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A Bohr topos (a notion going back to ideas of (Butterfield-Hamilton-Isham) and further expored in (Isham-Döring 07) and (Heunen-Landsman-Spitters 09)) is a topos associated with any quantum mechanical system which is such that the observables and states of the physical system are more or less naturally encoded in the internal logic of the topos. A detailed motivation/derivation of this from classical theorems on the foundations of standard quantum mechanics is at order-theoretic structure in quantum mechanics.

One might think of a Bohr topos as (part of) a formalization of the “coordination” of the physical theory of quantum mechanics, providing a formalized prescription of how to map the theory to propositions about (experimental) observables of the system. The internal logic of Bohr toposes has been argued (e.g. Heunen-Landsman-Spitters 09) to be a better formal context for such considerations than the old quantum logic going back to von Neumann.

The notion of Bohr topos is named after Niels Bohr, whose informal ideas about the nature of quantum mechanics (e.g. Scheibe 73) it is supposed to formalize, see at interpretation of quantum mechanics – Bohr’s standpoint.

Sometimes in the literature the discussion of Bohr toposes is referred to as “the topos-theoretic formulation of physics”, but actually Bohr toposes currently formalize but one aspect of quantum mechanics, namely “the quantum mechanical phase space” in the form of the quantum observables and the quantum states. The plain Bohr topos does not even encode any dynamics, though in the spirit of AQFT a certain presheaf of Bohr toposes on spacetime does encode dynamics (Nuiten 11). For other and more comprehensive usages of topos theory in the formalization of physics see at geometry of physics and at higher category theory and physics.


To every quantum mechanical system is associated its Bohr topos: a ringed topos which plays the role of the quantum phase space. The idea of this construction – Bohrification – is that it naturally captures the geometric and logical aspects of quantum physics in terms of higher geometry/topos theory.

One way to understand Bohrification is as a generalization of the construction of the Gelfand spectrum of commutative C-star algebras to a context of noncommutative C *C^*-algebras. It assigns to a noncommutative C *C^*-algebra AA essentially a system of Gelfand dual spaces to each of its commutative subalgebras. Together, this system yields a generalized Gelfand spectrum in the form of a locale Σ̲ A\underline{\Sigma}_A internal to the sheaf topos 𝒯 A\mathcal{T}_A over the poset of commutative subalgebras of AA, or equivalently its externalization Σ A𝒞(A)\Sigma_A \to \mathcal{C}(A) regarded as a locale over the locale of open subalgebras.

As a topos, the Bohr topos is just a presheaf topos, the topos of presheaves on this poset of commutative subalgebras of AA, but the point is that it is naturally a ringed topos with the original non-commutative algebra AA appearing as a commutative C *C^\ast-algebra internal to the Bohr topos. This allows to talk about quantum states on AA much like classical states, but internal to the Bohr topos.

In fact, under mild assumptions on AA, its poset of commutative subalgebras, and hence the Bohr topos over it, encodes precisely the Jordan algebra underlying AA. As discussed at Jordan algebra, this is precisely that part of AA which knows about the quantum observables themselves. In order to have the Bohr topos remember the full non-commutative algebra structure of AA, it needs to be equipped with the information about Hamiltonian flows induced on AA by automorphisms of the form exp(i[H,])\exp(i [H,-]), where [,][-,-] is the commutator (that gets discarded as one passes to the Jordan algebra). According to Andreas Döring (private communication at MPI Bonn, April 2013), this can be formulated nicely in topos theory.

More detailed

The formalization of the notion quantum mechanical system (see there for details) with its states and observables is the following.

  • The system as such is encoded by a C-star algebra AA;

  • a self-adjoint operator aAa \in A is thought of as being an observable of the system: a kind of observation that one can make about it;

  • a \mathbb{C}-linear map ρ:A\rho : A \to \mathbb{C} (which is “positive” and “normalized”) is thought of as being a state that the system can be in – a physical configuration (or rather: a probability distribution of such);

  • the pairing (a,ρ)ρ(a)(a, \rho) \mapsto \rho(a) \in \mathbb{C} is thought of as being the value of the observable aa made on the system in state ρ\rho: for instance the total energy of the system as measured in some chosen unit;

  • a one-parameter group of automorphisms Aut(A)\mathbb{R} \to Aut(A) (inner automorphisms for “localized” systems, see below) is thought of as being an evolution of the system, for instance in time or more general in spacetime.

(Often in the literature quantum mechanical systems are instead dually conceived of in terms of Hilbert spaces of pure states. The relation between these two descriptions is established by the GNS-construction which allows to pass from one to the other.)

Notice that these axioms are naturally thought of as exhibiting the C *C^\ast-algebra AA as a formal dual of the would-be quantum phase space of the physical system – not quite an ordinary topological space (which by Gelfand duality it would be if AA were commutative) but still a kind of generalized space. Traditionally it is common to regard this as a space in noncommutative geometry. Notice that if we do so – hence if we regard the object SpecAC *Alg opSpec A \in C^\ast Alg^{op} that is the incarnation of AC *AlgA \in C^\ast Alg in the opposite category – then the definition of observable above reduces to the evident notion of real-valued functions on quantum phase space: such a function is a morphism SpecASpec A \to \mathbb{R} in C *Alg opC^\ast Alg^{op}, which dually is a C *C^\ast-algebra homomorphism C() 0AC(\mathbb{R})_0 \to A (where on the left we have the unitization of functions with compact support). That such correspond to self-adjoint operators of AA is the statement of functional calculus for C *C^\ast-algebras.

