A finiteness space is a set together with a collection of its subsets that behave, in certain ways, like the collection of finite subsets.
Given a set and a subset of its power set, define
This defines a Galois connection on .
A finiteness space is a set equipped with a such that .
We refer to the elements of as finitary, and the elements of as cofinitary.
All finite subsets of belong to for any . Thus, in a finiteness space all finite subsets are both finitary and cofinitary.
Any set of the form is down-closed, i.e. contains all subsets of its elements. Thus, any subset of a finitary subset is finitary, and any subset of a cofinitary subset is cofinitary.
Any set of the form is closed under finite unions. Thus, any finite union of finitary subsets is finitary, and any finite union of cofinitary subsets is cofinitary.
Taking all finite subsets of gives a minimal finiteness structure on .
Dually, taking all subsets of gives a maximal finiteness structure.
In fact, if is a finiteness space, so is , its dual. The maximal and minimal finiteness structures are dual.
We cannot obtain any other finiteness spaces purely by cardinality restriction. For instance, if all countable subsets of are finitary, then all cofinitary subsets must be finite, hence is maximal. Thus, to obtain finiteness spaces other than the minimal or maximal ones, we must invoke some other structure on .
Let be a poset and the set of all subsets of that are noetherian, i.e. contain no infinite strictly increasing chains. Then this is a finiteness space. Its dual (at least, assuming the axiom of choice) consists of subsets that are artinian (contain no infinite strictly decreasing chains) and narrow (contain no infinite antichains).
Let be a linearly ordered set and let the set of all left-finite subsets, i.e. those such that is finite for all . Since this is of the form for the set of subsets , it is a finiteness structure.
A relation or morphism of finiteness spaces is a relation from to such that
An equivalent way to say this is that
Additionally, in the presence of condition (1), condition (2) is equivalent to its special case when is a singleton.
Let denote the category of finiteness spaces and relations, and and its wide subcategories in which the morphisms are restricted to be partial functions or functions, respectively.
Note that a partial function lies in if and only if the preimage of any cofinitary subset is cofinitary. For this is precisely the second condition above specialized to a partial function, while conversely if this holds then for any finitary and cofinitary , we have a surjection so that finiteness of the former implies finiteness of the latter.
Let denote the category of sets and partial functions. Note that is equivalent to the category of pointed sets, and hence in particular is complete and cocomplete.
The following theorem says that and are almost topological concrete categories over and respectively.
The (faithful) forgetful functors and have initial lifts for all (possibly large) sources having the property that for all there exists an such that .
The functor is simply the pullback along the inclusion , so it suffices to consider the former, which we denote simply .
Let be a set and a -structured source satisfying the given condition, where each is a finiteness space. Let , making a finiteness space for which is finitary if and only if is finite for all cofinitary . Thus each becomes a morphism in .
For universality, suppose given a finiteness space and a partial function such that each composite is a morphism in . The latter means that is cofinitary for all cofinitary , which is to say is finite for all finitary . It follows that is finite (being a surjective image of ), hence is finitary, i.e. condition (1) in the definition of morphism holds.
Thus it remains to show that for any the preimage is cofinitary in . By assumption, there exists an such that , which is to say . But then , hence is cofinitary.
In the case of , the extra condition reduces to the statement that if is inhabited then so is the set of indices . This condition is necessary: empty -structured sources on inhabited sets do not have initial lifts. The only candidate is the maximal finiteness structure in which all subsets are finitary and only finite sets are cofinitary, but not all functions into a maximal finiteness space are finiteness functions (only those with finite fibers).
In particular, does not have a terminal object, though it does have all inhabited limits, and the forgetful functor is a Grothendieck fibration.
A similar argument shows that has all inhabited limits. (The above is essentially the argument given for this in BCJS, unraveled into an explicit construction of limits in in terms of the equivalence to .) But also has a terminal object, namely the empty set with its unique finiteness structure, so it is a complete category.
In fact, is also a cocomplete category, as shown in BCJS. But its coequalizers are not preserved by , and in particular not constructed as final lifts of -structured sinks.
Let be any semiring; we can define a category as follows:
The composite of and is defined by
To see that this is well-defined, first note that by assumption, for any the set is finitary, and for any the set is cofinitary. Thus their intersection is finite, so the above sum is finite and thus makes sense. Now to check that is indeed a morphism in , note that if it must be that there exists a such thath , and therefore holds. In other words, , and therefore is also a morphism in .
It is straightforward to check associativity and unitality, so we have a category . We note the following properties:
is enriched over abelian monoids (abelian groups if is a ring), and indeed over -modules.
has a zero object (the empty finiteness space) and finite biproducts (disjoint unions of finiteness spaces).
Any distributive lattice is a semiring with join as addition and meet as multiplication. In the case the set of truth values, we have .
The proof above that , plus a simpler argument in the unary case, shows that for any we have an identity-on-objects colax functor . (For general , is not a 2-category, but it is when is a distributive lattice, including the case of ; thus this makes sense.)
For general , the category is star-autonomous, and the functor preserves this structure strictly.
Any morphism in induces a morphism in by setting if and otherwise. However, this does not define a functor since morphisms of the form need not be closed under composition, as the composite of two such might involve a finite sum of copies of . There are two cases when this does work: when is an additive idempotent in (such as when is a distributive lattice), and when we restrict the domain to . In particular, we have a strict symmetric monoidal functor for any that is a partial section of .
Let be a finiteness space and an abelian group (or more generally an abelian monoid), and define
where is the support of . Then is an abelian group called the linearization of (with coefficients in ).
Note that if , then is just the set of finitary subsets of .
It is shown in BCJS that if is a partial finiteness monoid (i.e. a monoid in ) and a ring, then is also a ring with
where is the set of pairs such that . In fact need only be a semiring (in which case so is ).
If is merely a monoid in , then the above multiplication is still defined, but may not be associative or unital. This can be explained more abstractly as follows. We have , where is the one-element finiteness space (the monoidal unit, which is terminal in but not in any of the other categories of finitenes spaces). Thus, inherits a multiplicative monoid structure as soon as is a monoid object in . As noted above, we have a monoidal functor , which therefore preserves monoids; but no such functor whose domain is .
If is a minimal finiteness space, then the linearization is then the copower of by the set in Ab. Any monoid (or partial monoid) is a finiteness monoid with the minimal finiteness structure, and when is a ring the resulting ring structure on is induced on the basis elements by the multiplication of . When is a group , this is the group algebra .
If is a poset whose finitary subsets are the artinian and narrow ones, and whose multiplication preserves the strict order on each side, then it is a finitenes monoid, and its linearization is the ring of Ribenboim power series from with coefficients in (see BCJS). Thus, this includes rings of formal power series, formal Laurent series?, polynomials, Laurent polynomials, and Hahn series.
If is a linearly ordered abelian group and the finitary subsets are the left-finite ones, then its linearization is the corresponding Novikov field.
As shown in BCJS, other examples include Puiseux series, formal power series over a free monoid, and polynomials of bounded degree.
Last revised on July 24, 2019 at 17:24:45. See the history of this page for a list of all contributions to it.