On the Categorification of the Möbius Function
Rafael Dı́az
Abstract
In this notes we study several categorical generalizations of the Möbius function and discuss the relation between the various approaches. We emphasize the
topological and geometric meaning of the our constructions.
1
Introduction
The Möbius function µ : N+ −→ Z is the map such that µ(1) = 1, µ(n) = 0 if n has
repeated prime factors, and µ(n) = (−1)k if n is the product of k distinct prime numbers. It was introduced by Möbius in 1832 and it has been since an important tool in
number theory, complex analysis, and combinatorics. Our goal in this notes is to gently
introduced several generalizations of the Möbius function originating from the viewpoint
of category theory, which have emerged thanks to the contribution of various authors.
We begin with a review of the classical Möbius theory where the main character is
the set of the positive natural numbers N+ , which for the purposes of the theory can
be regarded either as a poset or as a monoid. We adopt first the poset viewpoint. It
leads to the construction of a Möbius theory for locally finite posets (Rota), locally finite
directed graphs, locally finite categories (Haigh, Leinster), and essentially locally finite
categories. Then we adopt the monoid viewpoint which leads to the development of a
Möbius theory for finite decomposition monoids, Möbius categories (Leroux, Haigh), and
essentially finite decomposition categories.
Having Möbius theories for a variety of mathematical objects, a couple of natural
question arises. How the various theories relate to each other? We will address this
question as we develop the various theories along the notes. As a rule the relations are
quite subtle a non necessarily fully functorial. For example, one may wonder how can
one characterize the precise relation between the category of posets and the category of
monoids.
1
The second question asks for the prominent common features of a Möbius theory. In
the examples we have found the following structural features:
• The theory applies to some category C of objects and depends on a ring R. For
simplicity we mainly consider in this notes the characteristic zero case. In a fully
categorified approach one may even want R be a ring-like category.
• There is a construction that associates a R-algebra Ac to each object c ∈ C. The
algebra Ac often called the incidence or convolution algebra of c. The construction
of Ac need not be functorial. However, it is functorial under isomorphism, i.e.
we have a functor A : Cg −→ R-alg from the underlying groupoid of C to the
category of R-algebras. Often the category C comes equipped with a notion of
weak equivalences which are a suitable family of morphisms between objects of C.
The construction may fail to be functorial under equivalences even if is functorial
under isomorphism.
• There is a reasonable simple criteria to characterize the units (invertible elements)
of the R-algebra Ac . Moreover, there is a terminating recursive procedure that
finds the inverse u−1 for each unit u ∈ Ac . There are some canonical unit elements
say ξc ∈ Ac , perhaps with a rather straightforward definition, and such that t theirs
inverses µc are unexpectedly useful having interesting topological and geometrical
meaning.
• Often each algebra Ac comes equipped with a natural R-module, leading to the
Möbius inversion formula: n = mξc if and only if m = nµc .
We proceed to develop our main examples of Möbius theories, providing along the
notes references to some of main contributions in the field, and discussing the relation
between the various theories. We focus on the topological approach to understand the
meaning of the Möbius functions, in particular, the Euler characteristics for simplicial
groupoids will be useful for us. We emphasize that there is not a unique categorical
generalization for the Möbius functions, but rather several approaches, each with its on
advantages and applications. Although we explore here various routes, we do not mean
to be exhaustive. That would be a task exceeding the limits of these short notes. For
example, we do not cover the advances developed by Cartier and Foata [6], Dür [9], Fiore,
Lück and Sauer [10].
This notes were prepared for the CIMPA school ”Modern Methods in Combinatorics”,
San Luis, Argentina 2013. For the reader convenience we include a few exercises.
2
2
Classical Möbius Theory
In this section we review the basic elements of the classical Möbius theory. Let N+ be
the set of positive natural numbers and R be a commutative ring with identity.
Definition 1. The Möbius function is the map µ : N+ −→ R given by

if n = 1,
 1
0
if n has repeated prime factors,
µ(n) =

(−1)k if n = p1 ....pk , with pj distinct prime numbers.
Definition 2. The Dirichlet product ⋆ on [N+ , R], the set of maps from N+ to R, is
given on f, g ∈ [N+ , R] by:
∑
∑
∑
f ⋆ g(n) =
f (d)g(n/d) =
f (n/d)g(d) =
f (c)g(d).
d|n
cd=n
d|n
The unit for ⋆ is the map 1 : N+ −→ Z given by 1(1) = 1 and 1(n) = 0 for n ≥ 2.
The product ⋆ turns [N+ , R] into a commutative R-algebra. Next result is known as
the Möbius inversion formula.
Proposition 3. (Möbius Inversion Formula)
Let ξ ∈ [N+ , R] be the map constantly equal to 1. The Möbius function µ is the ⋆-inverse
of ξ. Thus for f ∈ [N+ , R], we have that g = f ⋆ ξ if and only if f = g ⋆ µ. Equivalently,
∑
∑
g(n) =
f (d)
if and only if
f (n) =
µ(n/d)g(d).
d|n
d|n
Proof. We show that ξ ⋆ µ = 1. As ξ ⋆ µ(1) = µ(1) = 1, we just have to check that
ξ ⋆ µ(n) = 0 for n ≥ 2. Given n = pa11 ...pakk , with pj distinct prime numbers, we have that
(
)
k
∏
∑
∑
∑
ξ ⋆ µ(pa11 ...pakk ) =
µ(d) = 1 +
µ
pi =
a
l=1 A⊆[k], |A|=l
a
d|p1 1 ...pkk
1+
k
∑
∑
l=1 A⊆[k], |A|=l
i∈A
( )
k
= (1 − 1)k = 0.
(−1) =
(−1)
l
l=0
l
k
∑
l
Exercise 4. Show that f ∈ [N+ , R] is a ⋆-unit if and only if f (1) is a unit in R.
Exercise 5. A map f ∈ [N+ , R] is multiplicative if f (ab) = f (a)f (b) for a, b coprime.
Show that 1, ξ and µ are multiplicative functions. Show that f ⋆g is multiplicative if f and
g are multiplicative. Show that the inverse of a multiplicative function is multiplicative.
3
Rooted in number theory the Möbius function also plays an important role in complex
analysis trough the theory of Dirichlet series.
Definition 6. A Dirichlet series with coefficients in R is a formal power series
∞
∑
an
n=1
ns
,
where an ∈ R.
We let DR be the R-algebra of Dirichlet series with the R-linear product given by
1 1
1
=
.
n s ms
(nm)s
For our present purposes we treat s as a formal variable. In the applications one
usually works over the complex numbers (R = C), the variable s is C-valued, and convergency issues are of the greatest importance. The product of convergent Dirichlet series
is just the usual product of functions.
The Dirichlet algebra DR is isomorphic to the R-algebra R << x1 , ..., xn , ... >> of
formal R-linear combinations of variables xn , with the product given by xn xm = xnm ,
via the map sending xn to n1s .
Note that n = pa11 ...pakk implies that xn = xap11 ...xapkk . Therefore DR is isomorphic to
the algebra R[[x2 , x3 , x5 , ...]] of formal power series with coefficients in R in the variables
xp with p a prime number.
Proposition 7. The map D : [N+ , R] −→ DR sending f ∈ [N+ , R] to its Dirichlet series
Df (s) =
∞
∑
f (n)
n=1
ns
is a ring isomorphism.
For example, the Dirichlet series associated with ξ is the Riemann zeta function
ζ(s) =
∞
∑
1
.
s
n
n=1
Thus we have that ζ −1 (s) =
∞
∑
µ(n)
n=1
ns
.
Exercise 8. Describe explicitly the product of Dirichlet series and prove Proposition 7.
Exercise 9. For p ≥ 2 a primer number, let fp ∈ [N+ , Z] be such that fp (n) = 1 if n = pl
for some l ∈ N, and fp (n) = 0 otherwise. Find fp−1 , Dfp and Dfp .
Exercise 10. Let η ∈ [N+ , Z] be such that η(1) = 1, and for n ≥ 2 we set η(p) = −1 for
p prime, and η(n) = 0 otherwise. Find η −1 , Dη and Dη−1 .
4
We close this section with a couple of facts (kind of beyond the main topic of these
notes) that show the importance of the Möbius function in number theory and complex
analysis. The first fact is a reformulation in terms of the Möbius function of a main
theorem in number theory on the asymptotic behaviour of prime numbers.
Theorem 11. (Prime Number Theorem and the Möbius Function)
Let π : N+ −→ N be the map such that π(n) is the number of prime numbers less than
or equal to n. We have that:
π(n)ln(n)
1∑
lim
= 1, or equivalently, lim
µ(k) = 0.
n→∞
n→∞ n
n
k≤n
The second fact relates the Möbius function to the Riemann Hypothesis. The Riemann zeta function ζ(s) can be analytically extended to the complex plane with a pole
at s = 1. The ζ function has zeroes at s = −2, −4, −6, ..., the so called trivial zeroes.
Any other zero is called nontrivial, and the Riemann Hypothesis claims that all of them
have real part equal to 21 .
