Available via license: CC BY 4.0
Content may be subject to copyright.
Introduction to Quantum Combinatorics
Tomasz Maszczyk
Abstract
We construct a topos of quantum sets and embed into it the classical topos of sets. We
show that the internal logic of the topos of sets, when interpreted in the topos of quantum
sets, provides the Birkhoff-von Neumann quantum propositional calculus of idempotents in
a canonical internal commutative algebra of the topos of quantum sets. We extend this
construction by allowing the quantum counterpart of Boolean algebras of classical truth
values which we introduce and study in detail. We realize expected values of observables in
quantum states in our topos of quantum sets as a tautological morphism from the canonical
internal commutative algebra to a canonical internal object of affine functions on quantum
states. We show also that in our topos of quantum sets one can speak about quantum quivers
in the sense of Day-Street and Chikhladze. Finally, we provide a categorical derivation of
the Leavitt path algebra of such a quantum quiver and relate it to the category of stable
representations of the quiver. It is based on a categorification of the Cuntz-Pimsner algebra
in the context of functor adjunctions replacing the customary use of Hilbert modules.
Contents
Acknowledgement 1
1 Introduction 3
2 Quantum Sets 5
2.1 Thetoposofquantumsets............................... 5
2.1.1 Quantumelements ............................... 5
2.1.2 Quantum elements in classical mathematics . . . . . . . . . . . . . . . . . 5
2.1.3 Quantum Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Setsasquantumsets.................................. 12
2.2.1 Birkhoff–von Neumann’s quantum propositional calculus . . . . . . . . . . 13
2.2.2 Quantumstates................................. 14
2.2.3 Positivity .................................... 14
2.2.4 Associative algebras as quantum algebras . . . . . . . . . . . . . . . . . . 15
1
arXiv:2505.11723v1 [math.CT] 16 May 2025
2.3 Quantum Universal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Quantum Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Quantum Quivers 23
3.1 Correspondences .................................... 24
3.2 A categorification of the Cuntz-Pimsner algebra . . . . . . . . . . . . . . . . . . . 25
3.3 A categorification of the quantum Leavitt path algebra . . . . . . . . . . . . . . . 27
2
1 Introduction
The history of the development of C*-algebras associated with combinatorial data began in the
context of a shift of finite type, which was associated by Cuntz and Krieger [11]with a transition
matrix. It was generalized by various authors, including Bates, Fowler, Kumjian, Laca, Pask,
and Raeburn [15], [22], to the context of more general subshifts associated with directed graphs.
In another direction, Exel and Laca [14] partially generalized Cuntz–Krieger algebras by allowing
for an infinite matrix with 0 and 1 entries. These two approaches were unified by Tomforde [38]
in the context of hypergraphs. Next, Carlsen [7], using the so-called extended Matsumoto’s
construction [26], addressed the case of a general one-sided subshift over a finite alphabet, while
Bates and Pask [4] associated so-called labeled graph algebras with any shift space over a finite
alphabet. Through the work of Carlsen, Ortega, and Pardo [9], it was shown that the case of
labeled graphs can be realized as a Boolean algebra of a family of subsets of vertices, with partial
actions given by the arrows. All these cases represent the 0-dimensional scenario of topological
graphs, as explored by Katsura [18] [19] [20], where the topology of actions can be reduced to their
combinatorial structure. The approach of Kumjian, Pask, Raeburn, and Renault [22] regarding
the groupoid presentation of graph C*-algebras has been extended by Exel [13] in the direction
of presenting quite general combinatorial C*-algebras via groupoids or inverse semigroups. Such
presentations are crucial for studying properties of these C*-algebras, such as ideal structure,
simplicity, and purely infinite properties, as well as for the computation of their K-theory.
Inspired by this development and motivated by Quantum Information Theory, efforts to
extend these concepts to the context of quantum graphs appeared recently. Similarly, the initial
datum is the adjacency matrix, quantized in an appropriate sense. Motivated by the work of
Musto-Reutter-Verdon [30], it associates a quantum version of the Cuntz-Krieger C*-algebra
with these structures. However, classically, this only covers the case of graphs with a finite set
of vertices and without multiple edges. Moreover, it is not clear what a quantum counterpart of
combinatorics might be in this context, nor how it relates to Boolean algebras, symmetries, and
dynamical systems defined by (partial) semigroup actions.
In addition, all such problems are convoluted because of a quite rich context, especially
the presence of a dagger structure related to the antilinear involutive antihomomorphism, C*-
algebras, Hilbert modules and hermitian scalar products together with a probabilistic interpre-
tation of a paring between states and projections. Therefore, to clarify the situation, it would
be helpful to take step back and start accumulating structures according to the usual classical
mathematical hierarchy which starts from the topos of sets, the classical logic as an internal
logic of this topos etc. As it will be shown here, surprisingly many constructions in the quan-
tum paradigm arises on the fundamental categorical level. The present paper begins with the
Yoneda full embedding of the category of counital coassociative coalgebras over a field which,
albeit being complete and cocomplete [2, 3], is not a topos, into its presheaf topos which we
call the topos of quantum sets. Better still, we construct a partial-monoidal structure on the
Yoneda functor, relating the tensor product of coalgebras with the argument-wise cartesian prod-
uct of presheaves. This connects a category with a structure of a topos, in which we can speak
the classical language, with a category in which we can speak the quantum language based
on coalgebraic calculus. While the linearization functor fully embeds the category of sets into
the category of coalgebras, which allows us to emulate classical combinatorics within the cat-
egory of coalgebras, the presence of non-cocommutative coalgebras is what allows us to model
quantum combinatorics. Although linearization functor is strong monoidal as transforming the
cartesian product of sets into the tensor product of coalgebras, and then composed with the
Yoneda functor goes to the argument-wise cartesian product of presheaves on coalgebras, when
3
extended to the tensor product of coalgebras, it encounters a non-trivial obstruction of an ab-
stract Eckmann-Hilton-type , to simultaneous realizability of a pair of coalgebra maps from a
coalgebra to two others coalgebras as a single coalgebra map to their tensor product. Our first
result says that this condition defines a subfunctor of the argument-wise cartesian product of
representable presheaves satisfying axioms of a partial-associative product. It is then natural to
speak about a (partial-associative) quantum Cartesian product of quantum sets. Note that in
opposite to the classical Cartesian product of sets, its quantum counterpart is not a categorical
product, but only a partial-monoidal structure on a (sub)category of quantum sets. This allows us
to construct quantum logic which, in particular, contains the Birkhoff–von Neumann’s quantum
propositional calculus interpreted through the image of classical logic in quantum logic internal
to the topos of quantum sets. In this internal language, the problems intractable by other ap-
proaches to quantum combinatorics can be sensibly addressed. In particular, quantum Boolean
algebras, quantum semigroups and their partial actions, quantum graphs, and their Leavitt path
algebras can be approached in this alternative way. The characteristic feature of all these alge-
braic constructions is that they, as presheaves on the category of coalgebras, take values in partial
algebraic structures. When, using the linearization functor, we fully embed the category of sets
into the category of quantum sets, all these constructions restrict to classical ones. The language
of coalgebras clarifies several technical problems encountered in the dual language of topological
algebras when trying to relate noncommutative theory with classical combinatorics. In the situ-
ation of (quantum, row finite) graphs, usually people apply the Cuntz-Pimsner construction for
a graph-induced self-correspondence between vertices of the graph to associate with it a graph
C*-algebra. In the parlance of noncommutative geometry, it can be inerpreted as associating to
a combinatorial datum a (locally compact Hausdorff) quantum space and relate combinatorics to
the thus induced noncommutative topology. In our approach, we categorify the Cuntz-Pimsner
construction [31] [18] [19] [20] for a self-correspondence, to an exact (dualizable) endofunctor on
the category of comodules over the coalgebra object of vertices of a quantum graph, to associate
with a graph its Leavitt path monad. We prove that the Eilenberg-Moore category of this monad
is equivalent to a category of quiver representations equipped with some retraction. When in
this result the latter category is an analog of the Yetter-Drinfeld module category for a (finite
dimensional) Hopf algebra, the Leavit path monad is an analog of the Drinfeld double algebra
whose modules form a category equivalent to the Yetter-Drinfeld module category. Since every
graph C*-algebra is a C*-completion of a Leavitt path algebra, it gives an insight into the deep
categorical roots of the bicategorical interpretation of Meyer-Sehnem [28] of the Cuntz-Pimsner
construction. In opposite to their interpretation which can be regarded as viewing it from above
and in the rich C*-algebraic and Hilbert module context, we propose a complementary derivation
of it from the bottom up explaining it as coming from canonical adjunctions associated with a
(quantum) quiver already on the (quantum) combinatorial level. We believe that understanding
these categorical point of view could be helpful in future more and more categorical questions
posed on the way of developing the theory also in the context of C*-algebras. The point is that
the construction of a graph C*-algebra of a quiver, in the parlance of Noncommutative Geometry,
associates with a quiver a locally compact quantum space represented by a C*-algebra, while we
associate with it only a quantum-combinatorial categorical object. Since both the quiver and the
resulting object are both combinatorial in nature, except one can be entirely classical and the
other is inherently quantum, difficulties created by a C*-algebraic structure cannot even arise.
