(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
To some extent one can think of a sheaf on a topological space as being like a Set-valued function on that space: to each point it assigns the stalk . 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 -toposes to general locally presentable (∞,1)-categories. Viewed this way, Lawvere distributions are the morphism in , the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories.
Throughout is some base topos or (∞,1)-topos and all notions are to be understood as indexed over this base.
Let and be (n,1)-toposes. A distribution on with values in is a (∞,1)-functor
that preserves small (∞,1)-colimits.
for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve finite colimits.
A point of a topos is a geometric morphism of the form
The left adjoint is therefore a Lawvere distribution. This sends any (∞,1)-sheaf to its stalk at the point . So this behaves like the Dirac distribution on functions.
The canonical distribution on a locally -connected topos
If is a locally ∞-connected (∞,1)-topos then its terminal global section (∞,1)-geometric morphism by definition has a further left adjoint
This left adjoint (the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos) is therefore a canonical -valued distribution on . It is also written
Multiplication of distributions by functions
For an -sheaf and a distribution, there is a new distribution
In the functional-notation this is the formula
Distributions on the point
The ∞Grpd-valued distributions on 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
- 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 Jonathan Funk, Singular coverings of toposes Lecture Notes in Mathematics, (2006) Volume 1890/2006
chapter 1 Lawvere Distributions on Toposes
For the -category theory generalization see Pr(∞,1)Cat and references given there.