Theorem 12. (Riemann Hypothesis and the Möbius Function)
The real part of the non-trivial zeroes of the Riemann zeta function is
for any ϵ > 0 there exists M > 0 such that
∑
1
µ(k) < M n 2 +ϵ .
1
2
if and only if
k≤n
3
Locally Finite Posets
In the 60’s a comprehensive Möbius theory for locally finite partially ordered sets (posets)
was developed by Gian-Carlo Rota and his collaborators, among them Crapo, Stanley,
Schmitt. Posets are actually a particular kind of categories, so this development may
already be regarded as a categorification of the classical Möbius theory. Almost all the
results in this section can be found in the first chapter of the book [21], where the reader
will find a nice combination of original and review papers. The newer results in this
section are the Leinster’s characterization of the relation between Möbius and matrix inversion for finite posets, and a discrete analogue for the Gauss-Bonnet theorem for finite
posets.
A partial order on a set X is a reflexive, antisymmetric, and transitive relation on
X. A poset is a pair (X, ≤) where X is a set and ≤ is a partial order on X. Morphism
between posets are order preserving maps. For x ≤ y in X, the interval [x, y] is the
subset of X given by [x, y] = {z ∈ X | x ≤ z ≤ y}. Let IX = {[x, y] | x ≤ y} be the set
of intervals of X.
5
Definition 13. A poset is locally finite if its intervals are finite sets.
Lemma 14. Let X be a locally finite poset. The free R-module < IX > generated by
the intervals IX of X, together with the R-linear maps ∆ :< IX >−→< IX > ⊗ < IX >
and ϵ :< IX >−→ R given, respectively, by
{
∑
1 if x = y,
[x, y] ⊗ [y, z] and ϵ([x, y]) =
∆([x, z]) =
0 if x ̸= y,
x≤y≤z
is a R-coalgebra.
Proof. The counit property (ϵ ⊗ 1)∆([x, y]) = [x, y] is clear. Coassociativity follows
from the identities
∑
(∆ ⊗ 1)∆[x, w] =
[x, y] ⊗ [y, z] ⊗ [z, w] = (1 ⊗ ∆)∆[x, w].
x≤y≤z≤z
Recall that if C is a R-coalgebra, then its dual C ∗ = HomR (C, R) is an R-algebra
with the dual product ⋆ : C ∗ ⊗ C ∗ −→ C ∗ given on f, g ∈ C ∗ by
f ⋆ g(c) = (f ⊗ g)∆(c).
The unit ϵ ∈ C ∗ is the counit for C.
IX
The dual algebra < IX >∗ may be identified with the algebra [IX , R] of maps from
to R with the product given by
∑
(f ⋆ g)[x, z] =
f [x, y] ⊗ g[y, z].
x≤y≤z
The R-algebra ([IX , R], ⋆) is called the incidence algebra of the poset X.
Theorem 15. Let (X, ≤) be a locally finite poset. A map f ∈ [IX , R] is invertible in
the incidence algebra if and only if f [x, x] is a unit for all x ∈ X. In particular, the map
ξ ∈ [IX , R] constantly equal to 1 is invertible; its inverse µ ∈ [IX , R] is called the Möbius
function of the poset (X, ≤).
Proof. If f ⋆ g = ϵ, then f [x, x]g[x, x] = 1 and thus necessarily f [x, x] is a unit for all
x ∈ X. Conversely, assume that f [x, x] is a unit for x ∈ X, then its inverse g, if it exists,
must satisfy g[x, x] = f [x, x]−1 for x ∈ X, and for x < z we must have that
∑
f [x, x]g[x, z] +
f [x, y]g[y, z] = 0,
x<y≤z
6
or equivalently
g[x, z] = −
∑ f [x, y]g[y, z]
.
f
[x,
x]
x<y≤z
The equations above give a terminating recursive definition for g since X is locally finite.
Solving the recursion we get for x < y ∈ X that
g[x, y] =
∑
∑
(−1)n
n≥1 x=x0 <x1 <...<xn =y
f [x0 , x1 ].................f [xn−1 , xn ]
f [x0 , x0 ]f [x1 , x1 ].......f [xn−1 , xn−1 ]f [xn , xn ]
Corollary 16. The Möbius function µ ∈ [IX , R] of a locally finite poset (X, ≤) is given
by µ[x, x] = 1 for x ∈ X, and for x < y ∈ X we have that
∑
µ[x, y] =
(−1)n |{x = x0 < x1 < ..... < xn−1 < xn = y}|.
n≥1
We need a few notions in combinatorial algebraic topology, the book [14] provides a
nice introduction to the subject and discusses homology theory in Chapter 3.
To a finite poset (X, ≤) we associate the simplicial complex CX of chains in X, with
vertex set X, and such that a subset c of X is in CX if an only if the restriction of ≤
to c is a linear order. For n ≥ −1, let Cn X be the subset of CX consisting of chains of
cardinality n + 1. Note that the empty set is the unique element of C−1 X. The geometric
realization of CX is the topological space
∑
|CX| = {a ∈ [X, [0, 1]] | support(a) ∈ C and
a(x) = 1},
x∈X
where [0, 1] is the unit interval in R, the space [X, [0, 1]] =
product topology, and |CX| is given the subspace topology.
∏
x∈X [0, 1]
is given the
e n (|CX|), for n ≥ −1, of |CX| coincide with the
The reduced homology groups H
homology groups of the differential complex (< CX >, d) such that
⊕
< CX > =
< Cn X >
n≥−1
where < Cn X > is the Z-module generated by Cn X. The differential map
d :< Cn X >−→< Cn−1 X >
7
is given on generators by
d{x0 < .... < xn } =
n
∑
(−1)i {x0 < ... < xbi < ... < xn }.
i=0
Therefore the reduced Euler characteristic of |CX| is given by
∑
∑
e n (|CX|).
χ
e|CX| =
(−1)n |Cn X| =
(−1)n rankH
n≥−1
n≥0
The Euler characteristic of |CX| is defined in terms of the homology groups of the
differential complex (< C≥0 X >, d) where
⊕
< C≥0 X > =
< Cn X >
by
n≥0
χ|CX| =
∑
(−1)k |Cn X| =
n≥0
∑
(−1)n rankHn (|CX|).
n≥0
The following topological results are due to P. Hall.
Theorem 17. (Homological Interpretation)
The Möbius function µ ∈ [IX , R] of a locally finite poset (X, ≤) is given for x < y ∈ X
by
∑
∑
e n (|C(x, y)|),
µ[x, y] = χ
e|C(x, y)| =
(−1)n |Cn (x, y)| =
(−1)n rankH
n≥−1
n≥0
where (x, y) is the interval {z | x < z < y} ⊆ X with the induced order.
Proof.
µ[x, y] =
∑
(−1)n |{x = x0 < x1 < ... < xn = y}| =
n≥1
∑
(−1)n−2 |Cn−2 (x, y)| = χ
e|C(x, y)|.
n≥1
Let X be a finite poset and X be the poset obtained from X by adjoining a minimum
b
0 and a maximum b
1 to X.
Corollary 18. Let µX ∈ [IX , R] be the Möbius function of X, then
∑
e k (|CX|).
µX [b
0, b
1] = χ
e|CX| =
(−1)k rankH
k≥0
8
Proof. Follows from Theorem 17 since (b
0, b
1) = X.
Proposition 19. The Euler characteristic χ|CX| of a finite poset (X, ≤) is given by
∑
χ|CX| =
µ(x, y).
x,y∈X
Proof. The result follows from the identity |C0 X| = |X| =
that for k ≥ 1 we have that
⊔
Ck X =
Ck−2 (x, y).
∑
x∈X
µ(x, x) and the fact
x<y
Thus
χ|CX| =
∑
x∈X
∑
(−1)k |Ck X| = |C0 X| +
k≥0
(
k≥1
x<y
∑
x∈X
µ(x, x) +
∑
(−1)k |Ck X| =
k≥1
)
∑ ∑
µ(x, x)+
(−1)k |Ck−2 (x, y)|
∑
=
∑
(
)
∑ ∑
µ(x, x)+
(−1)k−2 |Ck−2 (x, y)| =
x∈X
µ(x, y) =
x<y
∑
x<y
k≥1
µ(x, y).
x,y∈X
For a ∈ X, set X≥a = {x ∈ X | x ≥ a}. There is a structure of right [IX , R]-module
on [X≥a , R] via the R-bilinear map
⋆ : [X≥a , R] × [IX , R] −→ [X≥a , R]
sending a pair (f, g) to the map f ⋆ g ∈ [X≥a , R] given by
∑
f ⋆ g(y) =
f (x)g[x, y].
a≤x≤y
Theorem 20. (Möbius Inversion for Locally Finite Posets)
Given a ∈ X and f, g ∈ [X≥a , R], we have that f = g ⋆ ξ if and only if g = f ⋆ µ, or
equivalently
∑
∑
f (y) =
g(x) if and only if g(y) =
f (x)µ[x, y].
a≤x≤y
a≤x≤y
Remark 21. If X is a finite set, then [X, R] is a right [IX , R]-module and we have
f, g ∈ [X, R], that
∑
∑
f (y) =
g(x) if and only if g(y) =
f (x)µ[x, y].
x≤y
x≤y
9
Fix a finite poset (X, ≤). The set of maps [X × X, R] is naturally an algebra (square
matrices indexed by X) with the product
∑
f g(x, z) =
f (x, y)g(y, z).
y∈X
There is a natural embedding [IX , R] −→ [X × X, R] of algebras sending f ∈ [IX , R] to
the map f : X × X −→ R given by
{
f [x, y] if x ≤ y,
f (x, y) =
0
otherwise.