Therefore (quantum) combinatorial laboratory offers an opportunity of easy testing categorical
questions before they are to be generalized to the algebraic or C*-algebraic context.
4
2 Quantum Sets
2.1 The topos of quantum sets
Let kbe a fixed field of characteristic zero. By Vect we denote the symmetric monoidal category
of k-vector spaces under the tensor product over k, and by Coalg the category of counital
coassociative coalgebras over k, i.e. comonoids in Vect. In calculations for coalgebras we use
the Heyneman-Sweedler convention [35], e.g. for the comultiplication ∆ : C→C⊗Cwe use the
notation ∆(c) = c(1) ⊗c(1).
Definition 2.1. The topos of quantum sets qSet is the topos of presheaves on Coalg with
natural transformations as morphisms. It is a symmetric monoidal category with respect to an
argument-wise product as a monoidal product and the singleton-constant presheaf as a monoidal
unit.
Note that the coalgebras of rank zero and one represent presheaves which are the initial and
the terminal objects of qSet, respectively. Moreover, the initial object is monoidally absorbing
and the terminal one is a monoidal unit. These we will call the quantum empty set and
the quantum singleton and, for the reasons which will be explained in subsection 2.2, denote
classically as ∅and {•}, respectively.
Definition 2.2. The internal language and the internal logic [25] of the topos qSet will be
called quantum language and quantum logic, respectively.
Definition 2.3. A quantum set is called representable if it is representable as a presheaf [25].
2.1.1 Quantum elements
Definition 2.4. For every quantum set Xwe define the category of quantum elements, whose
objects are pairs (C, x)where Cis a finitely dimensional coalgebra, x∈X(C), and morphisms
(C′, x′)→(C, x)being coalgebra maps π:C′→Csuch that X(π)(x) = x′, with an obvious
composition.
This definition is based on the general notion of the category of elements of a presheaf [25].
The Fundamental Theorem on Coalgebras [35] saying that every coalgebra is a colimit of finite
dimensional coalgebras, the fact that finite dimensional ones form a small subcategory of compact
objects, and finally the Density Theorem [25] saying that every presheaf on a small category is
a colimit of representable ones, altogether can be summarised to what in this context should be
called the Quantum Extensionality Principle.
2.1.2 Quantum elements in classical mathematics
Example 2.1.1. For S={•} (a singleton), we recover the set Tas the set of its classical
elements
Y(kT)(k{•}) = T.
5
If T=∅(empty set), then Y(k∅)is an initial object in qSet.
If T={•} (a singleton), Y(k{•})is a terminal object in qSet.
By virtue of Theorem 6 it means that the singleton regarded as a quantum set (quantum
singleton) has a single quantum element. Since this element is classical, the quantum singleton
is an example of a quantum set with no non-classical elements.
Example 2.1.2. Group-like elements. Let kSbe the k−linearization of a set Swith its
standard coalgebra structure. For S={•} (a singleton), we obtain Y(D)(k{•}) = G(D),the
set of gruplike elements of the coalgebra D. They can be interpreted as classical elements of the
quantum set Y(D). In general,
Y(D)(kS) = ngs∈G(D)s∈So,
is a family of grouplike elements of the coalgebra D, i.e.
∆(gs) = gs⊗gs, ε(gs)=1,
indexed by the set S.
Example 2.1.3. Primitive elements. Let C=k{•,▶}be a coalgebra with
∆(•) = •⊗•,∆(▶) = • ⊗ ▶+▶⊗ •,
ε(•)=1, ε(▶) = 0
The canonical injective coalgebra map k{•} → k{•,▶}induces a surjective map on quantum
elements
Y(D)(k{•,▶})→Y(D)(k{•})
which coincides with the canonical projection
π:a
g∈G(D)
Pg(D)→G(D),
where the preimage of an element g∈G(D)
π−1(g) = Pg(D) := np∈D∆(p) = p⊗g+g⊗p, ε(p)=0o
is the subset of g−primitive elements of D.
In particular, a group and a Lie algebra can be seen as elements of (representable) quantum
sets as follows.
1. Y(kG)(k{•}) = G
2. Y(U(g))(k{•,▶}) = g
Example 2.1.4. Quantum elements of finite decomposition categories
Definition 2.5. If every morphism in a category Dhas finitely many decompositions, we say
that this category has finite decomposition property, f.d. in short.
6
For any small category with the finite decomposition property the linearization of the set of
morphisms D:= kDadmits a coalgebra structure with
∆(m) = X
m1◦m2=m
m1⊗m2, ε(m) = (1man identity
0otherwise
Proposition 1. For every small category with f.d. property and Das above,
Y(D)(C) = {x:D→F(C)|xafunctor of finite support, X
d∈Ob(D)
x
d= 1}.
where by F(C)we mean its multiplicative monoid with zero and we identify objects with identity
morphisms.
Proof. Writing x(c) = Pmxm(c)mand regarding m7→ xm∈F(C)as a function with finite
support, we see that xbeing compatible with comultiplication and the counit is tantamount to
xm1◦m2=xm1xm2,X
d∈Ob(D)
x
d= 1,
respectively.
Note that for a discrete category, we obtain the complete system of orthogonal idempotents,
as before.
Here are the three most important examples of categories with finite decomposition. In the
first two we regard a monoid as a category with a single object.
1. C= (N,+,0).D=kndnn∈No,
∆ : dn→X
n′+n′′=n
dn′⊗dn′′ , ε(dn) = δn,0,
Y(D)(C) = nx∈F(C)xnilpotento.
Here xis the coefficient of the coalgebra map C→Dat d1. Since F(D)∼
=k[[t]] is the
algebra of formal power series, and Y(D)admits a distinguished element {•} → Y(D)
corresponding to d0, dual to the augmentation F(D)∼
=k[[t]] →k,t7→ 0,Y(D)can be
regarded as a base parameterizing formal deformations of quantum sets. In Section ??, it
will be used to define formal deformation quantisations of classical sets.
2. C= (N>0,×,1), D =kndnn∈N>0o,
∆ : dn→X
n′×n′′=n
dn′⊗dn′′ , ε(dn) = δn,1,
Y(D)(C) = n{xp}p∈P⊂F(C){xp}p∈Pfinite family of commuting nilpotentso
Here xpis the coefficient of the coalgebra map C→Dat dp, while Pis a finite set of
primes.
7
3. C=I×I, where Iis a finite set, with objects i∈Iand arrows (i, j)∈I×I(we adopt the
convention saying that (i, j)is an arrow from jto i), the composition (i, j)◦(j, k) := (i, k)
and identities (i, i), is called the pair category on I. Then
D:= kndi,j i, j ∈ {1, . . . , n}o,
∆(di,k) = X
j
di,j ⊗dj,k, ε(di,k ) = δi,k
is called the comatrix coalgebra and represents a quantum set of systems of matrix
units in F, i.e.
Y(D)(C) = n{xij }i,j∈Ixijxj′k=δj,j ′xik ,X
i
xii = 1 in F(C).o.
Example 2.1.5. Finite-dimensional group representations as quantum elements.
For an affine group scheme Gover k, and a finitely dimensional k−vector space Vwe can
form a set Repk(G, V )of O(G)−comodule structures on V. Then
Repk(G, V ) = Y(O(G))(V∗⊗V).
2.1.3 Quantum Cartesian product
By the Yoneda embedding theorem [25] , Coalg and qSet are related by the fully faithful
Yoneda functor
Coalg Y
−→ PSh(Coalg) =: qSet,
Y(D) = Coalg(−, D).
Since both categories
Coalg,⊗,k{•}and PSh(Coalg),×,{•}
are symmetric monoidal one could ask how the Yoneda embedding interacts with those struc-
tures. To answer this question, we need Segal’s notion of a partial monoid [34] generalized and
adapted to our categorical context.
Definition 2.6. We call a full subcategory Cof qSet partial monoidal if it contains a
monoidal unit {•} and for any two objects X1,X2of Cin this subcategory there is a sub-
presheaf X1×CX2in Cof the argument-wise product X1×X2of presheaves in qSet, i.e.
X1×CX2(C)⊆X1(C)×X2(C),
such that for any three objects X1,X2,X3of C
(x1, x2)∈(X1×CX2)(C),((x1, x2), x3)∈((X1×CX2)×CX3)(C)(1)
if and only if
(x2, x3)∈(X2×CX3)(C),(x1,(x2, x3)) ∈(X1×C(X2×CX3))(C),(2)
and then, under the identification
X1(C)×X2(C)×X3(C)∼
=X1(C)×X2(C)×X3(C),
8
the equality
((x1, x2), x3) = (x1,(x2, x3)) (3)
holds, and for every object Xin Cthe containments {•}×CX⊆ {•}×Xand X×C{•} ⊆ X×{•}
are equalities.
Theorem 2. The full subcategory Cof representable quantum sets is a partial monoidal sym-
metric subcategory of qSet.
Proof. For Xi:= Y(Di), we define (X1×CX2)(C)as
{(x1, x2)∈X1(C)×X2(C)|x1(c(1))⊗x2(c(2)) = x1(c(2))⊗x2(c(1) )∈D1⊗D2}.(4)
To prove that X1×CX2is indeed representable, we define
(x1, x2)(c) := x1(c(1))⊗x2(c(2))(5)
and show that it defines a coalgebra map (x1, x2) : C→D1⊗D2.
