Discrete Euler integration over functions on finite categories
aa r X i v : . [ m a t h . C O ] A ug Discrete Euler integration overfunctions on finite categories
Kohei TanakaAugust 26, 2018
Abstract
This paper provides the theory of integration with respect to Eulercharacteristics of finite categories. As an application, we use sensors toenumerate the targets lying on a poset. This is a discrete analogue toBaryshnikov and Ghrist’s work on integral theory using topological Eulercharacteristics.
It has long been known that the Euler characteristic is an important homotopyinvariant of a space. By regarding it as a topological measure, we can derive thetheory of the integral with respect to the Euler characteristic (
Euler integration ).Baryshnikov and Ghrist developed the theory of Euler integration, and theyapplied it to sensor networks [BG09], [BG10]. They established a way to usesensors to enumerate targets in a filed. Let us review briefly a simple case thatthey considered.Consider a situation in which there are a finite number of targets T lying ona topological space X . Assume that each point of X has a sensor recording thenearby targets, and each target t ∈ T has a contractible target support: U t = { x ∈ X | the sensor at x detects t } . The sensors return the counting function h : X → N ∪ { } given by the numberof detectable sensors at each point: h ( x ) = { t ∈ T | x ∈ U t } ♯ . Then, we can enumerate the targets by integrating with respect to the Eulercharacteristic χ (Theorem 3.2 of [BG09]): T ♯ = Z X hdχ. definable ) functions on a finite category. Following this, wedefine the Euler integration of definable functions, and investigate its properties.Section 4 presents an application with sensor networks. The Euler characteristic of a finite category was introduced by Leinster [Lei08];it is a generalization of the concept of M¨obius inversion [Rot64] for posets.
Definition 2.1.
Suppose that C is a finite category consisting of a finite numberof objects and morphisms. We denote the set of objects of C by ob( C ), and theset of morphisms from x to y by C ( x, y ).1. The similarity matrix of C is the function ζ : ob( C ) × ob( C ) → Q , givenby the cardinality of each set of morphisms: ζ ( a, b ) = C ( a, b ) ♯ .2. Let u : ob( C ) → Q denote the column vector with u ( a ) = 1, for any object a of C . A weighting on C is a column vector w : ob( C ) → Q such that ζw = u , and dually, a coweighting on C is a row vector v : ob( C ) → Q such that vζ = u ∗ , where u ∗ is the transposition of the matrix u .Note that we have X i ∈ ob( C ) w ( i ) = u ∗ w = vζw = vu = X j ∈ ob( C ) v ( j ) , if both a weighting and a coweighting exist. Moreover, X i ∈ ob( C ) w ( i ) = u ∗ w = vζw = vζw ′ = u ∗ w ′ = X i ∈ ob( C ) w ′ ( i ) , w and w ′ on C . This guarantees the following definitionof the Euler characteristic. Definition 2.2 ([Lei08]) . Let C be a finite category. We say that C has Eulercharacteristic if it has both a weighting w and a coweighting v on C . Then, the Euler characteristic of C is defined by χ ( C ) = X i ∈ ob( C ) w ( i ) = X j ∈ ob( C ) v ( j ) . Proposition 2.3 (Example 2.3 (d) in [Lei08]) . If C has Euler characteristicand either an initial or a terminal object, then χ ( C ) = 1 . For a small category C , the classifying space BC is defined as the geomet-ric realization of the nerve of C . When C never has a nontrivial circuit ofmorphisms ( acyclic ), the following relation holds between the topological Eulercharacteristic and the combinatorial one defined in Definition 2.2. Proposition 2.4 (Proposition 2.11 in [Lei08]) . Every finite acyclic category C has Euler characteristic, and χ ( C ) = χ ( BC ) . For topological Euler characteristics, we have the following well-known inclusion-exclusion formula: χ ( A ∪ B ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) , for suitable subspaces A and B of a space. Unfortunately, in the case of Eulercharacteristics of categories, it does not hold in general. In [Tan], the authorintroduced two classes of categories satisfying the inclusion-exclusion formulastated above. Definition 2.5.