Thus we may regard [IX , R] as a subalgebra of [X × X, R].
A map f ∈ [X × X, R] is transitive if for n ≥ 0, and x0 , ..., xn ∈ X we have that
f (x0 , x1 ) = 0 implies that f (x0 , x1 )f (x1 , x2 )...f (xn−1 , xn ) = 0.
Leinster has shown in [16, 17] the following result.
Lemma 22. Let f ∈ [X × X, R] be invertible and transitive, then f (x, y) = 0 implies
that f −1 (x, y) = 0.
Thus the following result holds.
Theorem 23. Let f ∈ [IX , R] ⊆ [X × X, R] be transitive, then f is invertible in [IX , R]
if and only if f is invertible in [X × X, R].
So far we have regarded IX as a set. Note however that IX may be naturally regarded
as a full subcategory of the category of finite posets and increasing maps. Thus let IX be
the set of isomorphism classes of intervals in X. The coproduct and counit on < IX >
descent to a coproduct and counit on < IX >, giving an algebra structure ⋆ to
< IX >∗ = [IX , R],
the set of maps from IX to R, or equivalently, the set of maps from IX to R invariant
under isomorphisms. We call ([IX , R], ⋆) the reduced incidence algebra of (X, ≤).
Example 24. Let (N+ , |) be the poset of positive natural numbers with the order n|m
if and only if n divides m. The reduced incidence algebra ([I, R], ⋆) is isomorphic to the
∑
. Indeed, since
Dirichlet algebra DR via the map sending f ∈ [I, R] to Df = n∈N+ f [1,n]
ns
[d, n] ≃ [1, d/n] whenever d|n, we have for f, g ∈ [I, R] that


∑ f ⋆ g[1, n]
∑ ∑
1

Df ⋆g =
=
f [1, d]g[d, n] s =
s
n
n
n∈N
n∈N
+
+
10
d|n
∑
n∈N+


∑
1

f [1, d]g[1, n/d] s = Df Dg .
n
d|n
Example 25. The reduced incidence algebra of the poset (N, ≤) is isomorphic to R[[x]]
∑
n
via the map sending f ∈ [I, R] to fb = ∞
n=0 f [0, n]x . Indeed, since [k, n] ≃ [0, n − k] for
k ≤ n, we have for f, g ∈ [I, R] that:
(
)
∞
∑
∑
n
f[
⋆ g = sum∞
f [0, k]g[0, n − k] xn = fbgb.
n=0 (f ⋆ g)[0, n]x =
n=0
k≤n
Example 26. The reduced incidence algebra of the poset (Pf N, ⊆) of finite subsets of N
ordered by inclusion is isomorphic to the divided powers algebra R << x0 , x1 , ..., xn!n , ... >>
(
) xn+m
m
b = ∑∞ f [∅, [n]] xn . In(with xn!n xm!
= n+m
)
via
the
map
sending
f
∈
[I,
R]
to
f
n=0
i
(n+m)!
n!
deed, since [a, b] ≃ [∅, b \ a], we have for f, g ∈ [I, R] that:


∞
∞
∑
∑
∑
xn

f [∅, a]g[∅, [n] \ a]
f[
⋆g =
(f ⋆ g)[∅, [n]]xn =
=
n!
n=0
n=0
a⊆[n]
( n ( )
)
∞
∑
∑ n
xn
f [∅, [k]]g[∅, [n − k]]
= fbgb.
k
n!
n=0
k=0
Next we show that the Möbius function admits a Hopf theoretical interpretation.
This connection has been developed by Shmitt in a remarkable series of papers.
Theorem 27. For a locally finite poset (X, ≤) let R[[x[a,b] ]] be the R-algebra of formal
power series in the variables x[a,b] with a < b ∈ X, i.e. one variable for each interval in
X with different endpoints. The structural maps given below turn R[[x[a,b] ]] into a Hopf
algebra. Moreover we have for a < b ∈ X that µ[a, b] = S(x[a,b] )(1).
Proof. The counit, coproduct, and antipode are given, respectively, on generators by
ϵ(1) = 1
∆(1) = 1 ⊗ 1
S(1) = 1
and
and
and
ϵ(x[a,b] ) = 0,
∆(x[a,b] ) = 1 ⊗ x[a,b] +
S(x[a,b] ) =
∑
∑
∑
n≥1 {a=a0 <a1 <...<an =b}
11
x[a,c] ⊗ x[c,b] + x[a,b] ⊗ 1,
a<c<b
(−1)n x[a0 ,a1 ] ....x[an−1 ,an ] .
Corollary 28. For a locally finite poset (X, ≤) let R[[x[a,b] ]] be the R-algebra of formal
power series in the variables x[a,b] with a < b ∈ X and [a, b] ∈ I, i.e. one variable for each
isomorphism class of intervals with different endpoints in X. The structural maps from
Theorem 27 induce structural maps on R[[x[a,b] ]] turning it into a Hopf algebra. Moreover
for a < b ∈ X and [a, b] ∈ I we have that µ[a, b] = S(x[a,b] )(1).
Next we state and prove a discrete analogue of the Gauss-Bonnet theorem that applies for finite posets and show how it relates to the Möbius function. Recall that the
Gauss-Bonnet theorem for smooth manifolds describe the Euler characteristic as an integral of a top differential form. Related results for graphs have been develop by Knill [?].
We have already defined the space of linear k-chains on a finite poset (X, ≤) as the
free Z-module < Ck X > generated by the set Ck X of linearly subsets of X of length
k + 1. The discrete analogue Ωk X of the differential forms of degree k is simply the dual
module < Ck X >∗ , or equivalently, the Z-module [Ck X, Z]. We denote by
∫
ω
c
the natural pairing between c ∈< Ck X > and ω ∈ Ωk X. We use the same notation for
the trivially extended pairing
∫
:< C≥0 X > ×ΩX −→ Z, where
< CX > =
⊕
< Ck X >
and
ΩX =
⊕
Ωk X.
k≥0
k≥0
Let M be the set of all maximal linearly ordered subsets of X. The fundamental class
of X is the linear chain [X] ∈< CX > given by:
∑
[X] =
m.
m∈M
For c ∈ CX, we set M(c) = {m ∈ M | c ⊆ m}. The Euler class eX ∈ ΩX of X is such
that e(c) = 0 for c ∈
/ M, and for m ∈ M we have that
eX (m) =
∑ (−1)|c|+1
.
|M(c)|
∅̸=c⊆m
Similarly, the reduced Euler class eeX ∈ ΩX vanishes on non-maximal linear chains, and
is given on a maximal chain m ∈ M by
eeX (m) =
∑ (−1)|c|+1
c⊆m
12
|M(c)|
.
Theorem 29. (Gauss-Bonnet for Finite Posets)
Let (X, ≤) be a finite poset and |CX| its geometric realization. We have that
∫
∫
χ|CX| =
eX ,
and
χ
e|CX| =
eeX .
[X]
[X]
Proof. We show the latter identity, the proof of the former being analogous:
χ
e|CX| =
∑
(−1)k |Ck X| =
k≥−1
∑
∑
(−1)|c|+1 =
c∈CX
∑ (−1)|c|+1
=
|M(c)|
c∈CX m∈M(c)
∫
∑
∑ ∑ (−1)|c|+1
=
eeX (m) =
eeX .
|M(c)|
[X]
m∈M
m∈M c⊆m
Corollary 30. Let (X, ≤) be a poset and x < y ∈ X, then
∫
µ(x, y) = χ
e|C(x, y)| =
ee(x,y) ,
[(x,y)]
where (x, y) = {z | x < z < y}.
4
Locally Finite Directed Graphs
A directed graph is a pair (X, E) of sets together with a map (s, t) : E −→ X × X. For
x, y ∈ X, we set
E(x, y) = {e ∈ E | (s, t)(e) = (x, y)}.
A directed graph (X, E) is reflexive if E(x, x) ̸= ∅ for all x ∈ X.
A walk γ of length l(γ) = n ∈ N+ in (X, E) is a sequence γ = (γ1 , γ2 , ..., γn ) ∈ E n
such that t(γi ) = s(γi+1 ) for i ∈ [n − 1], and s(γi ) ̸= t(γi ) for i ∈ [n]. We say that the
walk γ begins at s(γ1 ) and ends at t(γn ).
For x, y ∈ X, we let W (x, y) (Wk (x, y)) be the set of walks from x to y (of length k.)
A circuit γ of length n ∈ N≥2 in (X, E) is a walk of length n such that t(γn ) = s(γ1 )
and t(γi ) ̸= t(γj ) for i ̸= j.
Definition 31. A graph (X, E) is locally finite if E(x, x) and W (x, y) are finite sets for
all x, y ∈ X.
