Consider a category $C$ whose objects are thought of as spaces and whose morphisms are regarded as structure-preserving morphisms between these spaces.
There is then a general notion of
spaces modeled on $C$ that are testable by objects of $C$;
quantities with values in $C$.
Very generally,
a space modeled on the objects of $C$ is a presheaf on $C$, i.e. a functor
$X : C^{op} \to$ Set:
we think of each such presheaf as being a rule that assigns to each test space $U \in C$ the set $X(U)$ of allowed maps from $U$ into the would-be space $X$;
a quantity modeled on $C$ is a copresheaf on $C$, i.e. a functor
$A : C \to Set$:
we think of each such copresheaf $A$ as a rule that assigns to each test space $U \in C$ the set $A(U)$ of allowed maps from a would-be space into $U$, hence as the collection of $U$-valued functions on $X$. Since a function on a point is a “quantity”, this are generalized quantities.
Typically the admissible (co)presheaves that are regarded as generalized spaces and quantities modeled on $C$ are required to respect certain properties of $C$:
If $C$ carries the structure of a site, one asks a generalized space to be a presheaf $X = PSh(C) = [C^{op},Set]$ that respects the way objects in $C$ are covered by other objects. These are the sheaves. The category of sheaves
$Sh(C) \hookrightarrow PSh(C)$
is the topos of spaces modeled on objects in $C$. 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 $U \times V$ is the same as a pair of functions, one with values in $U$, one with values in $V$. Hence one is typically interested in coprosheaves that preserve at least product
$CoSh(C) \hookrightarrow CoPSh(C)$.
As indicated in
from page 17 on, the general situation involving
(generalized) spaces;
(generalized) quantities (e.g. function algebras);
the duality between the two;
which underlies much of mathematics is at its heart controled by the following elementary category theoretic reasoning:
Let $S$ 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 $S = \Delta$, the simplicial category, or $S =$ CartSp.
An ordinary manifold, for instance, is a space required to be locally isomorphic to an object in $S = CartSp$. But more generally, a space $X$ modeled on $S$ need only be probeable by objects of $S$, giving a rule which to each test object $U \in S$ assigns the collection of admissible maps from $U$ to $X$, such that this assignment is well-behaved with respect to morphisms in $S$. Such an assignment is nothing but a presheaf on $S$, i.e. a contravariant functor
Therefore general spaces modeled on $S$ are nothing but presheaves on $S$:
Of course this is an extremely general notion of spaces modeled on $S$.
We have the Yoneda embedding $S \hookrightarrow Spaces_S$ and using this we can say that the collection of functions on a generalized space $X$ with values in $U \in S$ is
This assignment is manifestly covariant in $U$, and hence more generally we can consider the functions on $X$, $C(X)$ to be a copresheaf on $S$, namely a covariant functor
One can think of $C(X)$ as being a generalized quantity which may be co-probed by objects of $S$.
In this vein, one can say, generally, that co-presheaves on $S$ are generalized quantities modeled on $S$, and we write
Given any such generalized quantity $A \in Quantities_S$, we can ask which generalized space it behaves like the algebra of functions on. This generalized space should be called $Spec(A)$ and can be defined as a presheaf by the assignment
In total this yields an adjoint pair of functors between generalized spaces and generalized quantities:
(That this is an adjunction can be understood as a special case of abstract Stone duality induced by a dualizing object.)
Lawvere addresses 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 $S$ 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.
we describe the duality between space and quantity induced by forming
functions on spaces;
spectra of function algebras.
Let $V$ be a symmetric monoidal category and $C$ a $V$-enriched category. Write $[C^{op},V]$ for the enriched functor category and $j : C \to [C^{op},V]$ for the Yoneda embedding.
There is canonically a $V$-adjunction
the Isbell adjunction. Here
$\mathcal{O} := [C^{op},V](j(-), -)$ sends a presheaf $X$ to the copresheaf $U \mapsto [C^{op},V](X,j(U))$;
$Spec := [C,V]^{op}(j(-),-)$ sends a copresheaf $A$ to the presheaf $U \mapsto [C,V](A, j^{op}(U))$.
If we assume that $C$ is tensored over $V$, 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
where we used the Yoneda lemma $[C^{op},V](j(v),j(u)) \simeq V(v,u)$ and the co-Yoneda lemma $X \simeq \int^{v \in V} j(v) \cdot X(v)$ and the fact that the $V$-enriched hom sends colimits and coends in the first argument to limits and ends.
Analogously we find
Consider the category of test spaces $C =$ CartSp.
Then
spaces modeled on $C$ are generalized smooth spaces such as diffeological spaces;
quantities modeled on $C$ are smooth algebras ($C^\infty$-rings).
The adjunction $(\mathcal{O} \dashv Spec)$ sends a smooth space to its smooth algebra of functions and a smooth algebra of functions to its “spectrum”.
There are various specializations of interest on this
higher categorical version
∞-space modeled on $C$ is a simplicial presheaf on $C$, i.e. a functor $X : C^{op} \to$ SSet.
∞-quantity modeled on $C$ is a cosimplicial copresheaf on $C$, i.e. a functor $X : C \to CoSSet$ .
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 notion originates somewhere around
A kind of review is in