First, using (5), (4), coassociativity, the fact that xi:C→Diare coalgebra maps and the
diagonal comultiplication of D1⊗D2, we check that (x1, x2)respects comultiplication
(x1, x2)(c(1))⊗(x1, x2)(c(2))(6)
=x1(c(1)(1))⊗x2(c(1)(2))⊗x1(c(2)(1))⊗x2(c(2)(2) )(7)
=x1(c(1))⊗x2(c(2))⊗x1(c(3))⊗x2(c(4) )(8)
=x1(c(1))⊗x2(c(2)(1))⊗x1(c(2)(2))⊗x2(c(3) )(9)
=x1(c(1))⊗x2(c(2)(2))⊗x1(c(2)(1))⊗x2(c(3) )(10)
=x1(c(1))⊗x2(c(3))⊗x1(c(2))⊗x2(c(4) )(11)
=x1(c(1)(1))⊗x2(c(2)(1))⊗x1(c(1)(2))⊗x2(c(2)(2)) )(12)
=x1(c(1))(1) ⊗x2(c(2))(1) ⊗x1(c(1))(2) ⊗x2(c(2) )(2) (13)
= (x1(c(1))⊗x2(c(2))(1) ⊗(x1(c(1))⊗x2(c(2) )(2) (14)
= (x1, x2)(c)(1) ⊗(x1, x2)(c)(2).(15)
Next, using (5), the diagonal counit of D1⊗D2, the fact that xi:C→Diare coalgebra maps
and counitality, we check that it respects the counit
ε((x1, x2)(c)) (16)
=εx1(c(1))⊗x2(c(2))(17)
=εx1(c(1))εx2(c(2))(18)
=εc(1)εc(2)(19)
=εc(1)ε(c(2))(20)
=ε(c).(21)
Now, we are to prove that the conditions (1) and (2) are satisfied. First, we use the definition
(4) to rewrite them as follows
x1(c(1))⊗x2(c(2)) = x1(c(2))⊗x2(c(1) ),
9
(x1, x2)(c(1))⊗x3(c(2)) = (x1, x2)(c(2))⊗x3(c(1) )
and
x2(c(1))⊗x3(c(2)) = x2(c(2))⊗x3(c(1) ),
x1(c(1))⊗(x2, x3)(c(2)) = x1(c(2))⊗(x2, x3)(c(1) ).
By (5) they can be rewritten as
x1(c(1))⊗x2(c(2)) = x1(c(2))⊗x2(c(1) ),
x1((c(1)(1))) ⊗x2(c(1)(2))⊗x3(c(2)) = x1(c(2)(1) )⊗x2(c(2)(2))⊗x3(c(1) ),
and
x2(c(1))⊗x3(c(2)) = x2(c(2))⊗x3(c(1) ),
x1(c(1))⊗x2(c(2)(1))⊗x3(c(2)(2)) = x1(c(2) )⊗x2c(1)(1))⊗x3(c(1)(2) ).
By applying the counit to the utmost left tensor factor in the triple tensor in the first pair
of identities and to the utmost right tensor slot in the triple tensor of the second pair, using the
fact that xiare coalgebra maps and the Heyneman-Sweedler convention we get the following two
systems of identities
x1(c(1))⊗x2(c(2)) = x1(c(2))⊗x2(c(1) ),
x2(c(1))⊗x3(c(2)) = x2(c(2))⊗x3(c(1) ),
x1((c(1))) ⊗x2(c(2))⊗x3(c(3)) = x1(c(2) )⊗x2(c(3))⊗x3(c(1) ),
and
x2(c(1))⊗x3(c(2)) = x2(c(2))⊗x3(c(1) ),
x1(c(1))⊗x2(c(2)) = x1(c(2))⊗x2(c(1) ),
x1(c(1))⊗x2(c(2))⊗x3(c(3)) = x1(c(3) )⊗x2c(1))⊗x3(c(2) ).
To prove equivalence of the thus rewritten two systems, observe that first two identities in
both systems differ only by their order. hence it is enough to prove that assuming them allows
us to make the right hand sides of both remaining identities equal. To this end, it is enough to
use those first two identities and coassociativity in the Heyneman-Sweedler convention as follows
x1(c(2)))⊗x2(c(3))⊗x3(c(1))(22)
=x1(c(2)(1)))⊗x2(c(2)(2))⊗x3(c(1))(23)
=x1(c(2)(2)))⊗x2(c(2)(1))⊗x3(c(1))(24)
=x1(c(3))⊗x2(c(2))⊗x3(c(1))(25)
=x1(c(2))⊗x2(c(1)(2))⊗x3(c(1)(1))(26)
=x1(c(2))⊗x2(c(1)(1))⊗x3(c(1)(2))(27)
=x1(c(3))⊗x2(c(1))⊗x3(c(2)).(28)
10
Finally, assuming that this identities hold, we check (3) using (5) and associativity in the
Heyneman-Sweedler convention as follows.
((x1, x2), x3) (c)(29)
= (x1, x2)(c(1))⊗x3(c(2))(30)
=x1(c(1)(1))⊗x2(c(1)(2))⊗x3(c(2))(31)
=x1(c(1))⊗x2(c(2)(1))⊗x3(c(2)(2))(32)
=x1(c(1))⊗(x2, x3)(c(2))(33)
= (x1,(x2, x3))(c).(34)
Now, to check axioms related to the monoidal unit, observe that for X=Y(D), the
presheaves {•} ×CXand X×C{•} evaluated on Cbecome both X(C)by the counit iden-
tities, since the only element of {•}(C) = Y(k{•})(C)is the counit of C.
Corollary 3. The Yonneda embedding of Coalg into qSet = PSh(Coalg)admits a structure
of a symmetric partial monoidal functor.
Proof. Let Cbe the full subcategory of representable presheaves in qSet. It is the essential
image of the Yonneda embedding Y. First, note that the compatibility of monoidal units
morphism under Y
{•} ≃
−→ Yk{•}
is total, where its C−component reads as
{•}(C) = {•} → Y(k{•})(C) = Coalg(C, k{•}),
• 7→ εC.
However, the binatural transformation
Y(D1)×Y(D2)→Y(D1⊗D2)
is only partial with the domain Y(D1)×CY(D2), whose C−component reads as
Y(D1)×CY(D2)(C)→Y(D1⊗D2)(C),
i.e. consists of pairs
(x1, x2)∈(Y(D1)×CY(D2))(C)⊆Y(D1)(C)×Y(D2)(C)
such that the map
(x1, x2) := c7→ x1(c(1))⊗x2(c(2))∈Vect(C, D1⊗D2)
belongs in fact to Coalg(C, D ⊗D′).
Checking compatibility with associativity and symmetry, based on Theorem 2, is left to the
reader.
11
2.2 Sets as quantum sets
The following theorem, relating topos of sets and the topos of quantum sets, could be used to
model a classical observer in the quantum world.
Theorem 4. The composition of the functor of linearization and the Yoneda embedding
Set k−
−−−−→ Coalg Y
−−−→ PSh(Coalg) =: qSet
is a symmetric partial-monoidal full embedding.
Proof. First, we observe that the linearization functor k−:Set →Coalg is symmetric strong
monoidal when relating the cartesian product of sets with the singleton as a monoidal unit with
the tensor product of coalgebras with a final coalgebra as a monoidal unit. To prove that it is
fully faithful, we compute the set of morphisms Coalg(kS, kT)to be a set of maps S→kT,
s7→ Pt∈Txt(s)twhere
xt1(s)xt2(s) = δt1,t2xt1(s),
X
t∈T
xt(s)=1.(35)
Since char(k)=0and khas no nontrivial idempotents, for every s∈Sthere exists a unique
t0∈Tsuch that xt(s) = δt,t0and then such αis a unique solution to (35). Let us define f:S→T
so that f(s) := t0. This is the natural inverse to the map Set(S, T )→Coalg(kS, kT),f7→
(s7→ f(s) = Pt∈Tδt,f(s).Since the Yoneda embedding is also fully faithful, the composition is
fully faithful as well. Since the linearization functor is (strong) monoidal and the Yoneda functor
is partial-monoidal, their composition is partial-monoidal as well. Checking compatibility with
symmetry is left to the reader.
This partial-monoidal full embedding Set →qSet can be described as follows in terms of
a distinguished internal algebra Fin the topos qSet where F(C) := Vect(C, k)is the dual
convolution algebra of the coalgebra Cbeing its predual. Since for every set Sthe set F(kS) =
Set(S, k)is the algebra of k-valued functions on Swith the argument-wise multiplication and
the constant 1 as a unit, we call Fthe internal quantum function algebra. Note that it is a
purely algebraic analog of the von Neumann algebra, since a von Neumann C*-algebra is defined
as one having the predual [33]. Note also that, by the existence and the universal property of the
cofree coalgebra C(k)on the vector space k[35], Fis representable by C(k), i.e. F=Y(C(k)).
Proposition 5. The internal quantum function algebra Fis partial commutative, i.e. the re-
stricted multiplication
F×CF→F, (36)
where Cis the partial monoidal full subcategory in qSet consisting of representable presheaves,
is commutative.
