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Consider a category $C$ 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 $C$ that are testable or probe-able by objects of $C$;
quantities with values in $C$.
Very generally, following Lawvere 86:
a generalized space modeled on the objects of $C$ is a presheaf on $C$, i.e. a functor of the form
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 $X$ (this is really the perspective of functorial geometry, originally due to Grothendieck 65);
a generalized quantity modeled on $C$ is a copresheaf on $C$, i.e. a functor of the form
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 the would-be space $A$ into $U$, hence as the collection of $U$-valued functions on $A$. 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 $X$ by a test space $U$ turns out to coincide with the actual maps from $U$ to $X$, when $U$ 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 $C$ 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
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 copresheaves that preserve at least product
$CoSh(C) \hookrightarrow CoPSh(C)$.
As indicated in Lawvere 86, from p. 17 on
(generalized) spaces;
(generalized) quantities (e.g. algebras of functions);
the duality between the two;
which underlies much of mathematics is at its heart controlled by the following elementary category theoretic reasoning:
Let $C$ 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 $C = \Delta$, the simplicial category, or $C =$ CartSp, the category of $\mathbb{R}^n$ and smooth maps between them.
An ordinary manifold, for instance, is a space required to be locally isomorphic to an object in $C = CartSp$. But more generally, a space $X$ modeled on $C$ need only be probeable by objects of $C$, giving a rule with which, to each test object $U \in C$, we assign the set of probing maps from $U$ to $X$, such that this assignment is well-behaved with respect to morphisms in $C$. Such an assignment is nothing but a presheaf on $C$, i.e. a contravariant functor
Therefore general spaces modeled on $C$ are nothing but presheaves on $C$:
Of course this is an extremely general notion of spaces modeled on $C$.
For example, any smooth manifold $M$ is a presheaf on CartSp by $F_M:= Hom_{Man}(-, M): CartSp^{op} \to Set$, where we consider CartSp as a full subcategory of Man, the category of smooth manifolds and smooth maps between them. $F_M$ maps each $\mathbb{R}^n$ to the set of all the smooth ways that $\mathbb{R}^n$ can probe $M$.
In particular, every object in $C$ is a space modeled on $C$, by the Yoneda embedding $C \hookrightarrow Spaces_C$, whereby every object $c$ in $C$ is embedded as $Hom_C(-, c)$. That is, any object in $C$ is nothing but a consistent way to be probed by all the objects in $S$.
Now take a space $X$ modeled on $S$, and consider the set of quantities on $X$ with values in $U \in S$. It should be
This defines a covariant functor $C(X) := Hom_{Spaces_S}(X,-): S \to Set$. More generally, we can consider the S-valued quantities on $X$ to be a copresheaf on $S$, namely a covariant functor
One can think of $C(X)$ as a generalized quantity which may be co-probed by objects of $S$.
In this vein, one can say, generally, that copresheaves 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 contravariant 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 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 $S$ 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 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)
See also
Last revised on January 17, 2024 at 19:08:43. See the history of this page for a list of all contributions to it.