More subtle is the interpretation of the axiom for states. Historically this had been been subject to some discussion: by the spectral theorem two different observables a 1,a 2Aa_1, a_2 \in A have a compatible set of observable values if and only if these elements commute with each other in AA. Generally, a set {a i} i\{a_i\}_i of observables has a jointly consistent set of observable values if and only if the sub-C *C^\ast-algebra {a i} iA\langle \{a_i\}_i\rangle \subset A generated by them is commutative. Therefore for the phenomenological interpretation of the axioms it seems to make no sense to demand that a state ρ:A\rho : A \to \mathbb{C} be linear on non-commuting observables: if a 1a_1 and a 2a_2 do not commute, it is not a-priori clear that it makes sense to require that ρ(a 1+a 2)=ρ(a 1)+ρ(a 2)\rho(a_1 + a_2) = \rho(a_1) + \rho(a_2). This might experimentally fail, and hold only for commuting a 1,a 2a_1, a_2.

Therefore the notion of quasi-state was introduced: a quasi-state on AA is defined to be a (positive and normalized) function ρ:A\rho : A \to \mathbb{C} which is required to be \mathbb{C}-linear only on all commutative subalgebras of AA. Operationally, quasi-states should be the genuine states!

One would therefore tend to think that the terminology has been chosen in an unfortunate way. While maybe true, it turns out – non-trivially – that in a major class of cases of interest the distinction does not matter: namely Gleason's theorem states that for HH a separable complex Hilbert space with dimH>2dim H \gt 2 and A=B(H)A = B(H) the C *C^\ast-algebra of bounded operators on H\H, all quasi-states on AA are automatically states: a function that is linear on all commutative subalgebras is automatically also linear on all of AA.

While this means that the distinction between states and quasi-states disappears in a major case of interest, it does not disappear in all cases of interest. In particular, other foundational theorems about quantum mechanics concern the collection of commutative subalgebras, too.

Notably, one may wonder about the evident strengthening of the notion of quasi-states to that of a map ρ:A\rho : A \to \mathbb{C} which is not just linear but also an algebra homomorphism on each commuting subalgebra. Notice that by Gelfand duality every commutative C *C^\ast-algebra CC is the algebra of continuous functions on some topological space sp(C)sp(C). Under this duality a state on CC is a probability distribution on sp(C)sp(C), while an algebra homomorphism CC \to \mathbb{C} is a point of X CX_C. Therefore a quasi-state which is commutative-subalgebra-wise an algebra homomorphism may be thought of as encoding a collection of precise numerical values (as opposed to just expectation values) of all possible observables. Such a hypothetical quasi-state is sometimes called a collection of hidden variables of the quantum mechanical system: its existence would mean that despite the apparent probabilistic nature of quantum mechanics, there are “hidden” non-probabilistic states. But there are not. This is the statement of the Kochen-Specker theorem: under precisely the assumptions that make Gleason's theorem work, there is no quasi-state which is commutative-subalgebra-wise an algebra homomorphism.

In summary, this means:

The foundational characteristics of quantum physics are encoded in notions of functions on the algebra of observables AA which are linear and positive only commutative-subalgebra-wise .

Since the notion of commutative-subalgebra-wise homomorphism is at the heart of quantum physics, it seems worthwhile to consider natural formalizations of this notion. There is indeed a very natural and general abstract one: whenever any notion of function is defined only locally it is natural to consider the sheaf of such functions over all possible local patches.

The historically motivating example, and possibly still the most widely familiar one, is that of holomorphic functions on a complex manifold: there are in general very few holomorphic functions defined over all of a complex manifold, but plenty of them defined over any small enough subset. And it is of fundamental interest to consider the collections of holomorphic functions over each such subset, and how these restrict to each other under restriction of subsets. This collection of local data is a sheaf of functions on the complex manifold.

There is an evident analog setup of this situation that applies in the present case of interest, that of functions defined on commutative subalgebras:

For AA any C-star algebra, write 𝒞(A)\mathcal{C}(A) for the set of all its commutative C *C^\ast-subalgebras. This is naturally a poset under inclusion of subalgebras. A (co)presheaf of this set is a functor 𝒞(A)Set\mathcal{C}(A) \to Set. Any such functor we may think of as a collection of commutative-subalgebra-wise data on AA, consistent with restriction of subalgebras. The collection of all such functors – which we write [𝒞(A),Set][\mathcal{C}(A), Set] – is a category called a presheaf topos.

Inside this topos, all the above discussion of foundations of quantum mechanics finds a natural simple equivalent reformulation:

First of all, the non-commutative C *C^\ast-algebra AA naturally induces a commutative C-star algebra object A̲\underline{A} internal to [𝒞(A),Set][\mathcal{C}(A), Set]: namely the copresheaf defined by the tautological assignment

A̲:(C𝒞(A))C. \underline{A} : (C \in \mathcal{C}(A)) \mapsto C \,.

In words this means nothing but that the collection of all commutative subalgebras of AA may naturally be regarded as a single commutative C *C^\ast-algebra internal to the topos [𝒞(A),Set][\mathcal{C}(A), Set].

Below we shall discuss (here) that in a precise sense this commutative internal A̲\underline{A} captures precisely all the kinematical information encoded in the quantum mechanical system of AA – everything related to states and observables but not information about (time) evolution. So everything we have discussed so far.

The pair of these two ingredients

Bohr(A):=([𝒞(A),Set],A̲) Bohr(A) := ([\mathcal{C}(A), Set], \underline{A})

constitutes what is called a ringed topos – a special case of the notion of a locally ringed topos. This notion is a fundamental notion for generalized spaces in higher geometry. The most advanced general theory of higher geometry (Lurie09) is based on modelling spaces as ringed toposes.

We shall call this ringed topos the Bohr topos of AA.