13
Note that a locally finite directed graph has no circuits. Moreover, a finite directed
graph is locally finite if and only if it has not circuits.
Given a locally finite reflexive directed graph (X, E) consider the relation ≤ on X
such that: x ≤ y if and only if x = y or there is walk in (X, E) from x to y. Since the
concatenation of walks is a walk, and (X, E) has no circuits, the pair (X, ≤) is a locally
finite poset.
Thus for any locally finite reflexive directed graph (X, E) we have the incidence
algebra [I(X,≤) , R]. The adjacency map ξ ∈ [X × X, R] of (X, E) is given by
ξ(x, y) = |E(x, y)|.
Clearly, we may regard ξ as an element of the incidence algebra [I(X,≤) , R]. Note that in
general ξ(X,E) ̸= ξ(X,≤) .
Theorem 32. (Möbius Function for Locally Finite Directed Graphs)
Let (X, E) be a locally finite reflexive directed graph. The adjacency map ξ of (X, E) is
invertible in [I(X,≤) , R]; its inverse µ, called the Möbius function of (X, E), is such that
1
µ(x, x) = |E(x,x)|
and for x ̸= y ∈ X we have that
µ(x, y) =
∑
γ∈W (x,y)
(−1)l(γ)
.
|E(s(γ1 ), s(γ1 ))||E(t(γ1 ), t(γ1 )|......|E(t(γl(n) ), t(γl(n) )|
Proof. Follows directly from Theorem 15
Corollary 33. (Möbius Inversion for Locally Finite Directed Graphs)
Fix a ∈ X. For f, g ∈ [X≥a , R] we have that
g(y) =
∑
f (se)
if and only if
f (y) =
∑
g(x)µ(x, y).
a≤x≤y
e∈E, a≤se, te=y
Let (X, ≤) be a locally finite poset and consider the cover relation ≼ on X given by:
x ≼ y if x ≤ y and there is no z ∈ X such that x < z < y.
Consider the function η ∈ [I(X,≤) , R] given by η(x, x) = 1, η(x, y) = −1 if x ≺ y, and
η(x, y) = 0 otherwise. It follows from Theorem 51 that η is invertible and its inverse η −1
is such that
η −1 (x, x) = 1
and
η −1 (x, y) = |M[x, y]|,
14
where M[x, y] is the set of maximal totally ordered subsets of the interval [x, y] ⊆ X.
Fix a ∈ X, the finite difference operator
∆ : [X≥a , R] −→ [X≥a , R]
is given for f ∈ [X, R] and y ∈ X≥a by
∑
∆f (y) = f (y) −
f (x) = f ⋆ η(y).
a≤x≺y
The Möbius inversion formula tell us that
∆g = f
if and only if g(y) =
∑
|M[x, y]|f (x).
a≤x≤y
5
Locally Finite Categories
In this section we consider the coarse Möbius theory for categories (although we use
a different terminology) developed by Leinster in [16, 17]. This sort of Möbius theory
depends only on the underlying graph of the category.
Given a category C we let C0 be the collection of its objects. Abusing notation we
usually write x ∈ C instead of x ∈ C0 . Let C1 be the collection of morphisms in C,
and C(x, y) be the set of morphisms from x to y in C. The source and target maps
(s, t) : C1 −→ C0 × C0 together with the identity map 1 : C0 −→ C1 give C the structure
of a reflexive directed graph. Of course, in a category we have in addition the composition
maps
◦ : C(x, y) × C(y, z) −→ C(x, z)
sending a pair of morphisms (f, g) such that to its composition gf. Composition of
morphisms is associative and unital in the sense that
h(gf ) = (hg)f
and 1y f = f = f 1x ,
for f, g, h ∈ C(x, y) × C(y, z) × C(y, z).
f
The notation x −→ y means that f ∈ C(x, y). For x, y ∈ C we set
f
g
[x, y] = {z | there is a diagram x −→ z −→ y in C }.
Definition 34. A category C is locally finite if the following conditions hold:
• C(x, y) is a finite set for all x, y ∈ C.
15
• If there is a diagram x−→y−→x, then x = y.
• [x, y] is a finite set for x, y ∈ C.
Remark 35. Locally finite categories are skeletal.
Example 36. A finite category (i.e. a category with a finite set of morphisms) is locally
finite if and only if in any diagram x−→y−→x, we have that x = y.
Example 37. Let C be a category with C(x, y) finite for x, y ∈ C, let (X, ≤) be a locally
finite poset, and let F : X −→ C be a functor. The category XF with objects X and
morphisms given by XF (x, y) = C(F (x), F (y)) is locally finite.
Proposition 38. A category is locally finite if and only if its graph is locally finite.
Proof. Let C be a locally finite category. Then C(x, x) is a finite set for all x ∈ C. For
x ̸= y ∈ C, consider the set W (x, y) of walks from x to y. The objects in a walk
x−→x1 −→..... −→ xn −→ y
are all different because in any configuration of the form
x−→x1 −→..... −→ xk −→ x
we must have that x = x1 = ... = xk . Notice that if an object z ∈ C appears in a walk
from x to y, then z ∈ [x, y]. Since [x, y] is a finite set, then necessarily W (x, y) is also
finite. By the same argument we have that W (x, x) = ∅.
Conversely, assume that the graph of C is locally finite. The sets C(x, x) are finite
by definition. For x ̸= y, the sets C(x, y) and [x, y] are finite, since there is an injective
maps C(x, y) −→ W (x, y) and a surjective map
{x} ⊔ W2 (x, y) ⊔ {y} −→ [x, y].
Recall that a locally finite graph has no circuits, thus in any configuration x−→y−→x
we must have that x = y. In particular we get that [x, x] = {x}.
Lemma 39. Let C be a category with C(x, y) finite for x, y ∈ C. Let ≤ be the relation
on C0 given by x ≤ y if and only if there is a morphism x −→ y . Then C is a locally
finite category if and only if x ≤ y defines a locally finite partial order on C0 .
Proof. Immediate.
16
Thus for any locally finite category C we have the partially order set (C0 , ≤) and the
corresponding incidence algebra ([I(C0 ,≤) , R], ⋆). The adjacency map ξ : C0 × C0 −→ R
of C given by ξ(x, y) = |C(x, y)| lies naturally [I(C0 ,≤) , R].
Theorem 40. (Möbius Function for Locally Finite Categories)
Let C be a locally finite category. The adjacency map ξ of X is invertible in [I(C0 ,≤) , R];
1
its inverse µ, called the Möbius function of C, is such that µ(x, x) = |C(x,x)|
and for
x ̸= y ∈ C we have that
µ(x, y) =
∑
∑
(−1)n
n≥1 x=x0 <x1 <...<xn =y
∑
∑
n≥1 x=x0 →x1 →...→xn
|C(x0 , x1 )|.................|C(xn−1 , xn )|
=
|C(x0 , x0 )||C(x1 , x1 )|.......|C(xn−1 , xn−1 )||C(xn , xn )|
(−1)n
,
|C(x
,
x
)||C(x
,
x
)|.......|C(x
,
x
)||C(x
,
x
)|
0
0
1
1
n−1
n−1
n
n
=y
where the second sum above range over all diagrams in C of the form
x = x0 −→ x1 −→ ...... −→ xn = y.
Fix a ∈ C. Then [(C0 )≥a , R] is a right [I(C0 ,≤) , R]-module. From this action one
derives the following result.
Corollary 41. (Möbius Inversion for Locally Finite Categories)
Fix a ∈ X. For f, g ∈ [(C0 )≥a , R] we have that
g(y) =
∑
f (x)|C(x, y)|
if and only if
f (y) =
a≤x≤y
∑
g(x)µ(x, y).
a≤x≤y
Example 42. Let C be a locally finite category, fix a ∈ C, and let F : C −→ set be a
functor. Consider the functor G : C −→ set given by on objects by
⊔
G(y) =
F (x) × C(x, y).
a≤x≤y
By the Möbius inversion formula we have that
∑
|F (y)| =
|G(x)|µ(x, y).
a≤x≤y
17
6
Essentially Locally Finite Categories
The Möbius theory for locally finite categories developed in Section 5 although functorial
under isomorphisms of categories, fails to be functorial under equivalences of categories.
We recall that categories C and D are equivalent if there is a functor F : C −→ D
that is essentially surjective, full and faithful. Indeed a category may fail to be locally
finite but be equivalent to a locally finite category. For example, the category with two
points, a unique isomorphism between them, and identities as the only endomorphism,
is equivalent to the category with one morphism. The latter is locally finite whereas the
former is not.
Given a category C and objects x, y ∈ C, the notation x ≃ y means that x and y are
isomorphic. Let C be set of isomorphism classes of objets in C, i.e. C is the quotient set
C0 / ≃. A typical element of C is denoted by by x, meaning that we have an equivalence
class x ∈ C and that we have chosen an object x ∈ x. For x, y ∈ C we set
[x, y] = {z | there is a diagram x −→ z −→ y in C }.
Definition 43. A category C is essentially locally finite if the following conditions hold:
• C(x, y) is a finite set for all x, y ∈ C.
• If we have a diagram x−→y−→x in C, then x ≃ y.
