space and quantity






Consider a category CC whose objects are thought of as spaces of sorts (“test spaces”), and whose morphisms are regarded as homomorphisms between these spaces.

There is then a general notion of

  • spaces modeled on CC that are testable or probe-able by objects of CC;

  • quantities with values in CC.

Very generally, following Lawvere 86:

  • a generalized space modeled on the objects of CC is a presheaf on CC, i.e. a functor of the form

    X:C opX \;\colon\; C^{op} \to Set:

    we think of each such presheaf as being a rule that assigns to each test space UCU \in C the set X(U)X(U) of allowed maps from UU into the would-be space XX (this is really the perspective of functorial geometry, originally due to Grothendieck 65);

  • a generalized quantity modeled on CC is a copresheaf on CC, i.e. a functor of the form

    A:CSetA \;\colon\; C \to Set:

    we think of each such copresheaf AA as a rule that assigns to each test space UCU \in C the set A(U)A(U) of allowed maps from a would-be space into UU, hence as the collection of UU-valued functions on XX. Since a function on a point is a “quantity”, these are generalized quantities.

One may view the Yoneda lemma and the resulting Yoneda embedding as expressing consistency conditions on this perspective: The Yoneda lemma says that the prescribed rule for how to test a generalized space XX by a test space UU turns out to coincide with the actual maps from UU to XX, when UU is itself regarded as a generalized space, and the Yoneda embedding says that, as a result, the nature of maps between test spaces does not depend on whether we regard these as test spaces or as generalized spaces.

Beyond this automatic consistency condition, guaranteed by category theory itself, typically the admissible (co)presheaves that are regarded as generalized spaces and quantities are required to respect one more consistency condition:

  • If CC carries the structure of a site, one asks a generalized space to be a presheaf X=PSh(C)=[C op,Set]X = PSh(C) = [C^{op},Set] that respects the way objects in CC are covered by other objects. These are the sheaves. The category of sheaves

    Sh(C)PSh(C)Sh(C) \hookrightarrow PSh(C)

    is the topos of spaces modeled on objects in CC. More details on how to think of sheaves as generalized spaces is at motivation for sheaves, cohomology and higher stacks.

  • Given any generalized spaces, functions out of it are expected to respect products of coefficient objects, in that a function with values in U×VU \times V is the same as a pair of functions, one with values in UU, one with values in VV. Hence one is typically interested in copresheaves that preserve at least product

    CoSh(C)CoPSh(C)CoSh(C) \hookrightarrow CoPSh(C).


As indicated in Lawvere 86, from p. 17 on

which underlies much of mathematics is at its heart controlled by the following elementary category theoretic reasoning:

Let SS be some category whose objects we want to think of as certain simple spaces on which we want to model more general kinds of spaces. For instance S=ΔS = \Delta, the simplicial category, or S=S = CartSp.

An ordinary manifold, for instance, is a space required to be locally isomorphic to an object in S=CartSpS = CartSp. But more generally, a space XX modeled on SS need only be probeable by objects of SS, giving a rule which to each test object USU \in S assigns the collection of admissible maps from UU to XX, such that this assignment is well-behaved with respect to morphisms in SS. Such an assignment is nothing but a presheaf on SS, i.e. a contravariant functor

X:S opSet. X : S^{op} \to Set \,.

Therefore general spaces modeled on SS are nothing but presheaves on SS:

Spaces S:=PSh(S). Spaces_S := PSh(S) \,.

Of course this is an extremely general notion of spaces modeled on SS.

We have the Yoneda embedding SSpaces SS \hookrightarrow Spaces_S and using this we can say that the collection of functions on a generalized space XX with values in USU \in S is

C(X,U):=Hom Spaces S(X,U). C(X,U) := Hom_{Spaces_S}(X,U) \,.

This assignment is manifestly covariant in UU, and hence more generally we can consider the functions on XX, C(X)C(X) to be a copresheaf on SS, namely a covariant functor

C(X):=Hom(X,):SSets. C(X) := Hom(X,-) : S \to Sets \,.

One can think of C(X)C(X) as being a generalized quantity which may be co-probed by objects of SS.

In this vein, one can say, generally, that co-presheaves on SS are generalized quantities modeled on SS, and we write

Quantities S:=CoPSh(S). Quantities_S := CoPSh(S) \,.

Given any such generalized quantity AQuantities SA \in Quantities_S, we can ask which generalized space it behaves like the algebra of functions on. This generalized space should be called Spec(A)Spec(A) and can be defined as a presheaf by the assignment

Spec(A):UHom Quantities S(A,C(U)). Spec(A) : U \mapsto Hom_{Quantities_S}(A, C(U)) \,.

In total this yields an adjoint pair of functors between generalized spaces and generalized quantities:

Spaces SSpec()C()Quantities S. Spaces_S \stackrel{\stackrel{C(-)}{\to}}{\stackrel{Spec(-)}{\leftarrow}} Quantities_S \,.