This terminology is meant to indicate that one may think of this construction as formalizing faithfully and usefully a heuristic that has been emphasized by Nils Bohr – one of the founding fathers of quantum mechanics – and is known as the doctrine of classical concepts (Scheibe) in quantum mechanics. This states that nonclassical/noncommutative as the logic/geometry of quantum mechanics may be, it is to be probed and detected by classical/commutative logic/geometry.

Namely in terms of the Bohr topos we have the following equivalent reformulations of the foundational facts about quantum physics discussed above, now internally in Bohr(A)Bohr(A).

Consistent quantum mechanical states. A quasi-state on AA is precisely an ordinary classical state on A̲\underline{A}, internal to Bohr(A)Bohr(A).

In particular (Gleason's theorem): if A=B(H)A = B(H) for dimH>2dim H \gt 2 then a quantum state on the external AA is precisely a classical state on the internal A̲\underline{A}.


Non-existence of hidden quantum variables. The Gelfand spectrum sp(A̲)sp(\underline{A}) of A̲\underline{A} internal to the Bohr topos has no global point. (Kochen-Specker).

These two statements might be taken as suggesting that a quantum mechanical system AA is naturally regarded in terms of its Bohr topos Bohr(A)Bohr(A) – somewhat more naturally than as a C *C^\ast-algebra AA. (The second, in a slightly different setup, was emphasized in IshamHamiltonButterfield, which inspired all of the following discussion, the first in Spitters). In fact, thinking of ringed toposes as generalized spaces in higher geometry, it suggests that the Bohr topos Bohr(A)Bohr(A) itself is the quantum phase space of the quantum mechanical system in question.

To which extent this perspective is genuinely useful is maybe still to be established. For pointers to the literature see the references below. Discussion along the above lines may suggest that this perspective is indeed useful, but what is probably still missing is a statement about quantum physics that can be formulated and proven in terms of Bohr toposes, while being hardly conceivable or at least more unnatural without. It is probably currently not clear if such statements have been found.

One potential such statement has been suggested in (Nuiten) after discussion with Spitters:

In the formalization of quantum field theory by the Haag-Kastler axioms – called AQFT – every quantum field theory is entirely encoded in terms of its local net of observables over spacetime XX. This is a copresheaf of C-star algebras

A:Op(X)C *Alg A : Op(X) \to C^\ast Alg

which assigns to every open subset UXU \subset X of spacetime the quantum subsystem A UA_U of quantum fields supported in that region. By the above, we may consider for each of these quantum systems their associated quantum phase spaces given by the correspondong Bohr toposes Bohr(A U)Bohr(A_U). This yields a presheaf

Bohr(A):Op(X) opC com *TopSpaceC com *Topos Bohr(A) : Op(X)^{op} \to C^\ast_{com} TopSpace \hookrightarrow C^\ast_{com}Topos

of ringed toposes whose internal ring object has the structure of a commutative C *C^\ast-algebra. With the copresheaf thus turned into a presheaf it is natural to ask under which conditions this is a sheaf: under which conditions this presheaf satisfies descent.

In (Nuiten) the following is observed: if AA satisfies what is called the split property (a strong form of the time slice axiom) then the Bohr-presheaf of quantum phase spaces satisfies spatial descent by local geometric morphisms precisely if the original copresheaf of observables A:Op(X)C *AlgA : Op(X) \to C^\ast Alg is indeed local – spatially and causally. So this means that a natural property of quantum physics – spatial and causal locality – corresponds from the perspective of Bohr toposes to a natural property of presheaves of quantum phase spaces: descent.

One can probably view this as further suggestive evidence that indeed quantum physics is naturally regarded from the point of view of the Bohr topos. But for seeing where this perspective is headed, it seems that more insights along these lines would be useful.


The discussion below proceeds in the following steps (following (Nuiten11))

  1. Bohr topos of a quantum mechanics system

    This discusses the Bohr topos incarnation of a quantum mechanical system – the topos-theoretic quantum phase space – and its functoriality.

  2. Kinematics in a Bohr topos

    This discusses how the classical kinematics internal to a Bohr topos is the external quantum kinematics of the underlying quantum mechanical system.

  3. (Pre-)Sheaf of Bohr toposes of a quantum field theory

    This discusses how the presheaves of Bohr toposes obtained by applying Bohrification to a local net of observables of a quantum field theory.

Bohr topos of a quantum mechanical system

We discuss the definitions and some basic properties of Bohr toposes: certain ringed toposes – in fact ringed spaces – associated with any (possibly non-commutative) algebra. We formulate the construction for C-star algebras, since this is the standard model for quantum mechanical systems, but actually much of it does not depend on either the topology or the star-algebra structure until we come to the discussion of the kinematics in a Bohr topos below.

C *C^*-algebras


A C-star algebra is (…)


For AA a star-algebra, an element aAa \in A is called a normal element if a *a=aa *a^* a = a a^*.


Every element of a C *C^*-algebra is the sum of two normal elements, because

a=12((a+a *)+(aa *)). a = \frac{1}{2} \left( \left( a + a^* \right) + \left( a - a^* \right) \right) \,.

This means whenever a linear morphism between the vector spaces underlying two C *C^*-algebras is defined on normal elements, it is already defined on all elements. This will be used in several of the arguments below.


A partial C *C^\ast-algebra is a set AA equipped with

  • a symmetric and reflexive binary relation CA×AC \subset A \times A;

  • elements 0,1A0,1 \in A;

  • an involution *:AA\ast : A \to A;

  • a function ()():×AA(-)\cdot (-) : \mathbb{C} \times A \to A;

  • a function :A\Vert-\Vert : A \to \mathbb{R};

  • (partial) binary operations +,×:CA+, \times : C \to A

such that every set SAS \subset A of elements that are pairwise in CC is contained in a set TAT \subset A whose elements are also pairwise in CC and on which the above operations yield the structure of a commutative C-star algebra.