Proof. Take (x1, x2)∈(F×CF)(C). By (4) and (5),
x1(c(1))⊗x2(c(2)) = x1(c(2))⊗x2(c(1) )∈C(k)⊗C(k).(37)
12
Applying the universal linear map C(k)→kto both slots of the tensor product on either side and
then the commutative multiplication in kwe get, regarding now coalgebra maps xi:C→C(k)
as linear maps xi:C→k,
x1(c(1))x2(c(2)) = x1(c(2))x2(c(1) ) = x2(c(1))x1(c(2) )(38)
in kwhich proves that in F(C)
x1x2=x2x1.(39)
Proposition 6. The image of a set Tunder the above embedding Set →qSet is the quantum
complete set of orthogonal idempotents in Findexed by T, i.e.
Y(kT)(C) = n{xt}t∈T⊂F(C)xt1xt2=δt1,t2xt1,X
t∈T
xt= 1o.(40)
Proof. The proof repeats the proof of (35) with kSreplaced by an arbitrary C.
It does make sense to introduce the following notion of quantization. Given a set T, we call
the quantum set qT := Y(kT), the quantization of a set T. Note that the evaluation of a
quantum set on the final coalgebra k{•} defines a functor in the opposite direction qSet →Set
which can be ragarded as a retraction of a quantum set to its classical part, since qT (k{•}) = T.
2.2.1 Birkhoff–von Neumann’s quantum propositional calculus
Corollary 7. The quantization qΩof the subobject classifier Ω = {⊥,⊤}(a.k.a. the set of
truth values) in the topos Set under the above embedding Set →qSet is the quantum set of
idempotents in the internal quantum function algebra F=qkin qSet, i.e.
Y(kΩ})(C) = x∈F(C)x2=x.(41)
Proof. By Proposition 6, the right hand side of (40) consists now of two elements x⊥, x⊤∈F(C)
satisfying
x2
⊥=x⊥, x⊥x⊤=x⊤x⊥= 0, x2
⊤=x⊤, x⊥+x⊤= 1,(42)
and hence it is uniquely determined by x:= x⊤satisfying x2=x.
Note that Ωis a Boole algebra. The linearization functor applied to Boolean operations ∧,
∨transforms them into coalgebra maps, and after applying the Yoneda embedding, a partial
distributive lattice structure under the induced operations
Y(kΩ) ×CY(kΩ) →Y(kΩ⊗kΩ) →Y(k(Ω ×Ω)) →Y(kΩ).(43)
This realizes the Birkhoff-von Neumann idea of quantum propositions as projections in
a von Neumann algebra. In our formalism however, it is not enough to have them commuting
before one aplies the binary Boolean operations ∧and ∨, we need them first to form an admissible
pair (x1, x2)∈(Y(kΩ) ×CY(kΩ))(C)before we multiply them in the von Neumann-like partial
commutative internal quantum function algebra F.
13
2.2.2 Quantum states
The quantum propositional calculus is not enough in physical applications. When one interpretes
elements of Fas quantum observables, the important question about their expected values in a
given state arises. Ignoring for the time being positivity part of the story, we define states as
follows.
Definition 2.7. The set of states St(C)of a given coalgebra Cis an affine space of elements
c∈Csatisfying ε(c) = 1. We will call the affine space ⟨−⟩(C)of affine functions St(C)→k
the space of expectations on C. It is easy to see that they form a presheaf of affine spaces on
Coalg, and hence an internal affine space in qSet. Moreover, there is a tautological linear map
of internal affine spaces in qSet (here we suppress the forgetful functor from algebras to affine
spaces)
F→ ⟨−⟩,(44)
defined as a natural transformation of presheaves, for x∈F(C)given as
(C∋c7→ x(c)∈k)7→ (St(C)∋c7→ x(c)∈k)(45)
which we call expectation on states of C. Its evaluation at a given state will be called the
expected value of a given observable x∈F(C)in the state c∈St(C), and denoted by ⟨x⟩c.
In particular, an evaluation of an idempotent element of F(C)at a given state c∈St(C)is
a purely algebraic counterpart of the notion of probability of a quantum proposition at a
given state on the von Neumann algebra. In the internal language of qSet it is a composition
Y(kΩ) −→ F−→ ⟨−⟩ (46)
transforming the truth value into its probability as follows ⊤ 7→ ⟨x⊤⟩c.
2.2.3 Positivity
In modelling the probabilistic quantum-physical measurement we have to restrict the above
constructions to the category of ∗-coalgebras over the field of complex numbers k=C, i.e.
coalgebras equipped with an antilinear, involutive, the comultiplication flipping and the counit
preserving map (−)∗:C→Cas objects and ∗-preserving coalgebra maps as morphisms. To the
previous purely algebraic condition ε(c)=1on c∈Cto be a state, we have to add the positivity
condition ∆(c) = Pic∗
i⊗ci. Note that such states form a convex subset of the above complex
affine space of purely algebraic states, since for every t∈[0,1]
∆((1 −t)c0+tc1) = (1 −t)∆(c0) + t∆(c1) = (1 −t)X
i
c∗
0,i ⊗c0,i +tX
i
c∗
1,j ⊗c1,j
=X
i
(√1−t c0,i)∗⊗(√1−t c0,i ) + X
j
(√t c1,j )∗⊗(√t c1,j ).(47)
Then the induced antilinear, involutive, the multiplication reversing and the unit preserving
map (−)∗:F(C)→F(C),x∗(c) := x(c∗)makes F(C)a∗-algebra. A state cbecomes now a
14
positive functional on F(C), as it follows from the following calculation.
⟨x∗x⟩c= (x∗x)(c) = x∗(c(1))x(c(2)) = X
i
x∗(c∗
i)x(ci)
=X
i
x(c∗∗
i)x(ci) = X
i
x(ci)x(ci) = X
ix(ci2≥0.(48)
When on the linearization C{⊥,⊤} we define ∗in the way that ⊥∗=⊥,⊤∗=⊤, we obtain
x∗
⊥=x⊥,x∗
⊤=x⊤. Therefore, by (48),
⟨x⊤⟩c=⟨x2
⊤⟩c=⟨x∗
⊤x⊤⟩c≥0.(49)
The same holds for x⊥= 1 −x⊤, hence by ⟨1⟩c=ε(c)=1we have ⟨x⊤⟩c∈[0,1]. Therefore
⟨x⊤⟩ccan be interpreted as probability of the proposition x⊤in the state c.
2.2.4 Associative algebras as quantum algebras
The construction of the internal partial commutative algebra Fcan be regarded as a base quan-
tum algebra obtained by quantizing the base field k, , i.e. F=qk.. Let us explain it and show
that it extends naturally to quantization of arbitrary k-algebras.
Definition 2.8. For every unital associative k-algebra Aits Sweedler cofree coalgebra C(A)[35]
represents a presheaf qA := Y(C(A)) on Coalg
C7→ Coalg(C, C(A)) = Vect(C, A)
of algebras (with convolution multiplication), regarded as free algebras for a monad Vect(C, −)
on the category of k-algebras [24]. We call such a presheaf object quantization of the algebra
A. In this way we obtain a presheaf qAlg of categories on Coalg where
qAlgqA′, qA(C) := AlgVect(C,−)(Vect(C, A′),Vect(C, A)).(50)
Remark 1. Note that the term deformation quantization splits then into two stages; first,
the quantization of a (typically commutative) algebra in our sense, and then a deformation of
the thus obtained quantum algebra.
One could also note that, despite we use the same prefix qfor quantization of sets and
algebras, the quantization of an algebra differs from the quantization of its underlying set. The
appropriate relation between quantum algebras and quantum sets is provided by the following
proposition where Meas(A′, A)denotes Sweedler’s universal measuring coalgebra [35].
Proposition 8. The category qAlg of quantum algebras provides a canonical enrichment of the
category Alg of algebras in the category of quantum sets qSet as follows
qAlgqA′, qA=YMeas(A′, A).
Proof. By Definition 2.8 and the existence of the universal measuring coalgebra [35] we get
qAlgqA′, qA(C)(51)
:= AlgVect(C,−)(Vect(C, A′),Vect(C, A)) (52)
=Alg(A′,Vect(C, A)) (53)
=CoalgC, Meas(A′, A)(54)
=YMeas(A′, A)(C)(55)
15
where the second equality follows from the equivalence between the full subcategory of free
algebras in the Eilenberg-Moore category and the Kleisli category [21] [24].
Lastly, since the same arguments as in the proof of Proposition 5 can be applied to any
commutative k-algebra instead of kitself, we get the following proposition.
Proposition 9. If an algebra Ais commutative, qA is partial commutative, i.e. the quantum
partial operation
qA ×CqA −→ qA,
induced by multiplication in Aand the symmetric monoidal structure of the Yoneda embedding,
is commutative.
2.3 Quantum Universal Algebra
Motivated by the above case of associative algebras, below we will consider representable quantum
sets, represented by objects of any algebraic kind compatible with its coalgebra structure. The
above partial symmetric monoidal structure of the Yoneda embedding will make them presheaves
of partial algebras of that kind.