• [x, y] is a finite set for x, y ∈ C.
Remark 44. In the applications we often find a stronger version of the second property:
the arrows in any configuration x−→y−→x are isomorphisms. We call categories with
such a property isocyclic, i.e. all cycles are formed by isomorphisms. Not all essentially
locally finite categories are isocyclic, e.g. a finite monoid considered as a category.
Lemma 45. Let C be a category with C(x, y) finite for x, y ∈ C. Let ≤ be the relation
on C given by x ≤ y if and only if there is a morphism x −→ y in C. Then C is a
essentially locally finite category if and only if ≤ is a locally finite partial order on C.
Lemma 46. A category C is essentially locally finite if and only if C is equivalent to a
locally finite category.
Proof. Assume C is essentially locally finite. Let S be a full subcategory of C whose objects include one and only representative of the isomorphism classes of C. S is equivalent
to C and skeletal, thus it is a locally finite category.
18
Conversely, let S be a locally finite category and F : S −→ C an equivalence of
categories. Then for objects y1 , y2 ∈ C there exist objects x1 , x2 ∈ S such that F (x1 ) ≃
y1 and F (x2 ) ≃ y2 . Moreover, we have bijective maps C(x1 , x2 ) −→ C(y1 , y2 ) and
[x1 , x2 ] −→ [y1 , y2 ]. Thus C(y1 , y2 ) and [y1 , y2 ] are finite sets. If there is a diagram
y1 −→ y2 −→ y1 in C, then there is a corresponding diagram x1 −→ x2 −→ x1 in S,
thus x1 = x2 and y1 ≃ y2 .
Example 47. Let C be a subcategory of the category of finite sets and maps. Consider
the subcategory IC of C with the same objects as C and such that f ∈ IC(x, y) if and
only if f is injective. Similarly, we have the subcategory SC of C with the same objects
as C and such that f ∈ SC(x, y) if and only if f is surjective. The categories IC and
SC are isocyclic and essentially locally finite categories.
Example 48. Let I be the category of finite sets and injective maps. A combinatorial
presheaf P is contravavariant functor P : I −→ set. The Grothendieck category of
elements EP has for objects pairs (x, a) where a ∈ P (x). A morphisms (x, a) −→ (y, b)
in EP is an injective map f : x −→ y such that Pf (b) = a. For any presheaf the category
EP is isocyclic and essentially locally finite.
Example 49. Let B be the category of finite sets and bijections, and let O be an operad
in the category set of finite sets. Roughly speaking O is a functor O : B −→ set with an
distinguished element 1 ∈ O([1]) and with suitable composition maps
∏
O(b) −→ O(x),
O(π) ×
b∈π
where π is a partition of the finite set x. Let SO be the category whose objects are finite
sets, and such that a morphism (f, a) ∈ SO (x, y) is a surjective map f : x −→ y together
∏
with an element a ∈ i∈y O(f −1 (i)). The composition of morphisms is defined with the
help of the operadic compositions.
Thus for any essentially locally finite category C we have the partially order set (C, ≤)
and the incidence algebra ([I(C,≤) , R], ⋆). The adjacency map ξ : C × C −→ R of C given
by ξ(x, y) = |C(x, y)| lives naturally [I(C,≤) , R].
Theorem 50. (Möbius Function for Essentially Locally Finite Categories)
Let C be an essentially locally finite category. The adjacency map ξ is invertible in
1
[I(C,≤) , R]; its inverse µ, called the Möbius function of C, is such that µ(x, x) = |C(x,x)|
and for x ̸= y ∈ C we have that
∑
∑
|C(x0 , x1 )|.................|C(xn−1 , xn )|
µ(x, y) =
.
(−1)n
|C(x0 , x0 )||C(x1 , x1 )|.......|C(xn−1 , xn−1 )||C(xn , xn )|
n≥1 x = x <x <...<x =y
0
1
n
19
Fix a ∈ C. Then [C ≥a , R] is a right [I(C,≤) , R]-module.
Corollary 51. (Möbius Inversion for Essentially Locally Finite Categories)
Fix a ∈ C. For f, g ∈ [C ≥a , R] we have that
∑
g(y) =
f (x)|C(x, y)|
if and only if
f (y) =
∑
g(x)µ(x, y).
a≤x≤y
a≤x≤y
The formula above for the Möbius function µ admits a nice conceptual understanding
in the case of isocyclic categories which we proceed to formulate.
First we introduce a few mathematical notions. The of cardinality of finite sets can
be viewed as map
| | : set −→ N
from the category of finite sets to the natural numbers invariant under isomorphisms and
such that |∅| = 0, |[1]| = 1, |x ⊔ y| = |x| + |y|, and |x × y| = |x||y|.
The notion of cardinality of finite sets admits a suitable extension [2, 3, 4] to the
category gpd of essentially finite groupoids via the map
| |g : gpd −→ Q
given by
|G|g =
∑
x∈G
1
.
|G(x, x)|
The map | |g is invariant under equivalence of groupoids and is such that
|∅| = 0,
|[1]| = 1,
|G ⊔ H| = |G| + |H|,
and |G × H| = |G||H|.
Another useful property of | |g is the following. Let a finite group G act on the finite set
X, then we have the groupoid X ≀ G with set of objects X and such that
X ≀ G(x, y) = {g ∈ G | gx = y}.
It is easy to check that
|X ≀ G|g =
|X|
.
|G|
Note that X ≀ G is the traditional quotient set X/G.
20
Our immediate goal is to understand the Möbius function in terms of the cardinality
of groupoids. Fix an isocyclic essentially locally finite category C. For x ∈ C, the set
C(x, x) is a group, moreover the group C(x, x) × C(x, x) acts on C(x, x) by pre and post
composition. We have that
µ(x, x) =
1
= |C(x, x) ≀ C(x, x) × C(x, x)|g .
|C(x, x)|
♮
For x < y ∈ C, consider the groupoid [[0, n], C]x,y
(the odd notation will be justified
below) whose objects are functors
F : [0, n] −→ C
from the interval [0, n] ⊆ N to C such that F (0) ≃ x, F (n) ≃ y, and F (i < i + 1) is
not an isomorphism for i ∈ [0, n−1]. Morphisms are isomorphic natural transformations.
Concretely, objects in [[0, n], C]♮x,y are diagrams in C of the form
x ≃ x0 −→ x1 −→ ...... −→ xn ≃ y,
♮
where none of the arrows is an isomorphism. Morphisms in [[0, n], C]x,y
are commutative
diagrams
/ xn−1
/ x1
/
/ xn
x0
....
z0
/ z1
/
/ zn−1
....
/ zn
where the vertical arrows are isomorphisms.
♮
Let us compute the groupoid cardinality of [[0, n], C]x,y
. Note that
⊔
♮
[[0, n], C]x,y
=
[[0, n], C]x,x1 ,....,xn−1 ,y
x≃x0 <x1 <......<xn−1 <xn ≃y
where the groupoids [[0, n], C]x,x1 ,....,xn−1 ,y are define just as [[0, n], C]x,y fixing beforehand
the isomorphism classes of the intermediate objects. The groupoids
[[0, n], C]x,x1 ,....,xn−1 ,y
are actually quite easy to understand. Objects and morphism in it are isomorphic to
diagrams of the form
x
f1
g0
/ x1
f2
/
....
fn−1
x
/ x1
fn
gn−1
g1
g1 f1 g0−1
/ xn−1
g2 f2 g1−1
/
....
21
−1
gn−1 fn−1 gn−2
/ xn−1
/ xn
gn
−1
gn fn gn−1
/ xn
for which the top horizontal arrows together with the vertical arrows uniquely determines
the bottom arrows. From this viewpoint is clear that we have an equivalence of groupoids
(n−1
) ( n
)
∏
∏
[[0, n], C]x,x1 ,....,xn−1 ,y ≃
C(xi , xi+1 ) ≀
C(xi , xi ) ,
i=0
thus we have that
[[0, n], C]x,x1 ,....,xn−1 ,y =
g
i=0
|C(x0 , x1 )|.................|C(xn−1 , xn )|
|C(x0 , x0 )||C(x1 , x1 )|.......|C(xn−1 , xn−1 )||C(xn , xn )|
and we obtain the following result.
Lemma 52. Let C be an isocyclic essentially locally finite category and x < y in C. We
have that:
∑
∑
∑
♮ µ(x, y) =
(−1)n [[0, n], C]x,x1 ,....,xn−1 ,y =
(−1)n [[0, n], C]x,y
.
g
n≥1 x=x0 <x1 <...<xn =y
n≥1
g
Below we need the theory on simplicial sets [20], in particular the formula above can
be understood in terms of augmented simplicial essentially finite groupoids. A simplicial
essentially finite groupoid is a functor G : ∆◦ −→ gpd, where ∆◦ is the oppositive category of ∆. Objects in ∆ are intervals [0, n] ⊆ N. Morphisms in ∆ are non-decreasing maps.
Thus a simplicial essentially finite groupoid G assigns a groupoid Gn to each n ∈ N,
and a functor
fb : Gm −→ Gn
to each non-decreasing map f : [0, n] −→ [0, m].