Let C be a small category. A filter D is a full subcategory of C such that an object y of C belongs to D whenever C ( x, y ) = ∅ for some object x of D . Dually, an ideal D of C is a full subcategory such that an object y of C belongs to D whenever C ( y, x ) = ∅ for some object x of D . in other words,a full subcategory D of C is an ideal if and only if the opposite category D op isa filter of C op .These are generalizations for posets of filters and ideals [Sta12], [Zap98]. Let D be a full subcategory of a small category C . The category of complements C \ D is defined as the full subcategory of C whose set of objects is ob( C ) \ ob( D ).If D is a filter (an ideal) of C , then C \ D is an ideal (a filter) of C . In otherwords, a filter or an ideal D determines a functor C → I , where I is the posetformed of 0 <
1. Hence, we have three bijective sets: the set F ( C ) of filters of C , the set I ( C ) of ideals of C , and the set C I of functors from C to I .The following theorem is shown in [Tan]. Theorem 2.6 (Corollary 3.4 of [Tan]) . Let both A and B be either filters orideals of a finite category C . If each of A , B , A ∩ B , and A ∪ B has an Eulercharacteristic, then we have χ ( A ∪ B ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) . Discrete Euler integration over functions oncategories
Throughout this paper, we will occasionally identify a full subcategory B of acategory C as the underlying set ob( B ) of objects. Moreover, for two categories C and D , a map on objects ob( C ) → ob( D ) will be simply denoted by C → D . Definition 3.1.
Let C be a small category.1. Two objects x and y of C are reflexible if C ( x, y ) = ∅ and C ( y, x ) = ∅ . Inthis case, let us denote x ∼ y . We then have an equivalence relation onthe set of objects of C . The quotient set P ( C ) is equipped with a partialorder that is defined by [ x ] ≤ [ y ] if C ( x, y ) = ∅ . Here, this order does notdepend on the choice of representing objects.2. Let ∨ x denote the prime filter generated from an object x of C , i.e., itconsists of ending objects of morphisms starting at x : ∨ x = { y ∈ ob( C ) | C ( x, y ) = ∅} . Dually, let ∧ x denote the prime ideal generated from x and consisting ofstarting objects of morphisms ending at x .3. An object x of C is called maximal if it satisfies either C ( x, y ) = C ( y, x ) = ∅ or C ( y, x ) = ∅ for any object y . Dually, a minimal object x of C satisfieseither C ( x, y ) = C ( y, x ) = ∅ or C ( x, y ) = ∅ for any object y .Baryshnikov and Ghrist introduced a class of integrable ( definable, con-structible ) functions with respect to the topological Euler characteristic [BG09],[BG10]. Below, we give the definition for definable maps in our discrete setting. Definition 3.2.
A map f : C → D on objects is called definable if it preservesthe reflexible relation. That is, f ( x ) ∼ f ( y ) in D for any reflexible pair x ∼ y in C . In particular, a definable map C → Q is called a definable function on C .Here, we regard the set of rational numbers Q as a totally ordered set in thecanonical order. Therefore, a definable function sends a reflexible pair to thesame value. Let DF( C ) denote the Q -vector space of definable functions on C .For a poset P , every Q -valued function on P is definable, and hence, the vectorspace DF( P ) consists of Q -valued functions on P . Remark 3.3.
The above definitions related to a category C can be describedin terms of the poset P ( C ) and the canonical functor π : C → P ( C ). • A full subcategory D of C is a filter (an ideal) of C if and only if π ( D ) isa filter (an ideal) of P ( C ). • An object x of C is maximal (minimal) if and only if π ( x ) is maximal(minimal) in P ( C ). 4 A map f : C → D on objects of two categories C and D is definable ifand only if it induces a map ˜ f : P ( C ) → P ( D ) (it is not necessary thatthe order be preserved) that makes the following diagram commute: C f / / π (cid:15) (cid:15) D π (cid:15) (cid:15) P ( C ) ˜ f / / P ( D ) . Definition 3.4.