Definition 2.9. Aquantum operation of arity n∈Non a quantum set represented by a
coalgebra Dis a coalgebra map
ω:D⊗n→D.
Note that quantum operations together with the above partial monoidal structure of the
Yoneda embedding induce following partial operations on representable quantum sets, defined
as compositions
Y(D)×C· ·· ×CY(D)−→ Y(D⊗ · ·· ⊗ D)−→ Y(D).
Definition 2.10. Aquantum identity is a commutative diagram of morphisms of repre-
sentable quantum sets induced by quantum operations and structural morphism of the symmetric
monoidal category Coalg.
Example 2.3.1 (Binary operations).Every binary operation
◦:B⊗B→B, b ⊗b′7→ b◦b′
allows us to define a partial binary operation on Y(B)as a partially defined composition
(Y(B)×CY(B))(C)→Y(B⊗B)(C)→Y(B)(C)
(β, β ′)7→ c7→ β(c(1))⊗β′(c(2))7→ β◦β′:= c7→ β(c(1))◦β′(c(2)).
2.3.1 Quantum Boolean Algebras
The aim of extending the notion of Boolean algebra to its quantum counterpart is to provide an
algebraic semantics [5] for quantum propositional calculus beyond the Birkhoff–von Neumann
propositional calculus, by allowing quantum truth values. Postponing for a while the question
of the negation in a Boolean algebra, we start with bounded distributive lattices.
16
Definition 2.11. A representable quantum bounded distributive lattice is a quantum set
represented by a coalgebra Bequipped with coalgebra maps
k{•} ⊥
−→ B, k{•} ⊤
−→ B, B ⊗kB∧
−→ B, B ⊗kB∨
−→ B
satisfying commutativity of the following diagrams in Coalg, where τdenotes the transposition
of tensor factors.
Associativity
B⊗kB⊗kB B ⊗kB
B⊗kB B
∧⊗kB
B⊗k∧
∧
∧
B⊗kB⊗kB B ⊗kB
B⊗kB B
∨⊗kB
B⊗k∨
∨
∨
Identity
B⊗kB B
B⊗kk{•}
∧
∼
=
B⊗k⊤
B⊗kB B
B⊗kk{•}
∨
∼
=
B⊗k⊥
Commutativity
B⊗kB B ⊗kB
B
τ
∨ ∨
B⊗kB B ⊗kB
B
τ
∧ ∧
17
Absorbtion
B⊗kB⊗kB B ⊗kB
B⊗kB B
B⊗k∨
∧
∆⊗kB
B⊗kε
B⊗kB⊗kB B ⊗kB
B⊗kB B
B⊗k∧
∨
∆⊗kB
B⊗kε
Distributivity
B⊗kB⊗kB B ⊗kB
B⊗kB⊗kB⊗kB
B⊗kB⊗kB⊗kB
B⊗kB B
B⊗k∧
∆⊗kB⊗kB
∨
B⊗kτ⊗kB
∨⊗k∨
∧
B⊗kB⊗kB B ⊗kB
B⊗kB⊗kB⊗kB
B⊗kB⊗kB⊗kB
B⊗kB B
B⊗k∨
∆⊗kB⊗kB
∧
B⊗kτ⊗kB
∧⊗k∧
∨
b∨(b′∧b′′)=(b(1) ∨b′)∧(b(2) ∨b′′ ), b ∧(b′∨b′′)=(b(1) ∧b′)∨(b(2) ∧b′′ )
Definition 2.12. We say, that a representable quantum bounded distributive lattice is a repre-
sentable quantum Boolean algebra if there exists an involutive coalgebra map ¬:B→Bo
making the following diagrams in Vect commute.
B⊗kB B ⊗kB
Bk{•} B
B⊗kB B ⊗kB
B⊗k¬
∨
∆
ε
∆
⊤
¬⊗kB
∨
B⊗kB B ⊗kB
Bk{•} B
B⊗kB B ⊗kB
B⊗k¬
∧
∆
ε
∆
⊥
¬⊗kB
∧
We will refer to it as Complement axiom. It reads as the following system of identities
(¬(b(1))) ∨b(2) =ε(b)⊤,(¬(b(1))) ∧b(2) =ε(b)⊥,
b(1) ∨ ¬(b(2)) = ε(b)⊤, b(1) ∧ ¬(b(2) ) = ε(b)⊥.
18
where for simplicity of notation we use the following conventions:
b∨b′:= ∨(b⊗kb′)b∧b′:= ∧(b⊗kb′)
⊥:= ⊥(•)⊤:= ⊤(•)
The following proposition is the sanity check for these axioms, saying that they restrict to
classical ones for classical Boolean algebras.
Proposition 10. The linearization kBof a Boolean algebra Brepresents a quantum Boolean
algebra.
Proof. Let (B , ⊥,⊤,¬,∧,∨)be a Boolean algebra and kBits linearization.
It is enough to check the axioms on the set Bof generators of its linearization kB, and the
rest will follow from linearity.
Identity. Since k{•} is a monoidal unit we have have the natuaral isomorphism
kB⊗kk{•} ∼
=
−→ kB
b⊗k• 7→ b
On the other hand we have
B⊗k⊥:b⊗k• 7→ b⊗k⊥.
But from the definition of ∨and ⊥we get
∨(b⊗k⊥) = b∨ ⊥ =b
so the identity diagram commutes.
Commutativity. Let us chase the commutativity diagram:
b⊗kb′b′⊗kb
∨(b⊗kb′)∨(b′⊗kb)
τ
∨ ∨
But we know that
∨(b⊗kb′) = b∨b′=b′∨b=∨(b′⊗kb),
so the diagram commutes.
Distributivity. Again, we chase the relevant diagram:
19
b⊗kb′⊗kb′′ b⊗k∧(b′, b′′)
b⊗kb⊗kb′⊗kb′′
b⊗kb′⊗kb⊗kb′′
∨(b, ∧(b′, b′′))
∨(b, b′)⊗k∨(b, b′′)∧(∨(b, b′),∨(b, b′′ )).
B⊗k∧
∆⊗kB⊗kB
∨
B⊗kτ⊗kB
∨⊗k∨
∧
The distributive law of the classical Boolean algebra Bimplies that
∨(b, ∧(b′, b′′)) = b∨(b′∧b′′ )=(b∨b′)∧(b∨b′′ ) = ∧(∨(b, b′),∨(b, b′′))
so the diagram commutes. Commutativity of the second diagram follows similarly.
Complements.: Finally, let us look at complements: Thanks to the commutativity axiom, in the
first diagram it is enough to check only the upper half.
b⊗kb b ⊗k¬(b)
∨(b⊗k¬(b))
b• ⊤
B⊗k¬
∨
∆
ε⊤
From the complement axiom for the Boolean algebra Bit follows that
∨(b⊗k¬(b)) = b∨ ¬b=⊤,
hence this upper half does commute. Commutativity of the second diagram will follow in a
similar fashion.
Better still, many properties, when appropriately generalized, of classical Boolean algebras
also apply to representable quantum Boolean algebras.
Theorem 11. Negation in a representable quantum bounded distributive lattice is unique.
20
Proof. Assume we have two negation coalgebra maps ¬,e
¬:B→Bop. Using the following
axioms of a representable quantum bounded distributive lattice and the complement axiom for
the negation morphism we get
e
¬(b)
(Identity) = e
¬(b)∧ ⊤
(counit) = e
¬(b(1)ε(b(2))) ∧ ⊤
(linearity) = e
¬(b(1))∧ε(b(2))⊤
(complement) = e
¬(b(1))∧(b(2)(1) ∨¬(b(2)(2)))
(distributivity) = ( e
¬(b(1))(1) ∧b(2)(1))∨(e
¬(b(1))(2) ∧¬(b(2)(2)))
(anticoalg) = ( e
¬(b(1)(2))∧b(2)(1))∨(e
¬(b(1)(1))∧¬(b(2)(2))))
(renumerate) = (( e
¬(b(2))∧b(3))∨(e
¬(b(1))∧¬(b(4))))
(renumerate) = ( e
¬(b(2)(1))∧b(2)(2))∨(e
¬(b(1))∧¬(b(3)))
(complement) = (ε(b(2) )⊥ ∨ (e
¬(b(1))∧¬(b(3))))
(complement) = ((b(2)(1) ∧¬(b(2)(2) )) ∨(e
¬(b(1))∧¬(b(3))))
(renumerate) = ((b(2) ∧¬(b(3)(1))) ∨(e
¬(b(1))∧¬(b(3)(2))))
(anticoalg) = ((b(2) ∧¬(b(3))(2))) ∨(e
¬(b(1))∧¬(b(3))(1)))
(commutativity) = (( e
¬(b(1))∧¬(b(3))(1))∨(b(2) )∧¬(b(3))(2) ))
(commutativity) = ((¬(b(3))(1))∧e
¬(b(1))∨∧(¬(b(3))(2) ∧b(2)))
(distributivity) = (¬(b(3) )∧(e¬(b(1))∨b(2)))
(renumerate) = (¬(b(2))∧(e¬(b(1)(1))∨b(1)(2)))
(complement) = (¬(b(2) )∧ε(b(1))⊤)
(linearity) = (¬(ε(b(1) )b(2))∧ ⊤)
(counit) = (¬(b)∧ ⊤)
(identity) = ¬(b)
(56)
In the classical Boolean setting, by the complement axiom, the first de Morgan law
¬(b∨b′) = ¬(b)∧¬(b′),
implies its apparently weaker version
(b∨b′)∨(¬(b)∧ ¬(b′)) = (b∨b′)∨(¬(b∨b′)) = ⊤
(b∨b′)∧(¬(b)∧ ¬(b′)) = (b∨b′)∧(¬(b∨b′)) = ⊥.(57)
However, by uniquness of the complement argument, these two versions are equivalent. Similarly,
the second de Morgan law
¬(b∧b′) = ¬(b)∨¬(b′)
is equivalent to its weak version
(b∧b′)∧(¬(b)∨ ¬(b′)) = (b∧b′)∧(¬(b∧b′)) = ⊥
(b∧b′)∨(¬(b)∨ ¬(b′)) = (b∧b′)∨(¬(b∧b′)) = ⊤.(58)
21
In the next theorem we prove a quantum counterpart of the weak de Morgan laws.