A homomorphism of partial C *C^\ast-algebra is a function preserving this structure. This defines a category PCstarPCstar of partial C *C^\ast algebras.

This appears as (vdBergHeunen, def. 11,12).


For AA a C-star algebra, write

N(A):={aAaa *=a *a} N(A) := \{a \in A | a a^* = a^* a\}

for its set of normal operators. This is naturally a partial C-star algebra with CN(A)×N(A)C \subset N(A) \times N(A) the set of pairs of elements that commute in AA.

The Bohr site


For AA a partial C-star algebra write 𝒞(A)\mathcal{C}(A) for the poset of total (not partial) commutative sub C-star algebras. We call this the poset of commutative subalgebras.


This construction extends to a functor

𝒞:C *AlgPoset. \mathcal{C} : C^* Alg \to Poset \,.

to the category Poset of posets: for f:ABf : A \to B a homomorphism we let 𝒞(f)\mathcal{C}(f) be over any C𝒞(A)C \in \mathcal{C}(A) be the image im(f C)C *Algim(f|_C) \in C^\ast Alg of the restriction of ff to CC.


By standard properties of C-star algebras (see here), this image im(f C)im(f|_C) is simply the set-theoretic image f(C)f(C).

Notice the following fact about Alexandroff spaces:


The functor

Alex:PosetTop Alex : Poset \to Top

from posets to topological spaces that sends a poset PP to the topological space whose underlying set is the underlying set of PP and whose open subsets are the upward closed subsets Up(P)Up(P) exhibits an equivalence of categories

PosetAlexAlexTopTop Poset \underoverset{\simeq}{Alex}{\to} AlexTop \hookrightarrow Top

of Poset with the full subcategory of Alexandroff spaces.


For AC *AlgA \in C^* Alg we call Alex𝒞(A)Alex \mathcal{C}(A) the Bohr site of AA.


A morphism f:ABf : A \to B in C *AlgC^* Alg is called commutativity reflecting if for all a 1,a 2Aa_1, a_2 \in A we have that if f(a 1)f(a_1) commutes with f(a 2)f(a_2) in BB then already a 1a_1 commutes with a 2a_2 in AA.


C *Alg crC *Alg C^* Alg_{cr} \subset C^* Alg

for the subcategory of C-star algebras on the commutativity-reflecting morphisms.


Every monomorphism ABA \hookrightarrow B in C *AlgC^* Alg is commutativity reflecting.


A morphism f:ABf : A \to B in C *AlgC^* Alg is commutativity reflecting precisely if the morphism 𝒞(f)\mathcal{C}(f) has a right adjoint

(𝒞(f)R f):𝒞(A)𝒞(f)R f𝒞(B). (\mathcal{C}(f) \dashv R_f) \;\colon\; \mathcal{C}(A) \stackrel{\overset{R_f}{\leftarrow}}{\underset{\mathcal{C}(f)}{\to}} \mathcal{C}(B) \,.

This appears as (Nuiten, lemma 2.6).



C *Alg incC *Alg crC *Alg C^* Alg_{inc} \subset C^* Alg_{cr} \subset C^* Alg

for the subcategories of C *AlgC^* Alg on the monomorphisms and on the commutativity-reflecting morphisms, respectively.

The Bohr topos



C com *TopC com *Topos C^*_{com} Top \hookrightarrow C^*_{com} Topos

for the categories of ringed spaces and ringed toposes, where the internal ring object is equipped with the structure of an internal commutative C-star algebra and the morphisms respect the C *C^*-algebra structure.

Hence a morphism (,A̲)(,B̲)(\mathcal{E}, \underline{A}) \to (\mathcal{F}, \underline{B}) in C com *ToposC^\ast_{com} Topos is

  1. a geometric morphism f:f : \mathcal{E} \to \mathcal{F} such that the inverse image f *B̲f^* \underline{B} is still an internal C *C^\ast-algebra in \mathcal{E};

  2. and a morphism of internal C *C^\ast-algebras

    f *B̲A̲ f^\ast \underline{B} \to \underline{A}

    in \mathcal{E}.


For AC *AlgA \in C^* Alg the Bohr topos of AA is the C *C^\ast-space/topos

Bohr(A):=(Sh(Alex(𝒞(A))),A̲)C com *TopC com *Topos Bohr(A) := ( Sh(Alex(\mathcal{C}(A))), \underline{A}) \in C^\ast_{com} Top \hookrightarrow C^\ast_{com} Topos

whose underlying topological space is (that corresponding to) the Bohr site, and whose internal C *C^\ast-algebra is the tautological copresheaf

A̲:(C𝒞(A))C \underline{A} : (C \in \mathcal{C}(A)) \mapsto C

in [𝒞(A),Set]Sh(Alex(𝒞(A)))[\mathcal{C}(A), Set] \simeq Sh(Alex(\mathcal{C}(A))) equipped with the evident objectwise commutative C *C^\ast-algebra structure.

Moreover, write

Bohr ¬¬(A):=(Sh ¬¬(Alex𝒞(A)),A̲) Bohr_{\not \not}(A) := (Sh_{\not \not}(Alex \mathcal{C}(A)), \underline{A})

for the C *C^*-topos whose underlying sheaf topos is that for the double negation topology on the plain Bohr topos.

The general notion of morphisms between toposes are geometric morphisms. But those that remember the morphisms of Bohr sites are essential geometric morphisms.


