nLab Lawvere distribution

Idea

A Lawvere distribution is a cosheaf valued in some topos, such as the category of sets Set. The concept of Lawvere distribution is a kind of categorification of the concept of distribution in functional analysis.

To some extent one may think of a sheaf $F$ on a topological space as being a Set-valued function on that space: to each point $x \in X$ it assigns the stalk $x^* F \in Set$. In this analogy a Lawvere distribution is the analog of a distribution in the sense of functional analysis: where the latter is a continuous 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 a higher/homotopical categorification of real-number valued functions. This yields a more general notion of Lawvere distributions on (∞,1)-toposes given by (∞,1)-colimit preserving (∞,1)-functors.

Still more generally one may 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

(Lawvere, see Definition 1.3.4 in Bunge and Funk.)

Let $\mathcal{E}$ and $\mathcal{K}$ be (∞,1)-toposes over Set (more generally, over an elementary topos $S$ with a natural numbers object). A distribution on $\mathcal{E}$ with values in $\mathcal{K}$ is an (∞,1)-cosheaf? on $E$ with values in $K$, i.e., an (∞,1)-functor

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

that preserves $S$-small (∞,1)-colimits. We write

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

for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that are (∞,1)-cosheaves?, i.e., preserve $S$-small colimits.

Remark

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

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

of ($S$-indexed) adjoint (∞,1)-functors.

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

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

This construction also works in the relative setting: if $e\colon E\to K$ locally ∞-connected geometric morphism of (∞,1)-toposes, then $e_!\colon E\to K$ is a $K$-valued Lawvere distribution on $E$.

Local toposes

If $E$ is a local (∞,1)-topos, more generally, $e\colon E\to K$ is a local (∞,1)-geometric morphism? of (∞,1)-toposes, then the right adjoint $e_*$ is a (left exact) Lawvere distribution on $E$.

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

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

Related entries

• symmetric topos
• bagdomain topos
• cosheaf

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

• Pr(∞,1)Cat

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)

See also:

• Marta Bunge, Cosheaves and Distributions on Toposes , Alg. Univ. 34 (1995) pp.469-484.

• Marta Bunge, Jonathon Funk, Spreads and the Symmetric Topos , JPAA 113 (1996) pp.1-38.

• Marta Bunge, Jonathon Funk, Spreads and the Symmetric Topos II , JPAA 130 (1998) pp.49-84.

• Anders Kock, Gonzalo E. Reyes, A Note on Frame Distributions , Cah. Top. Géom. Diff. Cat.40 (1999) pp.127-140.

• Andrew Pitts, On Product and Change of Base for Toposes , Cah. Top. Géom. Diff. Cat.26 (1985) pp.43-61.