nLab
Lawvere distribution

Idea

To some extent one can think of a sheaf $F$ on a topological space as being like a Set-valued function on that space: to each point $x\in X$ it assigns the stalk $x^{*}F\in \mathrm{Set}$. A Lawvere distribution is in this analogy the analog of a distribution in the sense of functional analysis: where the latter is a linear functional, the former is a colimit-preserving functor.

Here we think of a coproduct of sets as the categorification (under set cardinality) of the sum of numbers and hence read preservation of colimits as linearity .

Better yet, under ∞-groupoid cardinality we may think of tame ∞-groupoids as real numbers and hence of (∞,1)-sheaves as analogous to functions. This yields a notion of Lawvere distributions on (∞,1)-toposes given by (∞,1)-colimit preserving (∞,1)-functors.

More generally one can allow to generalize $(\mathrm{\infty },1)$-toposes to general locally presentable (∞,1)-categories. Viewed this way, Lawvere distributions are the morphism in $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$, the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories.

Definition

Throughout $𝒮$ is some base topos or (∞,1)-topos and all notions are to be understood as indexed over this base.

Definition

Let $ℰ$ and $𝒦$ be (n,1)-toposes. A distribution on $ℰ$ with values in $𝒦$ is a (∞,1)-functor

$$\mu :ℰ\to 𝒦$$\mu : \mathcal{E} \to \mathcal{K}

that preserves small (∞,1)-colimits.

Write

$$\mathrm{Dist}(ℰ,𝒦)\subset (\mathrm{\infty },1)\mathrm{Func}(ℰ,𝒦)$$Dist(\mathcal{E}, \mathcal{K}) \subset (\infty,1)Func(\mathcal{E}, \mathcal{K})

for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve finite colimits.

Remark

By the adjoint (∞,1)-functor theorem this is equivalently a pair

$$(\mu ⊣{\mu }^{*}):ℰ\to 𝒦$$(\mu \dashv \mu^*) : \mathcal{E} \to \mathcal{K}

of adjoint (∞,1)-functors.

Notation

To amplify the interpretation in analogy with distributions in functional analysis one sometimes write

$${\int }_{ℰ}(-)d\mu :ℰ\to 𝒦$$\int_{\mathcal{E}} (-) d\mu : \mathcal{E} \to \mathcal{K}

for a Lawvere distribution $\mu$.

Notably in the case that $𝒦=$ ∞Grpd and $F$ is an (∞,1)-sheaf such that $\mu (F)$ is tame, we may use

$${\int }_{ℰ}Fd\mu \in ℝ$$\int_{\mathcal{E}} F d \mu \in \mathbb{R}

for the corresponding ∞-groupoid cardinality.

Examples

Dirac $\delta$-distributions

A point of a topos is a geometric morphism of the form

$$(p^{*}⊣p_{*}):𝒮\overleftarrow{\to }ℰ.$$(p^* \dashv p_*) : \mathcal{S} \stackrel{\leftarrow}{\to} \mathcal{E} \,.

The left adjoint $p^{*}$ is therefore a Lawvere distribution. This sends any (∞,1)-sheaf to its stalk at the point $p$. So this behaves like the Dirac distribution on functions.

The canonical distribution on a locally $\mathrm{\infty }$-connected topos

If $ℰ$ is a locally ∞-connected (∞,1)-topos then its terminal global section (∞,1)-geometric morphism by definition has a further left adjoint

$$(\Pi ⊣\Delta ⊣\Gamma ):ℰ\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{\Delta }{\leftarrow }}{\underset{\Gamma }{\to }}}𝒮.$$(\Pi \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \mathcal{S} \,.

This left adjoint $\Pi$ (the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos) is therefore a canonical $𝒮$-valued distribution on $ℰ$. It is also written

$${\int }_{ℰ}(-)dx:ℰ\to \mathrm{\infty }\mathrm{Grpd}.$$\int_{\mathcal{E}}(-) d x : \mathcal{E} \to \infty Grpd \,.

Multiplication of distributions by functions

For $F\in ℰ$ an $(\mathrm{\infty },1)$-sheaf and $\mu :ℰ\to 𝒮$ a distribution, there is a new distribution

$$F\cdot \mu :G\mapsto \mu (F\times G).$$F \cdot \mu : G \mapsto \mu(F \times G) \,.

In the functional-notation this is the formula

$${\int }_{ℰ}Gd(F\times \mu )={\int }_{ℰ}G\times Fd\mu .$$\int_{\mathcal{E}} G d(F \times \mu) = \int_{\mathcal{E}} G \times F d \mu \,.

Distributions on the point

The ∞Grpd-valued distributions on $\mathrm{\infty }\mathrm{Grpd}\simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}(*)$ itself coincide with the value at the single point

$$\mathrm{Dist}(\mathrm{\infty }\mathrm{Grpd},\mathrm{\infty }\mathrm{Grpd})\simeq \mathrm{\infty }\mathrm{Grpd}.$$Dist(\infty Grpd, \infty Grpd) \simeq \infty Grpd.

References

The 1-categorical notion has been described by Bill Lawvere in a series of talks and expositions. For instance in the context of cohesive toposes in

• Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

A comprehensive discussion is in

• Marta Bunge and Jonathan Funk, Singular coverings of toposes Lecture Notes in Mathematics, (2006) Volume 1890/2006

  chapter 1 Lawvere Distributions on Toposes

For the $(\mathrm{\infty },1)$-category theory generalization see Pr(∞,1)Cat and references given there.