equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
Throughout $\mathcal{S}$ is some base topos or (∞,1)-topos and all notions are to be understood as indexed over this base.
Let $\mathcal{E}$ and $\mathcal{K}$ be (n,1)-toposes. A distribution on $\mathcal{E}$ with values in $\mathcal{K}$ is a (∞,1)-functor
that preserves small (∞,1)-colimits.
Write
for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve finite colimits.
By the adjoint (∞,1)-functor theorem this is equivalently a pair
To amplify the interpretation in analogy with distributions in functional analysis one sometimes write
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
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
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
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 $(\infty,1)$-category theory generalization see Pr(∞,1)Cat and references given there.