Contents

topos theory

cohomology

# Contents

## Idea

### In physics

In physics and especially in continuum mechanics and thermodynamics, a physical quantity associated with a physical system extended in space is called

• intensive if it is a function on (the physical system extended in) space;

• extensive if it is a density or linear distribution on (the physical system extended in) space.

For instance, for a solid body its temperature is intensive, but its mass is extensive: there is a temperature assigned to every point of the body (in the idealization of classical continuum mechanics anyway) but a mass is assigned only to every little “extended” piece of the body, not to a single point.

This terminology in physics appears vaguely in (Hegel 1812), Hegel 1817), more precisely in (Grassmann 1844) and in its fully modern form is maybe due to Richard Tolman (1917).

If we take into account that a physical system may or may not have a particular kind of quantity (for example, a thermodynamic system may or may not have a temperature), a system has a value of an intensive quantity if and only if all of its subsystems have a value of that quantity, and then all of these values are equal. However, it’s generally safe to assume that every system has a value of every extensive quantity.

To be precise, given a system $S_0$ made up of the subsystems $S_1$ and $S_2$:

• For an intensive quantity $q$, $S_0$ has a value $q_0$ of $q$ if and only if $S_1$ and $S_2$ also have values of $q$, and then we have $q_1 = q_0$ and $q_2 = q_0$.

• For an extensive quantity $Q$, all three systems have values of $Q$, and we have $Q_0 = Q_1 + Q_2$.

### In geometry and algebra

In (Lawvere 86) it is amplified that this duality is generally a fundamental one also in mathematics: given a topos $\mathbf{H}$ with a commutative ring object $R \in CRing(\mathbf{H})$, then

• the space of intensive quantities on an object $X \in \mathbf{H}$ is the mapping space $[X,R]_{\mathbf{H}} \in CRing(\mathbf{H})$ formed in $\mathbf{H}$;

• the space of extensive quantities on $X$ is the $R$-linear dual, namely the mapping space $[X,R]^\ast \coloneqq [[X,R], R]_{R Mod}$ formed in $R$-modules in $\mathbf{H}$.

(here the operation $[[-,R],R]_{R Mod}$ happens to be what is also called the “continuation monad” for $R$)

• the integration map is the canonical evaluation pairing

$\int_X \;\colon\; [X,R] \times [X,R]^\ast \longrightarrow R \,.$
$(x, f) \mapsto f(x)$

The elusive connection between integration and (co)ends can here be explained:

Proposition: Let $V$ be a monoidal closed category with tensor $\otimes_V$ and internal hom $[-, -]_V$. Suppose that the Kan extension $\text{Lan}_{\text{Id}_V} (\text{Id}_V)$ exists and is pointwise. Then

$X \cong \text{Id}_V (X) \cong \text{Lan}_{\text{Id}_V} (\text{Id}_V)(X) \cong \int^{Y \in V} [Y, X]_V \otimes_V Y$

and the structure maps are

$\epsilon_Y : [Y, X]_V \otimes_V Y \rightarrow X$

Occuring as counits in the supposed monoidal closed adjunction.

This is an instance of the coend formula for a Kan extension.

We may think of $V$ as the category of $R$-module objects in a topos $\mathbf{H}$, or, more generally, some category $\mathbf{H}$ resembling spaces. In that case, we get:

Corollary: Let $\mathbf{H}$ be a category with internal hom $[-, -]_{\mathbf{H}}$, and let $R$ be a commutative ring object in $\mathbf{H}$. Let $R \text{-mod}$ be the category of $R$-module objects in $\mathbf{H}$, with a chosen monoidal closed structure $\otimes_R$ (tensor product) and $[-, -]_R$ (internal Hom). Then, taking $V = R \text{-mod}$ above, and supposing that $\text{Lan}_{\text{Id}_{R \text{-mod}}} (\text{Id}_{R \text{-mod}})$ we have

$R \cong \int^{X \in R \text{-mod}} [X, R]_R \otimes_R X$

with structure maps:

$\epsilon_X : [X, R]_R \otimes_R X \rightarrow R$

In the above notation, $\epsilon_X = \int_X$. To make the analogy quite clear, consider the following example, which could naturally be called the example of “Lawvere Banach spaces”, in direct analogy with “Lawvere metric spaces”:

Example: Let $\mathbf{H}$ be the category of Lawvere metric spaces (Note: this category is not a topos, but the corollary above will still apply). Maps are short maps. $\mathbf{H}$ has an internal Hom consisting of bounded maps. Let $\mathbb{R}$ be the real numbers, with $d(x, y) = ||x - y||$ as Lawvere metric. $\mathbb{R} \text{-mod}$ is here the category of normed $\mathbb{R}$-modules, with short maps as maps, with the exception that $||x|| = 0$ does not imply that $x = 0$, and that $||x|| = \infty$ is possible. (This temporary notation is not to be confused with the category of real vector spaces over the real numbers, in which norms play no role).

