Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
(2,1)-quasitopos?
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 $F$ 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.
Throughout $\mathcal{S}$ 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 $\mathcal{K}$ 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
that preserves $S$-small (∞,1)-colimits. We write
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.
By the adjoint (∞,1)-functor theorem this is equivalently a pair
of ($S$-indexed) adjoint (∞,1)-functors.
To amplify the interpretation in analogy with distributions in functional analysis one sometimes writes
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
for the corresponding ∞-groupoid cardinality.
A point of a topos is a geometric morphism of the form
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.
If $\mathcal{E}$ is a locally ∞-connected (∞,1)-topos then its terminal global section (∞,1)-geometric morphism by definition has a further left adjoint
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
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 $E$.
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 $E$.
For $F \in \mathcal{E}$ an $(\infty,1)$-sheaf and $\mu : \mathcal{E} \to \mathcal{S}$ a distribution, there is a new distribution
In the functional notation this is the formula
The ∞Grpd-valued distributions on $\infty Grpd \simeq Sh_{(\infty,1)}(*)$ itself coincide with the value at the single point
For the $(\infty,1)$-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
A comprehensive discussion is in
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.
Last revised on October 12, 2022 at 12:11:20. See the history of this page for a list of all contributions to it.