Let B be a full subcategory of a category C , and let f : C → Q be a function on objects. The clipping of f on B is the function f B : C → Q defined by f B ( x ) = f ( x ) if x ∈ ob( B ), and f B ( x ) = 0 otherwise. This shouldnot be confused with the restriction f | B , whose domain is B . The clipping δ B of the constant function δ : C → Q onto 1 ∈ Q is called the incidence function on B . The incidence function on B is definable if B is either a filter or an idealof C .In Definition 2.3 of [BG09], a definable (constructible) function on a finitesimplicial complex is defined to be a linear form of the incidence functions onsimplices. Lemma 3.5.
A rational-valued function f on a finite poset P can be describedusing the following finite linear forms: f = n X i =1 a i δ A i = m X j =1 b j δ B j , where n, m ≥ ; a i , b j ∈ Q ; A i ∈ F ( P ) ; and B i ∈ I ( P ) .Proof. We use induction on the cardinality of P . When P consists of a singleelement x , it is obvious that f ( x ) = f ( x ) δ ∨ x = f ( x ) δ ∧ x . Assume that anyfunction on P has a linear form of incidence functions on filters, in the case of P ♯ ≤ n −
1. When P ♯ = n , take a maximal element x ∈ P . The inductiveassumption decomposes the restriction f | P −{ x } as f | P −{ x } = n X i =1 a i δ A i , for a i ∈ Q , A i ∈ F ( P − { x } ). The union A i ∪ { x } is a filter of P for each i . Thefunction f is described as f = n X i =1 a i δ A i ∪{ x } + f ( x ) − n X i =1 a i ! δ ∨ x . On the other hand, each incidence function δ Q on a filter Q is described as thelinear form δ Q = δ P − δ P \ Q of incidence functions on ideals, and this completesthe proof. 5 orollary 3.6. A definable function f on a finite category C can be describedusing the following finite linear forms: f = n X i =1 a i δ A i = m X j =1 b j δ B j , where n, m ≥ ; a i , b j ∈ Q ; A i ∈ F ( C ) ; and B i ∈ I ( C ) .Proof. The induced function π ( f ) on the poset P ( C ) has the desired liner forms: π ( f ) = n X i =1 a i δ A i = m X j =1 b j δ B j , where n, m ≥ a i , b j ∈ Q ; A i ∈ F ( P ( C )); and B j ∈ I ( P ( C )). For thecanonical projection π : C → P ( C ), the inverse images π − ( A i ) are filters and π − ( B i ) are ideals of C . We can describe the function f as f = n X i =1 a i δ π − ( A i ) = m X j =1 b j δ π − ( B j ) . For a category C , the corollary above shows the following equalities:DF( C ) = (X i a i δ A i | a i ∈ Q , A i ∈ F ( C ) ) = X j b j δ B j | b j ∈ Q , B j ∈ I ( C ) A filter A of a finite poset P can be written as the union of prime filters ∨ x .Hence, the vector space DF( P ) of rational-valued functions on P has two basesconsisting of δ ∨ x and δ ∧ x for each x ∈ P .In the case of a finite category C , we choose and fix objects x , · · · , x k suchthat P ( C ) = { [ x ] , · · · , [ x k ] } . The vector space DF( C ) of definable functionson C has two bases consisting of δ ∨ x i and δ ∧ x i for i = 1 , · · · , k . Definition 3.7.
A finite category C is called measurable if each filter and idealof C has Euler characteristic.For example, finite posets, acyclic categories, groups, and groupoids aremeasurable, since any full subcategory has Euler characteristic. Definition 3.8.
For a measurable category C , the Euler integration on filters is the linear map Z F C ( − ) dχ : DF( C ) −→ Q , which sends δ ∨ x to χ ( ∨ x ). Dually, the Euler integration on ideals is the linearmap Z I C ( − ) dχ : DF( C ) −→ Q , that sends δ ∧ x to χ ( ∧ x ). 6e should note that, in general, χ ( D ) = χ ( C ) − χ ( C \ D ) for a filter (anideal) D of C . This implies that Z F C f dχ = Z I C f dχ, for f = δ D = δ C − δ C \ D ∈ DF( C ). Remark 3.9.