There is a natural notion of completion on such spaces. The category of complete objects is essentially the category of Banach spaces over $\mathbb{R}$, except for the previously mentioned differences that $||x|| = 0$ does not imply that $x = 0$, and $||x|| = \infty$ is possible. Completion makes this category into a reflective subcategory of $\mathbb{R}$-mod. Write $\text{LawvBan}$ for the this category - it is a natural continuation of Lawvere metric spaces.

There is a natural choice of monoidal closed structure on $\text{LawvBan}$; the tensor product is projective tensor product (note: slight tweaking is necessary to account for the differences between Lawvere Banach spaces and Banach spaces). The internal Hom consists of bounded maps. The corollary above applies to this setup, so that

$R \cong \int^{X \in R \text{-mod}} [X, R]_R \otimes_R X$

where $\otimes_R$ is projective tensor product and $[X, R]_R$ consists of bounded maps from $X$ to $R$. There are structure maps

$\epsilon_X : [X, R]_R \otimes_R X \rightarrow R$

sending $f \otimes x$ to $f(x)$, for which we suggestively write

$\int_X : [X, R]_R \otimes_R X \rightarrow R$

### In higher geometry and higher algebra

Viewed this way, this naturally generalizes to the case where $\mathbf{H}$ is in fact an (∞,1)-topos and $R \in CRing(\mathbf{H})$ an E-∞ ring. In this case $[X,R]$ is called the $R$-cohomology spectrum of $X$ and $[X,R]^\ast$ is the corresponding generalized homology spectrum. In this form intensive and extensive properties appear in physics in the context of motivic quantization of local prequantum field theory.

More generally, for $\chi$ an $R$-(∞,1)-line bundle over $X$ then the corresponding extensive object is the $\chi$-twisted Thom spectrum $R_{\bullet + \chi}(X)$ and the intensive object is the $\chi$-twisted cohomology spectrum $R^{\bullet + \chi}(X) = [R_{\bullet+ \chi}(X),R]_{R Mod}$. See at motivic quantization for how this appears in physics.

## In modal homotopy type theory

Assume we are working in the context of a cohesive (∞,1)-topos, $\mathbf{H}$, with the three adjoint modalities, shape modality $\dashv$ flat modality $\dashv$ sharp modality $\int \dashv \flat \dashv \sharp$.

We may characterize the codomains of those functions which are intensive or extensive quantities in terms of $\sharp$.

• Intensive: functions whose value is genuinely given by their restriction to all possible points have as codomains types $X$ that are fully determined by their moment of continuity, that is those for which $X \to \sharp X$ is a monomorphism. In categorical semantics these are the concrete objects or equivalently the separated presheaves for $\sharp$: they are determined by their global points.

• Extensive: objects which have purely the negative moment of continuity $\overline{\sharp}$, or, in other words, which are maximally non-concrete, form codomains for “functions” which vanish on points and receive their contribution only from regions that extend beyond a single point. For example, the smooth moduli space of differential $n$-forms is maximally non-concrete. This concept of extension is precisely that which gave the name to Hermann Grassmann‘s Ausdehnungslehre that introduced the concept of exterior differential form.

So, the adjunction $(\flat \dashv \sharp)$ expresses quantity, discrete quantity and continuous quantity, and the latter is further subdivided into intensive and extensive quantity.

The concepts of intensive and extensive quantity are highlighted in

which states (p. xxiv, xxv) that intensive quantity is the topic of differential calculus and integration theory, while extensive quantity is the topic of this very Ausdehnungslehre.

General discussion includes

A formalization in categorical logic/topos theory is proposed in

See also the exposition of his ideas at Higher toposes of laws of motion.

Lawvere’s terminology is probably (see at objective logic) inspired by

and meant to be a formalization of this part of the “objective logic”, see also at Science of Logic.

The first use of the terms ‘intensive’ and ‘extensive’ appears to be

• Richard Tolman (1917). The Measurable Quantities of Physics. Physical Review 9 (3): 237–253.