An augmented simplicial essentially finite groupoid is a functor
G : ∆◦a −→ gpd
where ∆a is the category obtained from ∆ by adjoining an object [−1] and a unique
morphism [−1] −→ [0, n] for each n ∈ N. Thus an augmented simplicial groupoid G has
in addition a groupoid G−1 and a unique functor Gn −→ G−1 for each n ∈ N.
For G an augmented simplicial groupoid and n ≥ −1, we let G♮n be the full subgroupoid of Gn whose objects are the non-degenerated objects of Gn . An object x ∈ Gn
is degenerated if there is a non-decreasing map f : [0, n] −→ [0, m] with m < n, and an
object y ∈ Gm such that fb(y) ≃ x.
22
Definition 53. The reduced Euler characteristic χ
eG of an augmented simplicial groupoid
G : ∆a −→ gpd is given by
∑
χ
eg G =
|G♮n |g .
n≥−1
Let C be an isocyclic essentially locally finite category. For x < y in C, we define the
augmented simplicial groupoid C∗ (x, y) as follows. For n ≥ −1, we set
Cn (x, y) = [[0, n + 2], C]x,y .
Given a morphism f : [0, n] −→ [0, m] in ∆, consider the non-decreasing extension map
fe defined trough the commutative diagram
f
[0, n]
{0} ⊔ [1, n + 1] ⊔ {n + 2}
/ [0, m]
/ {0} ⊔ [1, m + 1] ⊔ {m + 2}
fe
where the vertical arrows arise from the increasing bijections [0, n] −→ [1, n+1], [0, m] −→
[1, m + 1], and fe (0) = 0, fe (n + 2) = m + 2. The functor
fb : [[0, m + 2], C]x,y −→ [[0, n + 2], C]x,y
is given by
fb(F ) = F ◦ fe .
If we have another map g : [0, m] −→ [0, k] in ∆, then
g[
◦ f (F ) = F ◦ (g ◦ f )e = F ◦ (ge ◦ fe ) = (F ◦ ge ) ◦ fe = (b
g ◦ fb)F.
The unique functor [[0, m + 2], C]x,y −→ [[0, 1], C]x,y sends a diagram
x0
z0
f1
g1
/ x1
/ z1
f2
/
....
g2
/
....
fn+1
gn+1
/ xn+1
/ zn+1
fn+2
gn+2
/ xn+2
/ zn+2
to the diagram
x0
z0
fn+2 fn+1 ....f2 f1
gn+2 gn+1 ....g2 g1
/ xn+2
/ zn+2
Thus C∗ (x, y) is in fact an augmented simplicial groupoid. Putting all together we
have the following result.
23
Theorem 54. Let C be an isocyclic essentially locally finite category and x < y in C.
We have that
µ(x, y) = χ
eg C∗ (x, y).
Proof. By Lemma 52 we have that
µ(x, y) =
∑
♮ (−1) [[0, n], C]x,y n
∑
♮ (−1) [[0, n + 2], C]x,y n
n≥−1
=
g
=
g
n≥1
∑
(−1) Cn (x, y)
n
n≥−1
g
= χ
eg C∗ (x, y).
There is another canonical element ξg in the incidence algebra [I(C,≤) , R] of an isocyclic
essentially locally finite category C. The map ξg is defined, for x, y ∈ C, by:
ξg (x, y) =
ξ(x, y)
|C(x, y)|
=
= C(x, y) ≀ C(x, x) × C(y, y) .
|C(x, x)||C(y, y)|
|C(x, x)||C(y, y)|
g
For x < y ∈ C, let D∗ (x, y) be the augmented sub simplicial groupoid of C∗ (x, y)
such that, for n ≥ −1, objects in Dn (x, y) are functors F : [0, n + 2] −→ C such that
F (0) = x and F (n + 2) = y. For F, G ∈ Dn (x, y), a natural isomorphism n : F −→ G is
a morphism in Dn (x, y) if and only if the morphisms n(x) : x −→ x and n(y) : y −→ y
are identities. Thus a morphism in D∗ (x, y) is a commutative diagram of the form
x
w; 1
ww
w
ww
x GG
GG
GG
# z1
/ x2
/
....
/ xn−2
/ xn−1
/ z2
/
....
/ zn−1
/ zn−2
JJ
JJ
JJ
$
t: y
t
t
tt
tt
where the diagonal and horizontal arrows are not allow to be isomorphisms, and the
vertical arrows are isomorphisms.
Theorem 55. The map ξg ∈ [I(C,≤) , R] is a unit and its inverse µg is given for x < y ∈ C
by µg (x, x) = |C(x, x)|, and
µg (x, y) =
∑
∑
(−1)n
n≥1 x = x0 <x1 <...<xn =y
|C(x0 , x1 )|.................|C(xn−1 , xn )|
.
|C(x1 , x1 )|.......|C(xn−1 , xn−1 )|
Moreover, we have that:
eg D∗ (x, y).
µ(x, y) = χ
24
7
Möbius Categories
In this section we adopt the viewpoint that regards N+ in the classical Möbius theory
as a monoid. The idea is to develop a Möbius theory that it applies to other monoids.
It turns that this can readily be done for the category of finite decomposition monoids.
Indeed one can go further and define a Möbius theory that applies to finite decomposition categories, better known in the literature as Möbius categories. Since a monoid
is just as a category with one object, the theory of locally finite monoids is embedded
in the theory of Möbius categories. The notion of Möbius categories was introduced by
Leroux in [7, 18], and since there have quite few publications in the field, among them
[11, 15, 17, 24]. Here we limits ourselves, to the most basic results on Möbius categories.
Let C be a category and f a morphism in C. A n-decomposition of f , for n ≥ 1,
is n-tuple (f1 , ..., fn ) of morphisms in C such that fn fn−1 ...f1 = f. Let Dn (f ) be the set
of n-decompositions of f . A decomposition is proper if none of its components is an
identity. For n ≥ 2, let PDn (f ) be the set of proper n-decompositions of f . Set
⊔
⊔
D(f ) =
Dn (f ) and PD(f ) =
PDn (f ).
n≥1
n≥1
Definition 56. A category C is Möbius if PD(f ) is a finite set for all morphisms f in
C, i.e. each morphism in C admits a finite number of proper decompositions.
Note that in Möbius category the only isomorphisms are the identities.
Lemma 57. If PD(f ) is a finite set, then Dn (f ) is a finite set for all n ≥ 1. Indeed we
have that:
( )
n ( )
n
∑
∑
n n
n−k
Dn (f ) =
PDk (f )
PDn (f ) =
(−1)
Dk (f ).
and
k
k
k≥1
k≥1
Proof. The identity on the left is shown as follows. Out of the n morphisms in a n( )
decomposition assume that n − k are identities, they can be placed in nk different
positions. Omitting the identity arrows a n-decomposition of the latter type reduces
to a proper k-decomposition. The identity on the right follows from Möbius inversion
formula.
Let f be a morphism in a category C. We say that f fixes a morphism g ∈ C if
f g = g or gf = g. Next result is due to Leroux [7].
Theorem 58. A category C is Möbius if and only if the following conditions hold:
• PD2 (f ) is a finite set for each f ∈ C.
25
• Identities admit no proper decomposition.
• If a morphism have fix morphisms, then it must be an identity.
Proof. Assume C is Möbius. By definition PD2 (f ) is a finite set for each f ∈ C. If
1x admits a proper decomposition fn ...f1 = 1, then 1X admits infinitely many proper
decompositions, indeed 1 = fn ...f1 = fn ...f1 fn ...f1 = fn ...f1 fn ...f1 fn ...f1 = ..... Let f be
a non-identity and g another morphism. If f g = g, then g ̸= 1 and admits infinitely
many proper decompositions g = f g = f f g = f f f g = f f f g = .... Thus f g ̸= g. Same
argument shows that gf ̸= g.
Suppose the conditions hold. An inductive argument shows that if PD2 (f ) is a finite
set, then PDn (f ) is a finite set for all n ≥ 2. Let k = |P D2 (f )|, we show that f can
not have proper decompositions of length greater or equal k + 2. Assume that we have
such a decomposition (f1 , ..., fk+2 ). Composing initial and final segments it generates
k + 1 proper 2-decompositions of f . They can not be all different, thus there exist g and
h, obtaining by composing some fi , such that gh = h. Then h must be an identity, in
contradiction with the fact that identities admit no proper decomposition.
Example 59. Let C be a graded category with finite graded components, i.e. for each
x, y ∈ C we have that
⊔
C(x, y) =
Cn (x, y), |Cn (x, y)| < ∞ and Cn (x, y) × Cm (x, y) −→ Cn+m (x, y)
n∈N
Assume in addition that C0 (x, x) = {1x }. For example, one can take C to be the category
of path (quotient by a length preserving equivalence relation) in a finite graph. Let (X, ≤)
be a locally finite poset and F : X −→ C a functor. The category XF with X as it set
of objects and such that XF (x, y) = C(F (x), F (y)) is a Möbius category.
A category is called one way if in any diagram x −→ y −→ x, we have that x = y.
Proposition 60. Let C a category with C(x, y) finite for all x, y ∈ C. Then C is a
Möbius category if and only if C is locally finite and one way.
