**OPEN PROBLEMS IN TOPOS THEORY**

Bill Lawvere

4 April 2009 * updated July 2016

*For Martin Hyland and Peter Johnstone in honor of their
sixtieth birthdays.*

Here are seven problems that I have not
yet been able to solve. Clarification on them would further advance work on
topos theory and its applications to thermomechanics, to algebraic geometry,
and to logic.

**(1) Quotient Toposes**

The number of "surjective" geometric
morphisms with domain a given Grothendieck topos is obviously not small. For
example, for every given monoid, there is the morphism to the topos formed by
the actions of that monoid on objects of the given topos. But how many are
those morphisms that, by analogy with the topological special case, are called "connected
surjective"? These actual quotients are determined by left exact comonads
that are moreover idempotent. Is there a Grothendieck topos for which the
number of these quotients is not small?

At the other extreme, could they be
parameterized internally, as subtoposes are?

**(2) Subqotients and Idempotents**

The (coherently) idempotent left exact
endofunctors parameterize the subquotients of a given topos, because the
splitting of idempotents in the left exact 2-category Lex can be analyzed into
two steps, as was shown by Pare, Rosebrugh and Wood. In one step the inclusion
is left-adjoint to the projection (corresponding to a quotient), and in the
other step the inclusion is right-adjoint to the projection (corresponding to
an inclusion). Since the system of left exact idempotents corresponds also to
certain left exact subcategories in their own right, it also has a lattice
structure. Thus, given any quotient (indeed any subquotient) there is a
smallest subtopos containing "it" (i.e. its inverse image), and
similarly given any subtopos there is a smallest quotient for which the
composite is an inclusion. The first of these closure operators clearly exists
because the class of subtoposes is parameterized by a single object; the question
of the existence of the second one is related to question (1) above.

What are the algebraic properties of
this lattice equipped with two closure operators, retracting respectively onto
the two kinds of special subquotients?

**(3) Boundaries of classes of models**

Because the system of subtoposes of any
topos is contravariantly parameterized by the points of an internal Heyting
algebra, it is itself a co-Heyting algebra and thus supports a boundary
operator that is defined by contradiction and that satisfies the Leibniz rule.
Because subtoposes represent classes of models of positive extensions of a
given theory, the effect of the boundary operator (and of the double negation
core operator) on classes of models needs to be determined both semantically
and syntactically.

**(4) The jump operator on levels within a topos**

The essential subtoposes of a given
topos were shown with Max Kelly to form a complete lattice. Each such "level"
receives a restriction or truncation functor from the top level and this functor
unites two opposite subcategories that, as categories in their own right, are
the same topos. For each level n, the essentiality gives an idempotent left
adjoint comonad sk(n) and the sheaf inclusion gives the idempotent right
adjoint monad cosk(n). The levels provide one systematic way of measuring the
complexity of objects; an object can be said to belong to level n if it is its
own n-skeleton. In the topos of simplicial sets, levels coincide with
dimensions. Besides the obvious ordering of levels, there is a stricter
ordering defined as follows: Level n is way below level m if every n-skeleton
is an m-coskeleton, i.e. every object belonging to level n is also in the
subtopos of m-sheaves. For the topos of simplicial sets Michael Zaks showed
over 20 years ago that above every level there is a smallest level that is way
above it; for the lowest dimensions this jump operator amounts to adding 1, but
for higher dimensions n it corresponds to 2n-1. *The proof is now published
(see below). There is another fundamental topos related to classical
constructions and combinatorial topology, namely the Boolean algebra classifier
that consists of presheaves on the category of finite non-empty sets. Here
again, the levels correspond to natural numbers (together with minus infinity),
so that there are sufficient chain conditions to ensure that the jump operator
exists. What is, in combinatorial or number theoretic terms, the way below
relation for this basic topos?

* Several key examples of this problem
were solved by the team of Carolyn Kennett, Emily Riehl, Michael Roy, and
Michael Zaks. *JPAA*
215 (2011), pp 949-961.

**(5) Coverings that admit averaging and microlinearity**

If K is a rig, then the category of
finitely-presented K-rigs is a co-extensive co-site, so that the
product-preserving functors from it to sets form a topos, inside of which we
can define a further subtopos. Namely, consider that an inclusion A -> B is
a (co)covering if there exists an averaging process from B onto A, i.e. a
retraction in the category of A-modules. For K = the field Q, the restriction
to extension fields of this covering notion applies to arbitrary inclusions,
yielding the Boolean topos that (as suggested by Galois) is the natural base
topos for algebraic geometry. More generally, if the natural numbers are
invertible in K and if G is a finite group operating on B with A as fixed
subalgebra, there is clearly such an averaging process. Similarly these
coverings imply that internally any nilpotent is a sum of first-order
nilpotents. The topos of sheaves thus has some of the good properties of
micro-linear objects, but it has the advantage of being a topos. The needed
stability property of these coverings follows from the fact that the coproduct
for A-algebras is actually a functor defined for the weaker structure of
A-modules.

In
general, what is the relation between micro-linearity and these coverings? Are
there other examples where a Grothendieck topology can be defined in this way
by averaging with respect to structure that is weaker than that classified by
the site? Can something similar be done with C-infinity functions and
distributions of compact support? How does the above particular example fit
into the family of Grothendieck topologies traditionally useful in algebraic
geometry?

**(6) How strong is the adjointness of fractional exponents?**

In a presheaf topos, the number of
ATOMs is small. In a general topos, there is for any finite set of ATOMs a
lower topos of relatively discrete objects over which it is defined and for
which the defining adjointness for those ATOMs and their corresponding
fractional exponents is enriched. Can the class of all ATOMs be suitably
parameterized and can we define a lower topos over which all these adjunctions
are enriched? (Recall that ATOMs are objects D for which ( )^D has a right
adjoint ( )^1/D; the term (distinct from Barr's above Boolean notion) was used
in Bunge's thesis to help characterize presheaf categories, but such objects
occur in more general contexts where they sometimes play the role of Amazingly
Tiny Objectified Motion, the fractional exponent being a special case of the
additional functor, characteristic of a local geometric morphism, that
Grothendieck denoted with a superscript exclamation point.)

**(7) The algebra of time**

Given a map D -> A in a topos where
D is an ATOM, the category of objects X, equipped with a prolongation operator
X^D -> X^A along this map, is a topos of laws of motion that receives an
essential geometric morphism from the given topos, enriched over the lower topos
of D-discrete objects. There is a further topos, consisting of actions of A^D,
and a functor ( )^D from the prolongation topos to this further topos,
corresponding to the classical representation of higher order ODE's in terms of
first-order ODE's on phase space. This process has a left-adjoint F enabling
the parameterization of any solutions of those equations over a time interval U
as a motion, i.e. as a morphism of prolongations with domain F(U).

Can the Algebra of Time F(U) be calculated in familiar terms ?