Every functor f:𝒞(A)𝒞(B)f : \mathcal{C}(A) \to \mathcal{C}(B) induces an essential geometric morphism

(f !f *f *):[𝒞(A),Set]f *:=Ran ff *:=()ff !:=Lan f[𝒞(B),Set] (f_! \dashv f^* \dashv f_*) : [\mathcal{C}(A), Set] \stackrel{\overset{f_! := Lan_f}{\longrightarrow}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f_* := Ran_f }{\longrightarrow}}} [\mathcal{C}(B), Set]

where Lan fLan_f and Ran fRan_f are left and right Kan extension along ff, respectively.

We also write [f,Set]:[𝒞(A),Set][𝒞(B),Set][f,Set] : [\mathcal{C}(A), Set] \to [\mathcal{C}(B), Set] for this. Notice that by the equivalence of copresheaves on posets and sheaves on the corresponding Alexandroff locales (see there for details) this is equivalently

Sh(Alex(f))):Sh(Alex𝒞(A))Sh(Alex𝒞(B)). Sh (Alex(f))) : Sh(Alex \mathcal{C}(A)) \to Sh(Alex \mathcal{C}(B)) \,.

The next proposition asserts that all essential geometric morphisms between Bohr toposes arise this way:


The 2-functor

[,Set]:PosetTopos ess [-,Set] : Poset \hookrightarrow Topos_{ess}

is a full and faithful 2-functor.

Analogously, essential geometric morphisms of the underlying toposes Bohr(A)Bohr(B)Bohr(A) \to Bohr(B) are precisely those in image under the functor ShAlexSh \circ Alex of functors 𝒞(A)𝒞(B)\mathcal{C}(A) \to \mathcal{C}(B).


By the discussion in the section In terms of essential geometric morphisms at Cauchy complete category we have a full and faithful embedding of Cauchy-complete catgeories [,Set]:Cat CauchyTopos ess[-,Set] : Cat_{Cauchy} \hookrightarrow Topos_{ess}. But posets are trivially Cauchy, complete, hence this restricts to an embedding [,Set]:PosetCat CauchyTopos ess[-,Set] : Poset \hookrightarrow Cat_{Cauchy} \hookrightarrow Topos_{ess}.

In terms of Alexandroff topologies: by the discussion of Alexandroff locales (in the entry Alexandroff space) we have that the functor Alex:PosetLocaleAlex\colon Poset \to Locale takes values precisely on those morphisms of locales whose inverse image has a left adjoint. The statement then follows using the properties of localic reflection, which says that the 2-functor Sh:LocaleToposSh : Locale \to Topos is a full and faithful 2-functor.

For such essential geometric morphisms to be parts of morphisms in C *ToposC^\ast Topos, def. 9 we need that their inverse images respect internal C *C^\ast-algebras:


For f:Bohr(A)Bohr(B)f : Bohr(A) \to Bohr(B) an essential geometric morphism, the inverse image f *B̲f^\ast \underline{B} is naturally a C *C^\ast-algebra in [𝒞(A),Set][\mathcal{C}(A), Set].


According to (HLS09, 4.8) every functor 𝒞(A)C *Alg\mathcal{C}(A) \to C^\ast Alg is an C *C^\ast-algebra internal to [𝒞(A),Set][\mathcal{C}(A), Set]. Here f *B̲f^* \underline{B} is such a functor, sending (C𝒞(A))im f(C)(C \in \mathcal{C}(A)) \mapsto im_f(C).

Using this we now discuss morphisms of Bohr toposes in C *ToposC^\ast Topos.


The construction of Bohr toposes from def. 10 extends to a functor of the form

Bohr:C *Alg cr opC com *TopSpaceC com *Topos Bohr : C^\ast Alg_{cr}^{op} \to C^\ast_{com} TopSpace \hookrightarrow C^\ast_{com} Topos

with the special property that any f:ABf : A \to B is sent to

  • an essential geometric morphisms (f *f *):Bohr(B)Bohr(A)(f^* \dashv f_*) : Bohr(B) \to Bohr(A) with an extra right adjoint

    Bohr(B)f !f *f *f !Bohr(A) Bohr(B) \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\stackrel{\overset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}} Bohr(A)
  • such that the corresponding internal homomorphism of internal algebras A̲f *B̲\underline{A} \to f_* \underline{B} is an epimorphism.

This is (Nuiten, lemma 2.7). (Essentially this argument also appears as (vdBergHeunen, prop. 33), where however the extra right adjoint is not made use of and instead the variances of the morphisms involved in the definition of C *ToposC^\ast Topos are redefined in order to make the statement come out.)


To a morphism f:ABf : A \to B in C *Alg crC^\ast Alg_{cr} which by prop. 3 corresponds to an adjoint pair

(𝒞(f)R f):𝒞(A)𝒞(f)R f𝒞(B) ( \mathcal{C}(f) \dashv R_f) : \mathcal{C}(A) \stackrel{\overset{R_f}{\leftarrow}}{\underset{\mathcal{C}(f)}{\to}} \mathcal{C}(B)

we assign the essential geometric morphism

(f !f *f *f !):=[𝒞(B),Set]f !:=Ran 𝒞(f)f *:=()𝒞(f)f *:=()R ff !:=Lan R f[𝒞(A),Set] (f_! \dashv f^* \dashv f_* \dashv f^!) := [\mathcal{C}(B), Set] \stackrel{\overset{f_! := Lan_{R_f}}{\longrightarrow}}{\stackrel{\overset{f^* := (-)\circ R_f}{\leftarrow}}{\stackrel{\overset{f_* := (-)\circ \mathcal{C}(f)}{\longrightarrow}}{\underset{f^! := Ran_{\mathcal{C}(f)}}{\leftarrow}}}} [\mathcal{C}(A), Set]

(wher LanLan and RanRan denote left and right Kan extension, respectively) equipped with the morphism of internal C *C^*-algebras

η f:A̲f *B̲ \eta_f : \underline{A} \to f_* \underline{B}

which over C𝒞(A)C \in \mathcal{C}(A) is the restriction of ff to CC and corestriction to 𝒞(f)(C)\mathcal{C}(f)(C)

f C:C𝒞(f)(C) f|_C : C \to \mathcal{C}(f)(C)

(to the C *C^\ast-completion of the algebraic image of f Cf|_C).