We now have two kinds of integration via Euler characteristicsof categories. For a measurable category C , the duality between the filters andideals implies that C F = C op I and Z F C f dχ = Z I C op f dχ, for any f ∈ DF( C ) = DF( C op ). These two integrals are not equal, but haveduality with the opposite category. Henceforward, we shall only consider thecase of Euler integration on filters. We call it Euler integration for short, and R F C will be simply denoted by R C . Euler integration on ideals satisfies the dualproperties of that described below. Lemma 3.10.
Euler integration does not depend on the linear representationof the definable functions with respect to the incidence functions on the filters.That is, if a definable function f can be described as P ni =1 a i δ A i for some a i ∈ Q and A i ∈ F ( C ) , then we have Z C f dχ = n X i =1 a i χ ( A i ) . Proof.
It suffices to show that Z C f B dχ = n X i =1 a i χ ( A i ∩ B ) (1)for any filter B of C . We use induction on the cardinality of the set of objectsof B . When B is a nonempty minimal filter, it is the prime filter ∨ b generatedfrom some maximal object b . Any pair of objects in B are reflexible, and theclipping f B = f ( b ) δ ∨ b is constant on B . Then, the following equality holds: Z C f B dχ = f ( b ) χ ( ∨ b ) = X i ; b ∈ A i a i χ ( ∨ b ) = n X i =1 a i χ ( A i ∩ ( ∨ b )) , since A i ∩ ( ∨ b ) = ∨ b if b ∈ A i , and A i ∩ ( ∨ b ) = ∅ otherwise.Next, assume that we have the equality stated in (1) for the clipping on B of any definable function, when B ♯ ≤ n −
1. When B ♯ = n , choose a minimalobject x of B . If the filter A i contains the object x , then A i ∩ B = ∨ x . Theclipping function f B can be represented as f B = f ( x ) δ ∨ x + f B \ ( ∧ x ) − f ( x ) δ ( ∨ x ) ∩ ( B \ ( ∧ x )) . f ( x ) δ ∨ x = X i ; x ∈ A i a i δ ∨ x ,f B \ ( ∧ x ) = X i ; x A i a i δ A i ∩ B + X i ; x ∈ A i a i δ A i \ ( ∧ x ) , and f ( x ) δ ( ∨ x ) ∩ ( B \ ( ∧ x )) = X i ; x ∈ A i a i δ ( ∨ x ) ∩ ( B \ ( ∧ x )) = X i ; x ∈ A i a i δ A i \ ( ∧ x ) . By applying the Euler integration to f B , we obtain the following equality: Z C f B dχ = X i ; x ∈ A i a i χ ( ∨ x ) + X i ; x A i a i χ ( A i ∩ B )= X i ; x ∈ A i a i χ ( A i ∩ B ) + X i ; x A i a i χ ( A i ∩ B )= n X i =1 a i χ ( A i ∩ B ) . The result corresponds to the case in which B = C . Proposition 3.11.
Let f be a definable function on a measurable category C .If C has a terminal object x , then we have Z C f dχ = f ( x ) . Proof.
It is sufficient to prove that Z C f B dχ = f ( x ) (2)for any filter B of C . Note that every filter of a category includes terminalobjects, if they exist. We use induction on the cardinality of B to prove this.When B is a nonempty minimal filter, it is the prime filter ∨ x generated fromthe terminal object x . For a definable function f on C , we have Z C f B dχ = Z C f ( x ) δ ∨ x = f ( x ) χ ( ∨ x ) = f ( x ) , by Proposition 2.3. We assume that our desired equality, stated in (2), holdsfor any definable function if B ♯ ≤ n −
1. When B ♯ = n , we take a minimalobject b of B . For a definable function f on C , the clipping function f B can berepresented as f B = f B \ ( ∧ b ) + f ( b ) δ ∨ b − f ( b ) δ ( B \ ( ∧ b )) ∩ ( ∨ b ) . B \ ( ∧ b ), ∨ b , and ( B \ ( ∧ b )) ∩ ( ∨ b ) is a filter of C . The inductiveassumption leads to the following equality: Z C f B dχ = f ( x ) + f ( b ) − f ( b ) = f ( x ) . The result corresponds to the case in which B = C .Let f be a definable function on a measurable category C , and let B be ameasurable full subcategory of C . For simplicity, the Euler integration R B (cid:0) f | B (cid:1) dχ of the restriction f | B is denoted by R B f dχ . Theorem 3.12.
