nLab Lawvere distribution

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Models

$(\infty,1)$-Topos Theory

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

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 $x^* F \in Set$. 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 $(\infty,1)$-toposes to general locally presentable (∞,1)-categories. Viewed this way, Lawvere distributions are the morphism in $Pr(\infty,1)Cat$, the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories.

Definition

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

Definition

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

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

that preserves small (∞,1)-colimits.

Write

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

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

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

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

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

for a Lawvere distribution $\mu$.

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

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

for the corresponding ∞-groupoid cardinality.

Examples

Dirac $\delta$-distributions

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

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

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 $\infty$-connected topos

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

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

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

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

Multiplication of distributions by functions

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

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

In the functional notation this is the formula

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

Distributions on the point

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

$Dist(\infty Grpd, \infty Grpd) \simeq \infty Grpd.$

For the $(\infty,1)$-category theory generalization and related references:

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 Jonathon Funk, Singular coverings of toposes , Lecture Notes in Mathematics vol. 1890 Springer Heidelberg (2006). (chap. 1)