Proof. Let C be locally finite. The map PD(f ) −→ C[x, y] sending (f1 , ..., fn ) to the
chain
{tf1 < tf2 .... < tfn−1 } ⊆ [x, y]
has finite range and finite fibers, thus PD(f ) is a finite set.
26
Let C be a Möbius category. The map PD(f ) −→ C[x, y] has finite domain and is
surjective. Thus [x, y] is a finite set. We show that any endomorphism in C is an identity.
Let f : x −→ x be an endomorphism. Since C(x, x) is a finite set, there is n ≥ 0 and
k ≥ 1 such f n = f n f k . Thus f k = 1, and since 1 can not be properly decompose we have
that f = 1.
We associate a convolution algebra for a category C such that D2 (f ) is a finite set
for all morphisms in C.
Definition 61. Let C be a category such that D2 (f ) is a finite set for all morphism
f ∈ C1 . The convolution algebra of C is the pair ([C 1 , R], ⋆) where for α, β ∈ [C1 , R] the
product α ⋆ β is given on f ∈ C1 by
∑
α ⋆ β(f ) =
α(f1 )β(f2 ).
(f1 ,f2 )∈D2 (f )
Proposition 62. ([C1 , R], ⋆) is an associative algebra with unit the map 1 ∈ [C1 , R] such
that 1(f ) = 1 if f is an identity, and zero otherwise.
Proof. The unit property is clear. Associativity follows from the identities
∑
α1 (f1 )α2 (f2 )α3 (f3 ) = (α1 ⋆ α2 ) ⋆ α3 (f ).
α1 ⋆ (α2 ⋆ α3 )(f ) =
(f1 ,f2 ,f3 )∈D3 (f )
More generally we have that
∑
α1 ⋆ α2 ⋆ .... ⋆ αn (f ) =
α1 (f1 )α2 (f2 )....αn (fn ).
(f1 ,...,fn )∈Dn (f )
Theorem 63. Let C be a Möbius category. A map α ∈ [C1 , R] is a unit if and only if
α(1x ) ∈ R is a unit for all x ∈ C. Thus the map ξ ∈ [C1 , R] constantly equal to 1 is a
unit in ([C1 , R], ⋆). Its inverse µ, called the Möbius function of C, is such that µ(1x ) = 1
for all x ∈ C, and if f is a non-identity we have that
∑
µ(f ) =
(−1)n PDn (f ).
n≥1
Proof. Assume α ⋆ β = 1. Then for all x ∈ C we have that
α(1x )β(1x ) = (α ⋆ β)(1x ) = 1,
thus
α(1x ) ̸= 0.
Conversely, if α(1x ) ̸= 0, then is inverse α−1 is given by α−1 (1x ) = α(11 x ) , and for f a
non-isomorphism we have that
∑
∑
α(f1 ).......α(fn )
.
α−1 (1) =
(−1)n
α(1sf1 )α(1tf1 )....α(1tfn )
n≥1
(f1 ,....,fn )∈PDn (f )
27
For f : x −→ y be a non-identity in C, we construct the augmented simplicial set
D∗ (f ) as follows. For n ≥ −1 we have inclusions
Dn (f ) ⊆ [[0, n + 2], C]x,y ,
where a functor F ∈ [[0, n + 2], C]x,y is in D∗ (f ) if and only if F0≤n+2 = f.
Note that, by definition, we have for n ≥ −1 that:
Dn (f ) = Dn+2 (f )
and
Dn (f )♮ = PDn+2 (f ).
So we have shown the following result.
Theorem 64. The Möbius function µ of a finite decomposition category C is given for
f a non-isomorphism by
∑
e n (|D∗ (f )|).
µ(f ) = χ
eD∗ (f ) =
(−1)n rankH
n≥0
e n (|D∗ (f )|) are obtain from the complex
Remark 65. The reduced homology groups H
generated in degree n by the (n + 2) decompositions of f , with the differential given by
d[f ] = 0 and
n+1
∑
(−1)k [f1 , ..., fk+1 fk , ...fn+2 ].
d[f1 , ..., fn+2 ] =
k=0
Remark 66. Let C be a Möbius category. The relation ≤ on C1 defined by
f ≤ g if and only if there is morphis h such that hf = h,
is a partial order. Reflexivity and transitivity are trivial. Let f ≤ g ≤ f, there are
morphisms h and l such that hf = g and lg = f . Thus (gh)f = f and since C is Möbius,
we have that gh = 1, and thus g = h = 1. So f = g.
A category C is (left) cancellative if for any morphisms gf = hf implies that g = h.
Theorem 67. Let C be a cancellative Möbius category. The convolution algebra [C1 , R]
is isomorphic to the subalgebra of the incidence algebra [I(C1 ,≤) , R] consisting of maps
α : I(C1 ,≤) −→ R such that α[f, gf ] = α[1, g] for all morphisms f, g ∈ C1 .
28
8
Essentially Finite Decomposition Categories
Let C be a category and C 1 be the set of isomorphisms classes of morphisms in C. Recall
that morphisms f, g ∈ C1 are isomorphic if they fit into a commutative diagram
x
z
f
g
/y
/w
where the vertical arrows are isomorphisms.
Fix f ∈ C 1 . A n-decomposition of f , for n ≥ 1, is n-tuple (f1 , ..., fn ) of morphisms
in C such that there are isomorphisms α and β for which
fn fn−1 ...f1 = βf α−1 ,
that is we have a commutative diagram of the form
f1
x0
/ x1
f2
/
....
fn−1
/ xn−1
fn
α
/ xn
β
/y
f
x
Let Dn f be the groupoid whose objects are n-decompositions of f and with morphisms commutative diagrams of the form
x0
/ x1
/
....
/ xn−1
/ xn
/ z1
/
....
/ zn−1
/ zn
z0
where the top and bottom part of the diagram are n-decompositions of f and the vertical
arrows are isomorphisms.
For n ≥ 2, a decomposition (f1 , ..., fn ) of f is called proper if none of the morphisms
fi is an isomorphism. For n ≥ 2, we let PDn f be the full subgroupoid of Dn f whose
objects are proper decompositions. For n = 1 we set PD1 f = D1 f . We also set
⊔
⊔
Df =
Dn f and PDf =
PDn f ,
n≥1
n≥1
i.e. (PDn f ) Dn f is the set of all (proper) decompositions of f .
29
Definition 68. An essentially finite decomposition category C is such that PDf is a
finite set for all morphisms f in C, i.e. each morphism in C admits a finite number of
equivalence classes of proper decompositions.
Lemma 69. If PDf is a finite set then Dn f
n ( )
∑
n Dn f =
and
PDk f k
k≥1
is a finite set for all n ≥ 1. We have that:
( )
n
∑
n n−k
PDn f =
(−1)
Dk f .
k
k≥1
Proof. The identity on the left is shown as follows. Out of the n morphisms in a n( )
decomposition assume that n − k are isomorphism, they can be placed in nk different
positions. Such a n-decomposition is isomorphic to a n-decomposition where the isomorphisms are replaced by identities and the remaining morphisms are not isomorphisms.
Omitting the identity arrows a n-decomposition of the latter type reduces to a proper
k-decomposition. The identity on the right follows from Möbius inversion formula.
Theorem 70. Let C be a category with C(x, y) finite for x, y ∈ C. Then C is an
essentially finite decomposition category if and only if C is an isocyclic essentially locally
finite category.
Proof. Let C be an essentially finite decomposition category. Let f : x −→ x be an
endomorphism in C, since C(x, x) is a finite set and
{1, f, f 2 , ...., f n , ...} ⊆ C(x, x),
there must be n ≥ 0 and k ≥ 1 such that f n = f n+k . If n = 0, then f k = 1 for some
k > 0 and f is an isomorphism. If n > 0, and f is not an isomorphism then the identities
f n = f n+k = f n+2k = f n+3k = .....
show that hn admits infinitely many decompositions, a contradiction. Thus f is an isomorphism, and x ≃ y.
Next we show that the intervals [x, y] are finite. The map
⊔
{x, y} ⊔
PD2 f −→ [x, y],
f ∈C(x,y)
being the identity on {x, y} and sending a 2-decomposition x −→ z −→ y of f to z,
is surjective and has a finite domain because C is a finite decomposition category and
C(x, y) is a finite set. Thus [x, y] is a finite set and C is an isocyclic essentially locally
finite category.
30
Assume that C is an isocyclic essentially locally finite category. Isomorphisms in C
admit a unique proper decompositions. Given a non-isomorphism f ∈ C(x, y) and a
chain
x < x1 < ... < xn−1 < y in C, we let PD(f , x1 , ..., xn−1 )
be the full subcategory of PDf whose objects are proper n-decompositions of f with
specified isomorphism classes for the intermediate objects. We have that
⊔
⊔
PDf =
PD(f , x1 , ..., xn−1 ).
n≥2 x<x1 <...<xn−1 <y
Since [x, y] is a finite set, it has a finite number of increasing chains. Since there are a
finite number of morphisms between objects of C, the sets PD(f , x1 , ..., xn−1 ) are finite.
