 jPj = 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 jXj = 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 2 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 jX j = ZX (x; y) 2 R x;y2X |the sum of all the entries of the inverse matrix of ZX . First examples • j;j = 0. • |•| = 1. −0 −`−1 ` ! e e 2 • • • = sum of entries of = e−` e−0 1 + e−` 2 1 0 ` • If d(x; y) = 1 for all x 6= y then jX j = 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; 1) ! R t 7! jtX j : E.g.: the magnitude function of X = (• ` !•) is jtX j 2 2=(1 + e−`t ) 1 0 t n A magnitude function has only finitely many singularities (none if X ⊆ R ). It is increasing for t 0, and lim jtX j = cardinality(X ). t!1 • 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; 1) ! R, the instantaneous growth of f at t 2 (0; 1) 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(jtX j ; 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 , jX j = supfjY j : finite Y ⊆ X g: Magnitude encodes geometric information Theorem (Juan-Antonio Barcel´o& Tony Carbery) n For compact X ⊆ R , jtX j voln(X ) = Cn lim t!1 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 jtX j = cn voln(X )t + cn−1 voln−1(@X )t + O(t ) as t ! 1, 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 jt(X [ Y )j + jt(X \ Y )j − jtX j − jtY j ! 0 as t ! 1. 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 jtB j is a known rational function of t over Z. Examples • tB1 = j[−t; t]j = 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 1 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;:::;xn2X 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).