Higher category theory
higher category theory
Extra properties and structure
(∞,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
There is a sense in which a sheaf is like a categorification of a function: the stalk-map from topos points to sets we may think of under decategorification as a cardinality-valued function.
Under this interpretation, many constructions in category theory have analogs in linear algebra: for instance products of numbers correspond to categorical products (more generally to limits) and addition of numbers to coproducts (more generally to colimits). Accordingly a colimit-preserving functor between sheaf toposes is analogous to a linear map or to a distribution: one also speaks of Lawvere distributions.
This categorification of linear algebra becomes even better behaved if we pass all the way to (∞,1)-sheaf (∞,1)-toposes. Under ∞-groupoid cardinality their stalks take values also in integers, in rational numbers, and in real numbers. See also the discussion at Goodwillie calculus.
A span of base change geometric morphisms between toposes behaves under this interpretation like the linear map given by a matrix. Such categorified integral transforms turn out to be of considerable interest in their own right: they include operations such as the Fourier-Mukai transform which categorifies the Fourier transform.
These analogies have been noticed and exploited at various places in the literature. See for instance the entries groupoidification or geometric ∞-function theory. Here we try to give a general abstract (∞,1)-topos theoretic description with examples from ordinary topos theory to motivate the constructions.
Every (∞,1)-topos is a locally presentable (∞,1)-category. More generally we may think of arbitrary locally presentable (∞,1)-categories as being analogous to vector spaces of linear functionals.
See locally presentable (∞,1)-category for details.
See profunctor for details.
For Pr(∞,1)Cat we have that is itself locally presentable.
See Pr(∞,1)Cat for details.
We consider from now on some fixed ambient (∞,1)-topos .
Notice that for each object the over-(∞,1)-topos is the little topos of -sheaves on . So to the extent that we think of these as function objects , and of locally presentable -categories as linear spaces, we may think of as the -vector space of -functions on
Let FinSet be the ordinary topos of finite sets. Then for a finite set, a function object on is a morphism of sets. Under the cardinality decategorification
we think of this as the function
where is the fiber of over .
Let ∞Grpd. By the (∞,1)-Grothendieck construction we have for an ∞-groupoid an equivalence of (∞,1)-categories
of the over-(∞,1)-category of all -groupoids over with the (∞,1)-category of (∞,1)-presheaves on . And since the -groupoid is equivalent to its opposite (∞,1)-category this is also equivalent to the (∞,1)-category of (∞,1)-functors from to ∞Grpd.
Products of function objects
For and in two function objects on , their product in we call the product of function objects.
This is computed in as the fiber product
and the morphism down to is the evident projection
In FinSet we have that the -valued function underlying the product function object is the usual pointwise product of functions
For every morphism in the ambient (∞,1)-topos there is the corresponding base change geometric morphism
between the corresponding over-(∞,1)-toposes. Here acts simply by postcomposition with :
while acts by (∞,1)-pullback along :
There is a further right adjoint . For the present purpose the relevance of its existence is that it implies that both as well as are left adjoints and hence both preserve (∞,1)-colimits. Therefore these are morphism in Pr(∞,1)Cat and hence behave like linear maps on our function spaces and .
When we think of base change in the context of linear algebra on sheaves, we shall write
and call the fiber integration of over the fibers of . In particular when is the terminal object we write simply
for the integral of with values in the ambient -topos. (See also the notation for Lawvere distributions).
Consider the ordinary topos FinSet and for any set the unique morphism to the terminal object.
For a function object with underlying function we have that the integral
has as underlying function the constant
If we are given an oriented span or correspondence
in it induces by composition of pullback and fiber integration operations a colimit-preserving -functor
We may always factor through the (∞,1)-product
We call the function object
on the integral kernel of .
We have the pull-tensor-push formula for :
This follows from the pasting law for pullbacks in :
In FinSet we have that is a -by--matrix with entries in natural numbers and the function
is the result of applying the familiar linear map given by usual matrix calculus on .
In the case ∞Grpd we have – as in the above example – by the (∞,1)-Grothendieck construction an equivalence
Since the -groupoid is equivalent to its opposite (∞,1)-category this may also be written as
The objects on the right we may again think of as -profunctors . So in particular the kernel is under this equivalence on the right hand identified with an -profunctor