Using prop. 4 the above prop. 6 has the following partial converse.


For A,BC *AlgA, B \in C^\ast Alg, morphisms f:Bohr(B)Bohr(A)f : Bohr(B) \to Bohr(A) in C *TopC^\ast Top for which

  1. the underlying geometric morphism has an extra right adjoint

  2. the morphism of internal algebras A̲f *B̲\underline{A} \to f_* \underline{B} is an epimorphism

are in bijection with functions

f:AB f : A \to B

that are homomorphisms on all commutative subalgebras and reflect commutativity.

In particular when AA is already commutative, morphisms Bohr(B)Bohr(A)Bohr(B) \to Bohr(A) with an extra right adjoint and epi ring homomorphism are in bijection with algebra homomorphisms ABA \to B.


By prop 4 every essential geometric morphism Sh(Alex𝒞(A))Sh(Alex𝒞(B))Sh(Alex \mathcal{C}(A)) \to Sh(Alex \mathcal{C}(B)) comes from a morphism of locales Alex𝒞AAlex𝒞AAlex \mathcal{C}A \to Alex \mathcal{C}A, which by the discussion at Alexandroff space is equivalently a morphism of posets 𝒞(f):𝒞(A)𝒞(B)\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B). By the assumption of the extra right adjoint we also have a geometric morphism the other way round, and hence, again by prop. 4, an adjoint pair

(𝒞(f)R f):𝒞(A)𝒞(B) (\mathcal{C}(f) \dashv R_f) : \mathcal{C}(A) \leftrightarrow \mathcal{C}(B)

that induces functors between toposes as in prop. 6. Then the fact that ff is a morphisms of C *C^\ast-toposes implies algebra homomorphisms

f C:Cf(C) f_C : C \to f(C)

natural in C𝒞(A)C \in \mathcal{C}(A).

By the assumption that this are the components of an epimorphism of copresheaves all these component morphisms are themselves epimorphisms and hence we have that indeed f(C)=image f C(C)f(C) = image_{f_C}(C).

Kinematics in a Bohr topos

A key aspect about the Bohr topos Bohr(A)Bohr(A) is that that classical kinematics of the commutative internal C *C^*-algebra A̲Bohr(A)\underline{A} \in Bohr(A) is the quantum kinematics of AA. In fact, the very definition of Bohr(A)Bohr(A) provides a formal context in which Gleason's theorem has a natural formulation:


(Gleason’s theorem)

For HH a Hilbert space of dimension dimH>2dim H \gt 2, and A=B(H)C *AlgA = B(H) \in C^\ast Alg its algebra of bounded operators, a state on AA is a function

ρ:A \rho : A \to \mathbb{C}

which is a \mathbb{C}-linear map when restricted to each commutative subalgebra CAC \subset A.

A function that preserves certain structure locally – here: over each commutative subalgebra – is precisely an internal fully structure preserving homomorphism in the presheaf topos over these local objects – here: over commutative subalgebras. Hence we have the following direct topos-theoretic equivalent reformulation of Gleason’s theorem.


For A=B(H)C *AlgA = B(H) \in C^\ast Alg as above, we have a natural bijection between the quantum states on AA and the (classical) states of A̲\underline{A} internal to Bohr(H)Bohr(H).

The phase space

The idea is that for AC *AlgA \in C^* Alg, the Bohr topos Bohr(A)=(Sh(Alex(𝒞(A))),A̲)C *TopSpaceC *ToposBohr(A) = (Sh(Alex(\mathcal{C}(A))), \underline{A}) \in C^* TopSpace \subset C^* Topos is the corresponding quantum phase space. More precisely, we may think of the internal commutative C *C^*-algebra A̲Bohr(A)\underline{A} \in Bohr(A) as the formal dual to the quantum phase space.

The following discussion makes this precise by exhibiting this formal dual as an internal locale. Since Bohr(A)Bohr(A) is a spatial topos, this internal locale is in fact an ordinary topological space bundle ΣAlex𝒞(A)\Sigma \to Alex \mathcal{C}(A) over the Alexandroff space Alex𝒞(A)Alex \mathcal{C}(A).


Write Σ̲ A\underline{\Sigma}_A and Σ A ¬¬\Sigma_A^{\not \not}, respectively for the corresponding internal locales associated to A̲\underline{A} by internal constructive Gelfand duality. Write

Σ A𝒞(A) \Sigma_A \to \mathcal{C}(A)


Σ A ¬¬𝒞(A) \Sigma^{\not \not}_A \to \mathcal{C}(A)

for the corresponding external locale, given under the equivalence of categories

Loc(Sh(𝒞(A)))Loc/𝒞(A) Loc(Sh(\mathcal{C}(A))) \simeq Loc/\mathcal{C}(A)

discussed at internal locale.


For AA a (noncommutative) C-star algebra, the assignment

AΣ A A \mapsto \Sigma_{A}


AΣ A ¬¬ A \mapsto \Sigma^{\not \not}_{A}

is called the Bohrification of AA.


Let Σ̲((̲A))\underline{\Sigma}(\underline(A)) be the internal locale from def. 11.