(That this is an adjunction can be understood as a special case of abstract Stone duality induced by a dualizing object.)

Lawvere refers to this adjoint pair as Isbell conjugation.

In conclusion, the grand duality between spaces and quantities is a consequence of the formal duality which reverses the arrows in the category SS of test spaces.

This story generalizes straightforwardly from presheaves with values in Set to presheaves with values in other categories. Of relevance are in particular presheaves with values in the category Top of topological spaces and presheaves with values in the category of spectra. See the examples below.

Isbell duality: global functions and spectrum

we describe the duality between space and quantity induced by forming

  • functions on spaces;

  • spectra of function algebras.

Let VV be a symmetric monoidal category and CC a VV-enriched category. Write [C op,V][C^{op},V] for the enriched functor category and j:C[C op,V]j : C \to [C^{op},V] for the Yoneda embedding.

There is canonically a VV-adjunction

(𝒪Spec):[C,V] opSpec𝒪[C op,V] (\mathcal{O} \dashv Spec) \;\;: \;\; [C, V]^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} [C^{op},V]

the Isbell adjunction. Here

  • 𝒪:=[C op,V](j(),)\mathcal{O} := [C^{op},V](j(-), -) sends a presheaf XX to the copresheaf U[C op,V](X,j(U))U \mapsto [C^{op},V](X,j(U));

  • Spec:=[C,V] op(j(),)Spec := [C,V]^{op}(j(-),-) sends a copresheaf AA to the presheaf U[C,V](A,j op(U))U \mapsto [C,V](A, j^{op}(U)).

If we assume that CC is tensored over VV, then that this is an adjunction may be seen in end/coend-calculus to express the hom-objects in the enriched functor category as follows. We compute

[C,V] op(𝒪(X),A) = uCV(A(u),[C op,V](X,j(u))) uCV(A(u),[C op,V]( vCj(v)X(v),j(u))) u,vCV(A(u)X(v),[C op,V](j(v),j(u))) u,vCV(A(u)X(v),V(v,u)), \begin{aligned} [C,V]^{op}(\mathcal{O}(X),A) & = \int_{u \in C} V(A(u), [C^{op},V](X,j(u))) \\ & \simeq \int_{u \in C} V(A(u), [C^{op},V](\int^{v \in C} j(v) \cdot X(v),j(u))) \\ & \simeq \int_{u, v \in C} V(A(u) \cdot X(v), [C^{op},V](j(v),j(u))) \\ & \simeq \int_{u, v \in C} V(A(u) \cdot X(v), V(v,u)) \end{aligned} \,,

where we used the Yoneda lemma [C op,V](j(v),j(u))V(v,u)[C^{op},V](j(v),j(u)) \simeq V(v,u) and the co-Yoneda lemma X vVj(v)X(v)X \simeq \int^{v \in V} j(v) \cdot X(v) and the fact that the VV-enriched hom sends colimits and coends in the first argument to limits and ends.

Analogously we find

[C op,V](X,SpecA) = vCV(X(v),[C,V](A,j op(v))) vCV(X(v),[C,V]( uCj op(u)X(v),j op(u))) u,vCV(A(u)X(v),[C,V](j op(v),j op(u))) u,vCV(A(u)X(v),V(v,u)), \begin{aligned} [C^{op},V](X,Spec A) & = \int_{v \in C} V(X(v),[C,V](A, j^{op}(v))) \\ & \simeq \int_{v \in C} V(X(v), [C,V](\int^{u \in C} j^{op}(u) \cdot X(v),j^{op}(u))) \\ & \simeq \int_{u, v \in C} V(A(u) \cdot X(v), [C,V](j^{op}(v),j^{op}(u))) \\ & \simeq \int_{u, v \in C} V(A(u) \cdot X(v), V(v,u)) \end{aligned} \,,


Cartesian test spaces: diffeological spaces and smooth algebras

Consider the category of test spaces C=C = CartSp.


The adjunction (𝒪Spec)(\mathcal{O} \dashv Spec) sends a smooth space to its smooth algebra of functions and a smooth algebra of functions to its “spectrum”.

Higher space and higher quantity

There are various specializations of interest on this

With the advent of Higher Topos Theory abstract concepts of space and quantity have been realized fully in the context of (∞,1)-toposes in terms of structured (∞,1)-toposes and generalized schemes. For a summary see the tables at notions of space.


The general perspective is due to

  • William Lawvere, Taking categories seriously, Revista Colombiana de Matematicas, XX (1986) 147-178, reprinted in: Reprints in Theory and Applications of Categories, No. 8 (2005) pp. 1-24 (TAC)

  • William Lawvere, Categories of space and quantity, in: J. Echeverria et al (eds.), The Space of mathematics, de Gruyter, Berlin, New York (1992) (pdf)

Last revised on June 12, 2018 at 07:15:41. See the history of this page for a list of all contributions to it.