Theorem 12. In every representable quantum Boolean algebra the following weak de Morgan
laws hold
•1st de Morgan law
(b(1) ∨b′
(1))∨(¬(b(2))∧ ¬(b′
(2))) = ε(b)ε(b′)⊤,
(b(1) ∨b′
(1))∧(¬(b(2))∧ ¬(b′
(2))) = ε(b)ε(b′)⊥.
•2nd de Morgan law
(b(1) ∧b′
(1))∧(¬(b(2))∨ ¬(b′
(2))) = ε(b)ε(b′)⊤,
(b(1) ∧b′
(1))∨(¬(b(2))∨ ¬(b′
(2))) = ε(b)ε(b′)⊥.
Proof. Using the axioms listed below we proceed as follows.
(b(1) ∨b′
(1))∨(¬(b(2))∧ ¬(b′
(2)))
(distr) = ((b(1) ∨b′
(1))(1) ∨ ¬(b(2))∧((b(1) ∨b′
(1))(2) ∨ ¬(b′
(2)))
(coalg) = ((b(1)(1) ∨b′
(1)(1))∨ ¬(b(2) )) ∧((b(1)(2) ∨b′
(1)(2))∨ ¬(b′
(2)))
(renumerate) = ((b(1) ∨b′
(1))∨ ¬(b(3))) ∧((b(2) ∨b′
(2))∨ ¬(b′
(3)))
(assoc &comm) = (b′
(1) ∨(b(1) ∨ ¬(b(3)))) ∧(b(2) ∨(b′
(2) ∨ ¬(b′
(3))))
(complement) = (b′
(1) ∨(b(1) ∨ ¬(b(3)))) ∧(b(2) ∨ε(b′
(2))⊤)
(linearity) = (b′
(1)ε(b′
(2))∨(b(1) ∨ ¬(b(3) ))) ∧(b(2) ∨ ⊤)
(counit) = (b′∨(b(1) ∨ ¬(b(3) ))) ∧(b(2) ∨ ⊤)
(⊤absorbs) = (b′∨(b(1) ∨ ¬(b(3)))) ∧(ε(b(2))⊤)
(linearity) = (b′∨(b(1) ∨ ¬(ε(b(2) )b(3)))) ∧ ⊤
(counit) = (b′∨(b(1) ∨ ¬((b(2) ))) ∧ ⊤
(identity) = b′∨(b(1) ∨ ¬(b(2)))
(complement) = b′∨ε(b)⊤
(linearity &⊤absorbs) = ε(b)ε(b′)⊤.
(59)
22
(b(1) ∨b′
(1))∧(¬(b(2))∧ ¬(b′
(2)))
(distr) = (b(1) ∧(¬(b(2))∧ ¬(b′
(2)))(1))∨(b′
(1) ∧(¬(b(2))∧ ¬(b′
(2)))(2))
(coalg) = (b(1) ∧(¬(b(2))(1) ∧ ¬(b′
(2))(1))) ∨(b′
(1) ∧(¬(b(2))(2) ∧ ¬(b′
(2))(2))))
(¬is anticoalg) = (b(1) ∧(¬(b(2)(2))∧ ¬(b′
(2)(2)))) ∨(b′
(1) ∧(¬(b(2)(1) ∧ ¬(b′
(2)(1))))
(ren, ass, com) = (b(1) ∧ ¬(b(2)(2) )) ∧ ¬(b′
(2))) ∨((b′
(1)(1) ∧ ¬(b′
(1)(2))∧ ¬(b(2)(1))))
(complement) = (b(1) ∧ ¬(b(2)(2) )) ∧ ¬(b′
(2)(2))) ∨(ε(b′
(1))⊥∧¬(b(2)(1))))
(⊥absorbs, linearity) = (b(1) ∧ ¬(b(2)(2))) ∧ ¬(ε(b′
(1))b′
(2))) ∨(ε(¬(b(2)(1)))⊥))
(counit) = (b(1) ∧ ¬(b(2)(2) )) ∧ ¬(b′)) ∨(ε(¬(b(2)(1)))⊥))
(linearity) = (b(1) ∧ ¬(ε((b(2)(1)))b(2)(2) )) ∧ ¬(b′)) ∨ ⊥
(counit) = (b(1) ∧ ¬(b(2) )) ∧ ¬(b′)) ∨ ⊥
(complement) = (ε(b)⊥∧¬(b′)) ∨ ⊥
(⊥absorbs) = ε(b)ε(b′)⊥(60)
For the weak second de Morgan law the proof is similar.
3 Quantum Quivers
Besides single coalgebras, one can consider diagrams of coalgebras, among which the most im-
portant are quantum quivers. Given a coalgebra D, the (co)opposite coalgebra is denoted by
Do. The following definition is adapted from the definition of the underlying quiver of a quan-
tum category explicitly defined by Chikhladze [10] inspired by Day-Street [12], to our context of
representable quantum sets.
Definition 3.1. Consider a pair of coalgebra maps ∂0:D1→Do
0,∂1:D1→D0satisfying for
every d1∈D1
∂0(d1(1))⊗∂1(d1(2)) = ∂0(d1(2))⊗∂0(d1(1) ).
Denote the representable presheaves E:= Y(D1),V:= Y(D0),Vo:= Y(Do
0)and call them
the representable quantum sets of edges,vertices and opposite vertices of the representable
quantum quiver, respectively.
Note that for every e∈E(C)we have sC(e) := ∂0◦e∈Vo(C)and tC(e) := ∂1◦e∈
V(C), which we call the source and the target of an edge e, respectively. Then, since the full
subcategory Cof qSet whose objects are representable presheaves is partial monoidal in the
sense of Definition 4, we have
(sC(e), tC(e)) ∈(V×CVo)(C).(61)
Now it is clear how to define quantum quiver, which is not necessarily representable. First, we
fix some partial-monoidal subcategory Cof qSet as a super-structure, next, given a quantum
set of vertices, we define the quantum set of opposite vertices as follows Vo(C) := V(Co), and
finally we complement it with natural transformations s:E→Voand t:E→Vsatisfying (61)
for every e∈E(C).
23
3.1 Correspondences
Definition 3.2. With any representable quantum set X=Y(C)as above we associate a cat-
egory CX:= Comodop
C, the opposite category of right C-comodules. With every morphism
f:X→Y,Y=Y(D)of representable quantum sets represented by a coalgebra map C→D
we associate a pair of adjoint functors
CX
⊣
CY.
f∗f∗
described as follows. The reader should beware that we work here with opposite categories of
comodules but we write the structural maps in the category of vector spaces without reversing
arrows. The right adjoint corestriction functor f∗is defined by the composition
f∗(M→M⊗C) := (M→M⊗C→M⊗D)
where the second composed arrow on the right hand side is induced by the coalgebra map C→D.
The left adjoint coinduction functor f∗is defined as
f∗(N→N⊗D) := (N□DC→N□DC⊗C)
where the arrow on the right hand side is induced by the comultiplication C→C⊗C.
Definition 3.3. We say that fis discrete if there is an adjunction f!⊣f∗.
Definition 3.3 is motivated by the following classical example.
Example 3.1.1. Let C:= kSfor a set S. Then the category CXis the opposite category
of vector spaces Mwith an S-decomposition M=⊕s∈SMs, with the C-comodule structure
M−→ M⊗Cof the form ms7→ ms⊗sfor every ms∈Ms, and morphisms being linear maps
preserving the S-decomposition. If f:S→Tis a map of sets, it is equivalent to the coalgebra
map f:C→D(therefore we use the same ffor denoting both) where D=kT, inducing the
following adjunctions f!⊣f∗⊣f∗for functors f!, f∗:CX→CY,f∗:CY→CX
(f!M)t=Y
s∈f−1(t)
Ms,(f∗N)s=Nf(s),(f∗M)t=M
s∈f−1(t)
Ms.
Note that when fibers of fare finite (such maps are called quasi-finite), f!=f∗hence f∗and
f∗form a Frobenius pair of functors [6].
Here is an example where we allow our sets to be quantum, representable by coalgebras.