Let f be a definable function on a measurable category C , andlet the category C be the union A ∪ B of two full subcategories A and B of C .If both A and B are either filters or ideals, then we have Z C f dχ = Z A f dχ + Z B f dχ − Z A ∩ B f dχ. Proof.
We will focus on the case in which both A and B are ideals. It sufficesto show the case of f = δ D for a filter D of C . Each D ∩ X is an ideal of D and a filter of X , for X = A , B , and A ∩ B . Theorem 2.6 induces the followingequality: Z C δ D dχ = χ ( D )= χ ( D ∩ A ) + χ ( D ∩ B ) − χ ( D ∩ A ∩ B )= Z A δ D dχ + Z B δ D dχ − Z A ∩ B δ D dχ. The case in which A and B are filters can be shown similarly. Definition 3.13.
Let C and D be categories. A map F : C → D is called measurable when the inverse image F − ( − ) preserves filters and ideals.For example, the underlying map on objects of a functor is measurable. Lemma 3.14.
A measurable map F : C → D is definable.Proof. Suppose that two objects a and b of C are reflexible. The inverse image F − ( ∨ F ( b )) of the filter generated from F ( b ) in D is a filter in C . The object b belongs to this filter and so does a . This implies that F ( a ) belongs to ∨ F ( b ).Dually, the filter F − ( ∨ F ( a )) contains the object b , and F ( b ) belongs to ∨ F ( a ).It follows that F ( a ) and F ( b ) are reflexible.At the rest of this section, we will examine the notion of pushforwards andFubini theorem, introduced in Definition 2.7 and Theorem 2.8 of [BG09]. Theyused the inverse image of each point of Y as the integral range, in order to definethe pushforward of a definable map X → Y . However, we use the inverse imageof the ideal generated by each point as integral range.9 efinition 3.15. Let F : C → D be a measurable map between measurablecategories C and D . The pushforward of F is the homomorphism F ∗ : DF( C ) → DF( D ) defined by F ∗ f ( d ) = Z F − ( ∧ d ) f dχ for f ∈ DF( C ) and d ∈ ob( D ).We need to verify that the pushforward described above is well-defined. Lemma 3.16.
For a measurable map F : C → D and a definable function f ∈ DF( C ) , the pushforward F ∗ f is definable.Proof. Suppose that two objects a and b of D are reflexible. We then have ∧ a = ∧ b , and F ∗ f ( a ) = Z F − ( ∧ a ) f dχ = Z F − ( ∧ b ) f dχ = F ∗ f ( b ) . Proposition 3.17.
Let F : C → D and G : D → E be two measurable mapfor measurable categories C , D , and E . If D is a poset, the pushforward iscompatible with the composition, i.e., ( G ◦ F ) ∗ = G ∗ ◦ F ∗ . Proof.