Therefore, PDf is a finite set, and C an essentially finite decomposition category.
Next we attempt to associate a convolution algebra for a category C such that D2 f
is a finite set for all morphisms in C. The convolution product ⋆ on [C 1 , R] is given on
α, β ∈ [C 1 , R] by
∑
α ⋆ β(f ) =
α(f 1 )β(f 2 ).
(f1 ,f2 )∈D2 f
Associativity of the product ⋆ is not obvious. Thus we impose a (fairly strong)
condition on C that guarantees associativity. The proof of Theorem 75 provides a weaker
(though less intuitive) condition that also guarantees associativity.
Definition 71. We say that a category C has the isomorphism filling property if any
commutative diagram
; y1 F
xx
xx
x
x
x FF
FF
FF
#
y2
FF
FF
F#
x; y
xx
x
xx
with y1 and y2 isomorphic objects in C, can be enhanced into a commutative diagram
; y1 FF
FF
xx
x
FF
x
xx
#
x FF
;y
x
x
FF
xx
FF
# xx
y2
where the vertical arrow is an isomorphism.
Example 72. The category I of finite set and injective maps is isomorphism filling.
31
Example 73. The category vect of finite dimensional vector spaces and injective linear
maps is isomorphism filling.
Example 74. Let B be the category of finite sets and bijections and F : B −→ set a
functor (i.e. a combinatorial species [1, 5, 8, 12]). Let ParF : B −→ set be the species of
F -colored partitions; it is such that ParF x is the set of pairs (π, s) where π is a partition
of x and s assigns to each b ∈ π an structure sb ∈ F b. Consider the category EParF whose
objects are triples (x, π, s) with (π, s) ∈ ParF x. A morphism f : (x, π, s) −→ (z, σ, u) is
an injective map such that there is (y, ρ, t) ∈ EParF such that
(f x ⊔ y, f π ⊔ ρ, f s ⊔ t) = (z, σ, u).
The category EParF is isomorphism filling.
Theorem 75. Let C be an isomorphism filling category with D2 f a finite set for f ∈ D2 f .
The product ⋆ turns [C 1 , R] into an associative algebra with unit the map 1 ∈ [C 1 , R]
such that 1(f ) = 1 if f is an identity, and zero otherwise.
Proof. The unit property is clear. Associativity follows from the identities
∑
(α1 ⋆ α2 ) ⋆ α3 (f ) =
α1 (f 1 )α2 (f 2 )α3 (f 3 ) = α1 ⋆ (α2 ⋆ α3 )(f ).
(f1 ,f2 ,f3 )∈D3 f
We show the left hand side identity, the other identity is proven similarly. By definition
we have that
∑
(α1 ⋆ α2 ) ⋆ α3 (f ) =
(α1 ⋆ α2 )(f 1 )α3 (f 2 ) =
(f1 ,f2 )∈D2 f
∑
∑
(f1 ,f2 )∈D2 f
(f3 ,f4 )∈D2 f 1
α1 (f 3 )α1 (f 4 )α3 (f 2 ).
Setting
E = {((f1 , f2 ), (f3 , f4 )) | (f1 , f2 ) ∈ D2 f and (f3 , f4 ) ∈ D2 f 1 },
the desired result will follow if there is a bijection τ : D3 f −→ E making the diagram
/
D3 f L
tE
t
LL
t
LL
tt
LL
tt
t
LL
t
LL
tt
LL
tt
L%
ytt
< C1 > ⊗ < C1 > ⊗ < C1 >
τ
commutative, where the diagonal arrows are given, respectively, by
(f1 , f2 , f3 ) −→ f 1 ⊗ f 2 ⊗ f 3 ,
32
((f1 , f2 ), (f3 , f4 )) −→ f 3 ⊗ f 4 ⊗ f 2 .
The desired map τ : D3 f −→ E is given by
(f1 , f2 , f3 ) −→ ((f2 f1 , f3 ), (f1 , f2 )).
We show that τ is well-defined, surjective, and injective.
Let α0 , α1 , α2 , α3 be isomorphisms from the appropriate objects so that
(f1 , f2 , f3 ) = (α1 f1 α0−1 , α2 f2 α1−1 , α3 f3 α2−1 ).
The following identities show that τ is well-defined:
τ (α1 f1 α0−1 , α2 f2 α1−1 , α3 f3 α2−1 ) = ((α2 f2 f1 α0−1 , α3 f3 α2−1 ), (α1 f1 α0−1 , α2 f2 α1−1 )) =
((f2 f1 , f3 ), (f1 , f2 )) = τ (f1 , f2 , f3 ).
To show that τ is surjective, take an element of E, it is given by a tuple ((f1 , f2 ), (f3 , f4 ))
and we may assume that
f2 f1 = f
and
f4 f3 = f1 .
So we have that
τ (f1 , f2 , f3 ) = ((f1 , f2 ), (f3 , f4 )).
For injectivity we proceed as follows. Suppose given are morphisms
f1
f2
f3
x0 −→ x1 −→ x2 −→ x3
and
g1
g2
g3
y0 −→ y1 −→ y2 −→ y3 ,
such that
τ (f1 , f2 , f3 ) = τ (g1 , g2 , g3 ) ∈ D3 f .
Then
((f2 f1 , f3 ), (f1 , f2 )) = ((f2 f1 , f3 ), (f1 , f2 )),
and thus we have commutative diagrams with vertical isomorphisms:
x0
f1
/ x1
f2
/ x2
f2
α2
α0
y0
g1
x0
y0
/ y1
f1
g1
g2
/ x1
/ y1
33
/ y2
f2
g2
/ x3
α3
g2
/ x2
/ y2
/ y3
In particular x1 ≃ y1 and thus we obtain the isomorphism filling α1 in the diagram
; x1 GG
GGf2
ww
w
GG
ww
GG
w
#
ww
α
1
x0 G
; x2
GG
w
GG
ww
ww −1
g1 α0 GGG
w
# ww α2 g2
f1
y1
So we get the diagram
f1
x0
α0
f2
/ x1
α1
g1
y0
f2
/ x2
α2
g2
/ y1
/ x3
α3
g2
/ y2
/ y3
Corollary 76. More generally by induction one shows that
∑
α1 ⋆ α2 ⋆ .... ⋆ αn (f ) =
α1 (f 1 )α2 (f 2 )....αn (f n ).
(f1 ,...,fn )∈Dn (f )
Theorem 77. Let C be an essentially finite decomposition isomorphism filling category.
A map α ∈ ([C 1 , R], ⋆) is a unit if and only if α(1x ) is a unit in R for all x ∈ C. Thus the
map ξ ∈ [C 1 , R] constantly equal to 1 is a unit in ([C 1 , R], ⋆); its inverse µ is called the
Möbius function of C and is such that µ(f ) = 1 if f is an isomorphism, and otherwise µ
is given by
∑
µ(f ) =
(−1)n PDn f .
n≥1
Proof. Assume α ⋆ β = 1. Then for all x ∈ C we have that
α(1x )β(1x ) = (α ⋆ β)(1x ) = 1,
thus
α(1x ) ̸= 0.
Conversely, if α(1x ) ̸= 0, then its inverse α−1 is given by α−1 (1x ) =
non-isomorphism we have that
α−1 (f ) =
∑
∑
n≥1
(f1 ,....,fn )∈PDn f
(−1)n
34
1
,
α(1x )
α(f 1 ).......α(f n )
.
α(1sf1 )α(1tf1 )....α(1tfn )
and for f a
Let f : x −→ y be a non-isomorphism in C1 . We construct the augmented simplicial
essentially finite groupoid D∗ f which is a full sub-simplicial groupoid of C∗ (x, y). Thus
for n ≥ −1, we have inclusions
Dn f ⊆ Cn (x, y) = [[0, n + 2], C]x,y .
A functor F ∈ Cn (x, y) is in Dn f if and only if F 0≤n+2 = f .
By definition we have for n ≥ −1 that:
Dn f = Dn+2 f
and
Dn f
♮
= PDn+2 f .
Taking isomorphisms classes of objects we get the simplicial complex D∗ f such that
Dn f = Dn+2 f
and Dn f
♮
= PDn+2 f .
So we have shown the following result.
Theorem 78. The Möbius function µ of a finite decomposition isomorphism filling category C is given for f a non-isomorphism by
µf = χ
eD∗ f .
Note that for x < y ∈ C we have the following identity of simplicial groupoids
⊔
C∗ (x, y) =
D∗ f ,
and thus
f ∈C(x,y)
∑
χ
eg C∗ (x, y) =
χ
eg D∗ f .
f ∈C(x,y)
Theorem 79. Let C be an essentially finite decomposition isomorphism filling category.
Then C is a essentially locally finite category. Its Möbius function µ(x, y) for x < y ∈ C
is given, according to Theorem 54, by
∑
χ
eg D∗ f
µ(x, y) =
f ∈C(x,y)
where
χ
eg D∗ f =
∑
∑
♮
(−1)n Dn f g =
(−1)n PDn f g .
n≥−1
n≥1
35
References
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Instituto de Matemáticas y sus Aplicaciones
Universidad Sergio Arboleda, Bogotá, Colombia
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