Example 3.1.2. Every coalgebra map f:C→Dinduces an adjunction f∗⊣f∗of functors
f∗:CX→CY,f∗:CY→CXas above. If Tis a (C, D)-bicomodule such that there is an
isomorphism of functors cohomD(T, −)∼
=
−−→ − □DC, by the definition of cohom as a left adjoint
functor between comodule categories [37], the left adjoint f!:CX→CY,f!M=M□CT, to f∗
exists implying that fis discrete. If, in addition, Tis isomorphic to Cas a (C, D)-bicomodule,
we have f!=f∗, i.e. f∗and f∗form a Frobenius pair of functors.
24
Definition 3.4. By a quantum correspondence from Xto Ywe mean a morphism of repre-
sentable quantum sets of the form F−→ Y×CXo. The associated diagram in qSet
F
Y Xo
t s
defines the source and the target morphisms, sand t, respectively. We say that the correspon-
dence is a quantum multivaled map from Xto Y, if sis discrete. Any quantum multivaled
map from Xto Ybetween representable quantum sets represented by coalgebras DXand DY,
respectively, with Frepresented by a coalgebra DF, defines adjunctions s!⊣s∗⊣s∗and t∗⊣t∗
where s!, s∗:CF→CX,s∗:CX→CF,t∗:CY→CFand t∗:CF→CYas in Example
3.1.2 with Tbeing DFregarded at first as a right DY⊗Do
X-comodule via the coalgebra map
DF→DY⊗Do
X, and next as a (DX, DY)-bicomodule by regarding the right Do
X-coaction as a
left DX-coaction.
3.2 A categorification of the Cuntz-Pimsner algebra
Let Qbe an adjunction Q∗⊣Q∗between two endofunctors on some category C, with the unit
η: Id ⇒Q∗Q∗and counit ε:Q∗Q∗⇒Id natural transformations.
Definition 3.5. We define the stable category StQof the adjunction Qas follows. The
objects of StQare objects Nof Cbeing equipped with a retraction of Q∗Nonto Ninducing a
retraction of Q∗Q∗Nonto Q∗Nleaving the counit of the adjunction stable. Morphisms of StQ
are morphisms of Cpreserving the retraction.
The definition of making an object Nbelong to the stable category of the adjunction Q
can be rewritten as the existence of two morphisms ωN:N→Q∗Nand an opposite-directed
morphism σN:Q∗N N (by using the dashed arrow we stress its wrong-way map nature)
satisfying the equations
σN◦ωN= IdN, εN◦Q∗(ωN◦σN) = εN,(62)
while the retraction preserving morphisms ϕ:N→N′are those satisfying
Q∗(ϕ)◦ωN=ωN′◦ϕ, ϕ ◦σN=σN′◦Q∗(ϕ).(63)
On the other hand, given an adjunction Qas above, we can form the following monad
according to the formalism of [36].
Definition 3.6. We define the Cuntz-Pimsner monad of the adjunction Qas the quotient of
the free monad generated by the coproduct Q∗⊔Q∗of endofunctors by the congruence generated
by the following system of identities
σN◦Q∗(αN)◦ηN= IdN, αN◦Q∗(σN) = εN(64)
satisfied by pairs of morphisms αN:Q∗N→N,σN:Q∗N N.
25
Example 3.2.1. For a unital ring Blet us consider a B-bimodule Pwhich is finitely generated
projective as a right B-module. Then we have an adjunction Q∗⊣Q∗between two endofunctors
Q∗:= (−)⊗BPand Q∗:= Mod
B(P, −)∼
=(−)⊗BP∗
on the category of right B-modules. The unit of that adjunction expresses in terms of the dual
basis η:B→EndB(P)∼
=P⊗BP∗,b7→ bpi⊗Bp∗
i=pi⊗Bp∗
ibsatisfying the property
p=pi·p∗
i(p)(summation over isuppresed). The counit of that adjunction expresses in terms of
the evaluation ε:P∗⊗BP→B,p∗⊗Bp7→ p∗(p).
Definition 3.7. With a bimodule Pas above, we introduce the following quotient algebra of
the tensor algebra TBP⊕P∗of a B-bimodule P⊕P∗
OP:= TBP⊕P∗/b−η(b), p∗⊗Bp−ε(p∗⊗Bp).(65)
Let us note that (65) is an algebraic Cuntz-Pimsner ring in the sense of [8] with respect to the
canonical evaluation pairing ε:P∗⊗BP→B. The next proposition relates this Cuntz-Pimsner
ring to our Cuntz-Pimsber monad, justifying our terminology, as follows.
Theorem 13. The Cuntz-Pimsner monad of the adjunction defined by a finitely generated pro-
jective from the right B-bimodule Pas above is isomorphic to the Cuntz-Pimsner B-ring OP.
Proof. After denoting the maps αN:Q∗N→Nand σN:Q∗N N as the right actions
n⊗Bp7→ np and n⊗Bp∗7→ np∗, respectively, and next after expressing the unit of the
adjunction by the dual basis, the counit of the adjunction by the evaluation and by applying the
dual basis property, identities (64) read as
(npi) p∗
i=n, (n p∗) p =n·p∗(p)(66)
which means that the structure of an object of stable category on Nis tantamount to being a
right module over OP. The verification that morphisms in the stable category are equivalent to
morphisms of right OP-modules is routine.
Finally, given an adjunction between two endofunctors, the following easy monadicity result
relates our Cuntz-Pimsner monad to the aforementioned stable category. This theorem can
be regarded as a conceptual justification of somewhat ad hoc definition of the Cuntz-Pimsner
monad.
Theorem 14. The Eilenberg-Moore category of the Cuntz-Pimsner monad is canonically equiv-
alent to the stable category of the adjunction. Conversely, forgetting the retraction is monadic
and the resulting monad is the Cuntz-Pimsner monad.
Proof. The presentation of our Cuntz-Pimsner monad in terms of generators and relations (64)
implies immediately that its Eilenberg-Moore category is isomorphic to the category of objects
Nequipped with the morphisms αN:Q∗N→N,σN:Q∗N N satisfying (64) and morphisms
ϕ:N→N′satisfying
ϕ◦αN=αN′◦Q∗(ϕ), ϕ ◦σN=σN′◦Q∗(ϕ).(67)
Now the argument is that if αN:Q∗N→Nand ωN:N→Q∗Nare mates [36] under the
adjunction Q∗⊣Q∗, they determine each other by the formulas
ωN=Q∗(αN)◦ηN, αN=εN◦Q∗(ωN).(68)
26
Therefore
σN◦ωN=σN◦Q∗(αN)◦ηN, εN◦Q∗(ωN◦σN) = αN◦Q∗(σN)(69)
what makes the conditions (62) and (64) equivalent. and hence that objects in the stable category
and the Eilenberg-Moore category are equivalent. Moreover, by the first equation of (68) and
naturality of the unit ηwe get
Q∗(ϕ)◦ωN=Q∗(ϕ)◦Q∗(αN)◦ηN(70)
=Q∗(ϕ◦αN)◦ηN(71)
=Q∗(αN′◦Q∗(ϕ)) ◦ηN(72)
=Q∗(αN′)◦Q∗Q∗(ϕ)◦ηN(73)
=Q∗(αN′)◦ηN′◦ϕ(74)
=ωN′◦ϕ, (75)
while by the second equation of (68) and naturality of the counit εwe get
ϕ◦αN=ϕ◦εN◦Q∗(ωN)(76)
=εN′◦Q∗Q∗(ϕ)◦Q∗(ωN)(77)
=εN′◦Q∗(Q∗(ϕ)◦ωN)(78)
=εN′◦Q∗(ωN◦ϕ)(79)
=εN′◦Q∗(ωN)◦Q∗(ϕ)(80)
=αN′◦Q∗(ϕ),(81)
what proves equivalence of conditions (63) and (67) and hence that morphisms in the stable
category and the Eilenberg-Moore category coincide.
The converse part of the theorem follows from the first part.
By virtue of this theorem, the notion of the Cuntz-Pimsner monad for an adjunction is
therefore a categorification of the Cuntz-Pimsner construction in terms of the stable category.
3.3 A categorification of the quantum Leavitt path algebra
Definition 3.8. For a given representable quantum quiver (E, V, s, t)with dicrete (e.g, quasi-
finite) the source morphism s, we have a self-correspondence on Vdefined by two adjunctions
s!⊣s∗and t∗⊣t∗, where s!:CE→CVand t∗:CE→CV, as follows. It is easy to see
that then we have an adjunction Q:= (Q∗⊣Q∗)where Q∗:= t∗s∗and Q∗:= s!t∗. We call
the Cuntz-Pimsner monad of that adjunction the combinatorial Leavitt path monad of
(E, V , s, t).
The following example justifies this terminology, when one replaces the category of modules
over the algebra with local units, which is generated by orthogonal idempotents corresponding
to vertices of a quiver, by the opposite category of comodules over the linearized set of vertices.
Example 3.3.1. Let Bbe a k-algebra generated by orthogonal idempotents 1vcorresponding
to vertices v∈Vof a quiver (V, E, s.t)with the set of edges E. Let us define a B-bimodule P
generated by elements pecorresponding to edges e∈Eand the bimodule structure given by
pe·1v=δs(e)
vpe,1v·pe=δt(e)
vpe,(82)
and call it the algebraic auto-correspondence defined by the quiver.