The pushforward ( G ◦ F ) ∗ : DF( C ) → DF( E ) is given by( G ◦ F ) ∗ ( δ B )( e ) = Z ( G ◦ F ) − ( ∧ e ) δ B dχ = χ ( F − ( G − ( ∧ e )) ∩ B ) , for B ∈ F ( C ) and e ∈ ob( E ). We need to prove that ( G ◦ F ) ∗ ( δ B ) = G ∗ ( F ∗ ( δ B ))for any B ∈ F ( C ). It suffices to show that Z A F ∗ δ B dχ = χ ( F − ( A ) ∩ B ) (3)for any e ∈ ob( E ) and ideal A ⊂ G − ( ∧ e ). We again use induction on thecardinality of the ideal A . When A consists of a single (minimal) element d , Z { d } F ∗ δ B dχ = F ∗ δ B ( d ) = Z F − ( d ) δ B dχ = χ ( F − ( d ) ∩ B ) . Assume that equation (3) holds in the case of A ♯ ≦ n −
1. When A ♯ = n , wetake a maximal element a ∈ A . By Proposition 3.11, Theorem 2.6, and theinductive assumption, we have the following equality: Z A F ∗ δ B dχ = Z A \ ( ∨ a ) F ∗ δ B dχ + Z ∧ a F ∗ δ B dχ − Z ( A \ ( ∨ a )) ∩ ( ∧ a ) F ∗ δ B dχ = χ ( F − ( A \ ( ∨ a )) ∩ B ) + F ∗ δ B ( a ) − χ ( F − (( A \ ( ∨ a )) ∩ ( ∧ a )) ∩ B )= χ ( F − ( A \ ( ∨ a )) ∩ B ) + χ ( F − ( ∧ a ) ∩ B ) − χ ( F − (( A \ ( ∨ a )) ∩ ( ∧ a )) ∩ B )= χ ( F − ( A ) ∩ B ) . A = G − ( ∧ e ), we obtain the desired result: G ∗ ( F ∗ ( δ B ))( e ) = Z G − ( ∧ e ) F ∗ ( δ B ) = χ ( F − ( G − ( ∧ e )) ∩ B ) = ( G ◦ F ) ∗ ( δ B )( e ) . In the proposition above, assuming D to be a poset is essential to prove.For example, if D is a category consisting of two objects a and b , and paralleltwo morphisms from a to b . The classifying space of D is homotopy equivalentto a circle S , and the Euler characteristic χ ( D ) = 0. We have (1 D ) ∗ ( δ D )( b ) = χ ( D ) = 0, however,(1 D ) ∗ ((1 D ) ∗ ( δ D ))( b ) = Z D ( δ ∨ a + ( − δ ∨ b ) dχ = χ ( D ) − χ ( { b } ) = − . Corollary 3.18.
The pushforward yields a functor from the category of posetsto the category of Q -vector spaces, and it is given by P DF( P ) and F F ∗ .Proof. For the identity map id P : P → P , the pushforward F ∗ (id P ) = id DF( P ) ,since (id P ) ∗ ( f )( a ) = Z ∧ a f dχ = f ( a )for any f ∈ DF( P ) and a ∈ P , by Proposition 3.11. Proposition 3.17 completesthe proof. Theorem 3.19.
Let F : C → D be a measurable map from a measurablecategory C to a finite poset D . For any definable function f on C , the Eulerintegration over f coincides with the Euler integration over the pushforward of f : Z C f dχ = Z D F ∗ f dχ. Proof.
Let pt denote the terminal category consisting of a single object andthe identity morphism. Note that the pushforward of the unique map C → pt coincides with the Euler integration Z C ( − ) dχ : DF( C ) → DF( pt ) = Q . By applying Proposition 3.17 to the composition C F → D → pt , we obtain thedesired formula.The canonical functor π : C → P ( C ) is a measurable map for a measurablecategory C . Theorem 3.19 implies that Z C f dχ = Z P ( C ) π ∗ f dχ. Hence, the Euler integration of a definable function f on a measurable category C can be calculated from the function π ∗ f on the poset P ( C ). Note that,in general, the pushforward π ∗ f does not coincide with the induced map ˜ f introduced in Remark 3.3. 11 Application of discrete Euler integration tosensor network theory
This section presents an application of Euler integration over a function on aposet. It is based on the work of Baryshnikov and Ghrist on using topologicalEuler integration to enumerate targets in a network of sensors in a field. In[BG09], they proved that the cardinality of the targets lying on a field can beobtained from the topological Euler integral of the counting function.We consider a discrete analogue of the above. Assume that our network hasthe following properties: • Our network consists of a finite node that flows in only one direction (forexamples, transmission of electricity, streams of water or river, and acyclictraffic). Hence, we can regard a model of such a network as a finite poset( P, ≤ ). Nodes are elements of P , and lines are ordered pairs ( p, q ), denotedby p ≺ q , that do not contain r , with p < r < q in P . • It contains finitely many targets T (for example, broken points, specialspots, and errors or bugs) on the lines or nodes. More precisely, the targets T form a discrete subset of the Hasse diagram of P , where we regard thediagram as a one-dimensional simplicial complex. Furthermore, let everynode have a sensor, and let it count the targets lying below the node.Here, a target t ∈ T lying on a line or a node p (cid:22) q is said to be belowa node r if q ≤ r in P . We denote this as t ≤ r . The sensors return thecounting function h : P → N ∪ { } , given by the cardinality of targets lying below itself: h ( p ) = { t ∈ T | t ≤ p } ♯ . Theorem 4.1.
