nLab Lawvere distribution



(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



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 FF on a topological space as being a Set-valued function on that space: to each point xXx \in X it assigns the stalk x *FSetx^* 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 (,1)(\infty,1)-toposes to general locally presentable (∞,1)-categories. Viewed this way, Lawvere distributions are the morphism in Pr(,1)CatPr(\infty,1)Cat, the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories.


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


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

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

μ:𝒦 \mu : \mathcal{E} \to \mathcal{K}

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

Dist S(,𝒦)(,1)Func(,𝒦) 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 SS-small colimits.


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

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

of (SS-indexed) adjoint (∞,1)-functors.


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

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

for a Lawvere distribution μ\mu.

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

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

for the corresponding ∞-groupoid cardinality.


Dirac δ\delta-distributions

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

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

The left adjoint p *p^* is therefore a Lawvere distribution. This sends any (∞,1)-sheaf to its stalk at the point pp. 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

(ΠΔΓ):ΓΔΠ 0𝒮. (\Pi \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \mathcal{S} \,.

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

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

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

Local toposes

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

Multiplication of distributions by functions

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

Fμ:Gμ(F×G). F \cdot \mu : G \mapsto \mu(F \times G) \,.

In the functional notation this is the formula

Gd(F×μ)= G×Fdμ. \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 GrpdSh (,1)(*)\infty Grpd \simeq Sh_{(\infty,1)}(*) itself coincide with the value at the single point

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

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

Last revised on October 12, 2022 at 12:11:20. See the history of this page for a list of all contributions to it.