Regarded as an object

(Σ A𝒞(A))Loc/𝒞(A) (\Sigma_A \to \mathcal{C}(A)) \in Loc/\mathcal{C}(A)

of external locales over 𝒞(A)\mathcal{C}(A), this is the topological space whose underlying set is given by the disjoint union

Σ A= C𝒞(A)Σ(C) \Sigma_A = \coprod_{C \in \mathcal{C}(A)} \Sigma(C)

over all commutative C *C^*-subalgebras of AA of the ordinary Gelfand spectra Σ(C)\Sigma(C) of these commutative C *C^*-algebras, and whose open subsets are defined to be those 𝒰Σ A\mathcal{U} \subset \Sigma_A for which

  1. 𝒰 C𝒪(Σ(C))\mathcal{U}|_C \in \mathcal{O}(\Sigma(C)) for all commutative subalgebras CC;

  2. For all CDC \hookrightarrow D, if λ𝒰 C\lambda \in \mathcal{U}|_C and λΣ(D)\lambda' \in \Sigma(D) such that λ C=λ\lambda'|_C = \lambda, then λ𝒰 D\lambda' \in \mathcal{U}|_D.

Regarded equivalently as an internal locale in Sh(𝒞(A))Sh(\mathcal{C}(A)) this

As a presheaf on the poset 𝒞(A)\mathcal{C}(A) this is given by

Σ̲(A̲):UΣ U, \underline{\Sigma}(\underline{A}) : U \mapsto \Sigma_U \,,

where for U𝒪(𝒞(A))U \in \mathcal{O}(\mathcal{C}(A)) we set

Σ U:= CUΣ(C) \Sigma_U := \coprod_{C \in U} \Sigma(C)

with the relative topology inherited from Σ A\Sigma_A.

This appears as (HLSW, theorem 1).


The Bohrification of AncCStarA \in ncCStar only depends on its partial C-star algebra N(A)N(A) of normal elements

Σ AΣ N(A). \Sigma_A \simeq \Sigma_{N(A)} \,.

This is highlighted in (vdBergHeunen).


For AA a commutative C *C^\ast-algebra and Σ A Gelf\Sigma_A^{Gelf} \in Loc its ordinary Gelfand spectrum, we have that Bohrification in the double negation topology reproduces this ordinary Gelfand spectrum:

Σ A ¬¬Σ A Gel. \Sigma^{\not \not}_A \simeq \Sigma_A^{Gel} \,.

This is (Spitters06, theorem 9, corollary 10).


Then the construction of the ringed topos over the poset of commutative subalgebras

A([𝒞(A),Set],A̲) A \mapsto ([\mathcal{C}(A), Set], \underline{A})

extends to a functor

C *Alg inclC *TopSpaceC *Topos, C^* Alg_{incl} \to C^*TopSpace \hookrightarrow C^* Topos \,,

where on the right the morphisms of internal rings are even morphisms of internal C* algebras.


With the components of the morphism of internal rings the evident objectwise inclusions, this is directly checked.


Let Cstar incCstar_{inc} be the category of C-star algebras and inclusions. Then Bohrification extends to a functor

Σ ():CStar inc opLoc. \Sigma_{(-)} : CStar_{inc}^{op} \to Loc \,.

This is effectively the functoriality of the internal constructive Gelfand duality applied to the above observation. The statement appears as (vdBergHeunen, theorem 35).

The observables

For AC *AlgA \in C^\ast Alg a C*-algebra, then in quantum physics a self-adjoint operator aAa \in A is a quantum observable. The following statement asserts that quantum observables on AA are in a precise sense the \mathbb{R}-valued “functions” on the Bohr topos of AA.

Write C()C(\mathbb{R}) for the C *C^\ast-algebra of continuous complex functions on the real line. We think of Bohr(C())Bohr(C(\mathbb{R})) as the incarnation of \mathbb{R} in the context of Bohr toposes.


Morphisms f:Bohr(A)Bohr(C() 0)f : Bohr(A) \to Bohr(C(\mathbb{R})_0) with an extra right adjoint and C() 0f *A̲C(\mathbb{R})_0 \to f_*\underline{A} epi are in bijection to the observables on AA.


By prop. 7 such morphisms are in bijection to algebra homomorphisms

C() 0A. C(\mathbb{R})_0 \to A \,.

By functional calculus: every self-adjoint operator aAa \in A provides such a homomorphism by ff(a)f \mapsto f(a). Conversely, given such an algebra homomorphism, its image of i:xxi : x \mapsto x is a self-adjoint operator in AA, and these two constructions are clearly inverses of each other.


In the different but related context of the spectral presheaf (Isham-Döring 07) the identification of quantum observables with a topos-theoretic construction, as far as possible, has been called “daseinisation”. This is a bit more involved than the above direct characterization in terms of maps of ringed toposes.


Let x t 1Ax_{t_1} \in A be an observable and write x t 1\langle x_{t_1} \rangle for the subalgebra generated by it. Then (by general properties of presheaf over-toposes) the slice topos Bohr(A) /x t 1Bohr(A)_{/\langle x_{t_1}\rangle} is equivalent to

Bohr(A) /x t 1[x t 1 /𝒞(A),Set], Bohr(A)_{/\langle x_{t_1}\rangle} \simeq [\langle x_{t_1}\rangle_{/\mathcal{C}(A)}, Set] \,,

where on the right we have the copresheaves over the under-category x t 1 /𝒞(A)\langle x_{t_1}\rangle_{/\mathcal{C}(A)}. This is precisely the sub-poset of commutative subalgebras on those commutative subalgebras that contain x tx_{t}. This means that a classically consistent observation in the slice topos is one that is consistent with the observation of x t 1x_{t_1}.