27
Theorem 15. Whenever the source map of a classical quiver is quasi-finite, the Cuntz-Pimsner
monad of the algebraic auto-correspondence defined by the quiver is isomorphic to the the Leavitt
path algebra of that quiver.
Proof. Let us introduce elements p∗
eof the right dual P∗corresponding to edges of the quiver by
declaring that for any two edges e, f ∈E
p∗
e(pf) = δf
e1s(e).(83)
Since elements pegenerate Pas a right B-module, the following calculation
X
e
pe·p∗
e(pf) = X
e
pe·δf
e1s(e)=pf·1s(f)=pf.(84)
shows that the system (pe, p∗
e)e∈Eis a dual basis for Pregarded as a right B-module, proving
its projectivity.
The contragredient B-bimodule structure on P∗reads as
1v·p∗
e=δs(e)
vp∗
e, p∗
e·1v=δt(e)
vp∗
e.(85)
Since for squasi-finite Pis finitely generated projective as a right B-module over a ring Bwith
local units,
Q∗:= (−)⊗BPand Q∗:= Mod
B(P, −)∼
=(−)⊗BP∗(86)
is a pair of adjoint functors Q∗⊣Q∗. By Theorem 14 the corresponding Cuntz-Pimsner monad
is isomorphic to B-balanced tensoring from the right by the B-ring OP. The verification, based
on (82), (83) and (85), that OPis then the Leavitt path algebra [1] is left to the reader.
Remark 2. By the definition of Q∗:= t∗s∗and adjunction t∗⊣t∗, the morphism ωN:N→Q∗N
is equivalent to a morphism t∗N→s∗Nand hence can be regarded as a representation of the
quiver [16] which is completed to a retraction (ωN, σN)leaving the counit of the adjunction Q∗⊣
Q∗stable, and hence can be called stable representation of the quiver. Moreover, in a given
stable representation of a quiver forgetting about arrows and restricting the representation
to vertices only forgets also about stability. In this terminology, Theorem 14 specializes to the
following result.
Theorem 16. The Eilenberg-Moore category of the Leavitt path monad of a (quantum) quiver
is canonically equivalent to the category of stable representations of that quiver. Conversely,
forgetting about arrows in a stable representation of the quiver is monadic and the resulting
monad is the Leavitt path monad.
Remark 3. Let us stress that when defining our Q-stable category of a quantum quiver, in
opposite to the construction of a graph C*-algebra of a quiver, the Leavitt path algebra of a
quiver as in [8] or of a quantum quiver in the sense of [17], instead of the data like an algebra-
valued scalar product of a Hilbert-module, pairing of bimodules or adjacency matrices or their
quantum generalizations, we use the machinery of adjunctions which is mathematically more fun-
damental. Note that in the coalgebraic version we can weaken the row-finite condition (of finite
emission) appearing in the algebraic Cuntz-Krieger family in the construction of the absolute
Cuntz-Pimsner ring.
28
Acknowledgement
This research is part of the EU Staff Exchange project 101086394 Operator Algebras That One
Can See. The project is co-financed by the Polish Ministry of Education and Science under the
program PMW (grant agreement 5448/HE/2023/2).
The author sincerely acknowledges Jakub Zarzycki’s contribution to the study of quantum
Boolean algebras.
References
[1] G. Abrams, P. Ara, M. Siles Molina. Leavitt path algebras, Lecture Notes in Mathematics,
vol. 2191, Springer Verlag, 2017.
[2] A. L. Agore. Categorical constructions for Hopf algebras, Communications in Algebra 39
(2011), no. 4, 1476–1481.
[3] A. L. Agore. Limits of coalgebras, bialgebras and Hopf algebras, Proceedings of the American
Mathematical Society 139 (2011), no. 03, 855–855.
[4] T. Bates, D. Pask. C∗-algebras of labelled graphs, J. Operator Theory 57 (1) (2007) 207–226.
[5] W.J. Blok; D. Pigozzi. Algebraizable logics. American Mathematical Society, (1989).
[6] S. Caenepeel, G. Militaru, S. Zhu. Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius
type properties, Trans. Amer. Math. Soc. 349 (1997), 4311-4342.
[7] T.M. Carlsen. Cuntz–Pimsner C∗-algebras associated with subshifts, Internat. J. Math. 19
(2008) 47–70.
[8] T.M. Carlsen, E. Ortega. Algebraic Cuntz–Pimsner rings. Proceedings of the London Mathe-
matical Society, 103 (2011), 601-653.
[9] T.M. Carlsen, E. Ortega, E. Pardo. C∗-algebras associated to Boolean dynamical systems, J.
Math. Anal. Appl. 450 (2017) 727–768.
[10] D. Chikhladze. A category of quantum categories, Theory and Applications of Categories,
Vol. 25, No. 1 (2011), 1–37.
[11] J. Cuntz, W. Krieger. A class of C∗-algebras and topological Markov chains, Invent. Math.
56 (1980) 251–268.
[12] B. Day, R. Street. Quantum categories, star autonomy, and quantum groupoids, (2003)
[13] R. Exel. Inverse semigroups and combinatorial C∗-algebras, Bull. Braz. Math. Soc. 39 (2)
(2008) 191–313.
[14] R. Exel, M. Laca. Cuntz–Krieger algebras for infinite matrices, J. Reine Angew. Math. 512
(1999) 119–172.
[15] N. Fowler, M. Laca, I. Raeburn. The C∗-algebras of infinite graphs, Proc. Amer. Math. Soc.
128 (2000) 2319–2327.
[16] P. Gabriel. Unzerlegbare Darstellungen I, Manuscripta mathematica 6 (1972), 71-104.
29
[17] J. Graham, R. Goswami, J. Palin. Leavitt path algebras of quantum quivers, Arch. Math.
124 (2025), 29–48.
[18] T. Katsura. A construction of C∗-algebras from C∗-correspondences, Advances in Quantum
Dynamics, 173-182, Contemp. Math., 335, Amer. Math. Soc., Providence, RI, 2003.
[19] T. Katsura. On C∗-algebras associated with C∗-correspondences, J. Funct. Anal. 217 (2004),
366-401.
[20] T. Katsura. A class of C∗-algebras generalizing both graph C∗-algebras and homeomorphism
C∗-algebras, I – fundamental results, Trans. Amer. Math. Soc. 356 (11) (2004) 4287–4322.
[21] H. Kleisli. Every standard construction is induced by a pair of adjoint functors, Proc. Amer.
Math. Soc. 16 (1965), 544-546.
[22] A. Kumjian, D. Pask, I. Raeburn, J. Renault. Graphs, groupoids, and Cuntz–Krieger alge-
bras, J. Funct. Anal. 144 (2) (1997) 505–541.
[23] A. Kumjian and D. Pask. C∗-algebras of directed graphs and group actions, Ergod. Th. &
Dynam. Sys. 19 (1999), 1503–1519.
[24] S. MacLane. §VI.5 of: Categories for the Working Mathematician, Graduate Texts in Math-
ematics 5, Springer 1971.
[25] S. Mac Lane , I. Moerdijk. Sheaves in Geometry and Logic. A First Introduction to Topos
Theory. Springer (1994).
[26] K. Matsumoto. On C∗-algebras associated with subshifts, Internat. J. Math. 8 (1997)
357–374.
[27] P. Muhly, B. Solel. On the Morita equivalence of tensor algebras. Proceedings of London
Mathematical Society, 81(1) (2000), 113–168.
[28] R. Meyer, C. F. Sehnem. A bicategorical interpretation for relative Cuntz-Pimsner algebras.
Math Scand. 125 (2019), 84–112.
[29] P. Muhly, M. Tomforde. Topological quivers, Internat. J. Math. 16 (2005), 693–755.
[30] B. Musto, D. Reutter, D. Verdon. A compositional approach to quantum functions, J. Math.
Phys. 59, 081706 (2018).
[31] M. Pimsner. A class of C∗-algebras generalizing both Cuntz-Krieger algebras and crossed
products by Z, Free probability theory, 189-212, Fields Inst. Commun., 12, Amer. Math. Soc.,
Providence, RI 1997.
[32] I. Raeburn. Graph Algebras, CBMS Regional Conference Series in Mathematics 103 Pub-
lished for the Conference Board of the Mathematical Sciences, Washington D.C. by the AMS,
Providence, RI (2005).
[33] S. Sakai. C∗-Algebras and W∗-Algebras, Springer Berlin, Heidelberg 1997.
[34] G. Segal. Configuration-Spaces and Iterated Loop-Spaces. Inventiones mathematicae 21
(1973), 213-222.
[35] M. E. Sweedler. Hopf Algebras, Benjamin New York, 1969.
30
[36] R. Street. The Formal Theory of Monads. Journal of Pure and Applied Algebra 2(2) (1972),
149-168.
[37] M. Takeuchi. Morita theorems for categories of comodules, J. Fac. Sci. Univ. Tokyo 24
(1977), 629–644.
[38] M. Tomforde. A unified approach to Exel–Laca algebras and C∗-algebras associated to
graphs, J. Operator Theory 50 (2003), 345–368.
31