Given the counting function h : P → N ∪ { } for a collection oftargets T in a network P , we have T ♯ = Z P hdχ. Proof.
Let each target t ∈ T lie on a line or a node p t (cid:22) q t . Then, the followingequality holds: Z P hdχ = Z P X t ∈ T δ ∨ q t ! dχ = X t ∈ T χ ( ∨ q t ) = T ♯ . xample 4.2. The following diagram represents a counting function h for tar-gets T on a network. 1 • • ❄❄❄❄❄❄❄ • • ❄❄❄❄❄❄❄ • • ❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧⑧ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ • ⑧⑧⑧⑧⑧⑧⑧ • ⑧⑧⑧⑧⑧⑧⑧ • ⑧⑧⑧⑧⑧⑧⑧ • ⑧⑧⑧⑧⑧⑧⑧❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Here, we describe the underlying poset P as the Hasse diagram. Let us calculatethe Euler integration of h to enumerate the targets. The counting function is aposet map, hence h − ( ∨ i ) = { p ∈ P | h ( p ) ≥ i } is a filter of P for each i ∈ N .Then, T ♯ = Z P hdχ = Z P ∞ X i =1 δ h − ( ∨ i ) ! dχ = ∞ X i =1 χ ( h − ( ∨ i )) . The telescope sum above suggests that the Euler integration of h can be obtainedfrom the Euler characteristic of each level subposet h − ( ∨ i ). i = 1 i = 2 •• ❄❄❄❄❄❄❄ •• ❄❄❄❄❄❄❄ ◦• ⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧⑧ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ◦• ⑧⑧⑧⑧⑧⑧⑧ • ⑧⑧⑧⑧⑧⑧⑧ • ⑧⑧⑧⑧⑧⑧⑧❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ◦◦ ◦• ◦• ◦• • ⑧⑧⑧⑧⑧⑧⑧ • ⑧⑧⑧⑧⑧⑧⑧❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ i = 3 i = 4 ◦◦ ◦◦ ◦• ◦• • • ⑧⑧⑧⑧⑧⑧⑧❄❄❄❄❄❄❄ ◦◦ ◦◦ ◦◦ ◦◦ • ◦ The classifying space B ( h − ( ∨ S , and B ( h − ( ∨ S . Proposition 2.4 leads to thefollowing result: T ♯ = χ ( h − ( ∨ χ ( h − ( ∨ χ ( h − ( ∨ χ ( h − ( ∨ . Indeed, this counting function was given by the following five targets de-13cribed as “+”. +1 • • ❄❄❄❄❄❄❄ +1 • • ❄❄❄❄❄❄❄ • • + ❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧⑧ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ • ⑧⑧⑧⑧⑧⑧⑧ • + ⑧⑧⑧ ⑧⑧⑧ • + ⑧⑧⑧ ⑧⑧⑧ • ⑧⑧⑧⑧⑧⑧⑧❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ References [BD01] John C. Baez and James Dolan. From finite sets to Feynman di-agrams.
Mathematics unlimited–2001 and beyond , 29–50, Springer,Berlin, 2001.[BG09] Yuliy Baryshnikov and Robert Ghrist. Target enumeration via Eulercharacteristic integrals.
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