The states


For AC *AlgA \in C^\ast Alg write (Sh(Alex𝒞(A)),)(Sh(Alex \mathcal{C}(A)), \mathbb{R}) for the ringed topos as indicated, where \mathbb{R} denotes the copresheaf constant on \mathbb{R}.

The internal C *C^\ast-algebra A̲Bohr(A)\underline{A} \in Bohr(A) is an internal \mathbb{R}-module. Forgetting the algebra structure and only remembering the \mathbb{R}-module structure, we get a category of ”\mathbb{R}-module toposes”.


There is a canonical morphism of ringed toposes

π:Bohr(A)(Sh(Alex𝒞(A)),) \pi : Bohr(A) \to (Sh(Alex \mathcal{C}(A)), \mathbb{R})

whose underlying geometric morphism is the identity (and whose morphism of internal ring objects is the unique one).


This bundle is the C *C^\ast-topos incarnation of the morphism ΣAlex𝒞(A)\Sigma \to Alex \mathcal{C}(A) of locales discussed above.


A state ρ\rho on AA is a section of π\pi in the category of \mathbb{R}-moduled toposes that is positive and normalized.


(Pre-)Sheaf of Bohr toposes of a quantum field theory

Notice that in the context of AQFT a quantum field theory is encoded by a local net of C-star algebras on spacetime.


Let XX be a Lorentzian manifold and

A:𝒪(X)C *Alg inc A : \mathcal{O}(X) \to C^* Alg_{inc}

be a local net of algebras. Notice that by definition this indeed takes values in C *C^\ast-algebras and inclusions . Then postcomposition with the Bohr topos-functor yields a presheaf of ringed spaces

Bohr(A):𝒪(X) opC *Alg inc opC *Alg cr opBohrC *TopSpaceC *Topos. Bohr(A) : \mathcal{O}(X)^{op} \to C^* Alg_{inc}^{op} \hookrightarrow C^* Alg_{cr}^{op} \stackrel{Bohr}{\to} C^* TopSpace \hookrightarrow C^* Topos \,.

This appears as (Nuiten, def. 17).

Assume that a net of observables A:Op(X)C *Alg incA : Op(X) \to C^\ast Alg_{inc} satisfies the split property?. Then it is strongly local precisely if the corresponding presheaf of Bohr toposes Bohr(A):Op(X) opAC *Alg incBohrC *TopBohr(A) : Op(X)^{op} \stackrel{A}{\to} C^\ast Alg_{inc} \stackrel{Bohr}{\to} C^\ast Top satisfies spatial descent by local geometric morphisms (meaning that for every spatial hyperslice ΣX\Sigma \subset X the induced presheaf Bohr(A) Σ:Op(Σ) opC *TopBohr(A)|_\Sigma : Op(\Sigma)^{op} \to C^\ast Top satisfies descent by local geometric morphisms.)


This appears as (Nuiten, theorem 4.2).

Contravariant functors on open subsets

Above is discussed the notion of Bohr topos given by covariant functors on the poset of commutative subalgebras of a C*-algebra. The fact that the functors here are covariant is related to the fact that the algebra itself naturally exists inside the presheaf topos.

Alternatively one can explore the situation for contravariant functors on the poset of commutative subalgebras (Isham-Döring 07). The resulting presheaf topos then does not directly contain the given C *C^\ast-algebra, but by Gelfand duality, does directly contain an internal locale which is its Gelfand spectrum. This is called the “spectral presheaf”.


The assertion by Bohr that all experiments in quantum mechanics must be possible to describe in “classical terms” is in

  • Nils Bohr Discussion with Einstein on Epistemological Problems in Atomic Physics in P. A. Schilpp (ed.) Albert Einstein, Philosopher-Scientist (Evanston: Library of Living Philosophers) pp. 201–241. (1949)

however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms .

Niels Bohr’s views on quantum mechanics that give the construction of Bohrification its name are reviewed further in

  • Erhard Scheibe, The logical analysis of quantum mechanics . Oxford: Pergamon Press, 1973.

For more see at interpretation of quantum mechanics the section Bohr’s standpoint.

Maybe the first article to propose to use intuitionistic logic/topos theory for the description of quantum physics is

  • Murray Adelman, John Corbett, A Sheaf Model for Intuitionistic Quantum Mechanics Appl. Cat. Struct. 3: 79-104. 1995

The term Bohrification and the investigations associated with it are initiated in

See also

The computation of the internal Gelfand spectrum Σ̲\underline{\Sigma} was initiated in

with some results in section 5 and 6 of

  • Martijn Caspers, Gelfand spectra of C *C^*-algebras in topos theory (pdf)

and completed in

  • Sander Wolters, Contravariant vs covariant quantum logic: A comparison of two topos-theoretic approaches to quantum theory (arXiv:1010.2031)

An complete outline of the full proof is given in

Applications and examples for AA a matrix algebra are discussed in

The functoriality of Bohrification is observed in

The application of the double negation topology to make Bohrification coinicide with ordinary Gelfand duality on commutative C *C^*-algebras is discussed in

  • Bas Spitters, The space of measurement outcomes as a spectrum for non-commutative algebras in Cooper, Kashefi, Panangaden (eds.) Developments in computational models (DCM 2010)(arXiv:1006.1432)

The generalization of Bohrification from quantum mechanics to quantum field theory (AQFT) is discussed in

The original suggestion to interpret the Kochen-Specker theorem in the topos over the poset of commutative subalgebras (there taken to be presheaves instead of copresheaves) is due to

Disucssion of aspects of the process of quantization in terms of Bohr toposes is in

  • Kunji Nakayama, Sheaves in Quantum Topos Induced by Quantization (arXiv:1109.1192)

Revised on January 11, 2014 12:52:30 by Urs Schreiber (