nLab finiteness space

1. Idea

A finiteness space is a set together with a collection of its subsets that behave, in certain ways, like the collection of finite subsets.

2. Definition

Given a set $X$ and a subset $\mathcal{U} \subseteq P(X)$ of its power set, define

$$\mathcal{U}^\perp = \{ v \in P(X) \mid u \cap v \;\text{ is finite for all }\; u\in \mathcal{U} \}.$$

This defines a Galois connection on $P(X)$.

Definition. A finiteness space is a set $X$ equipped with a $\mathcal{U} \subseteq P(X)$ such that $\mathcal{U} = \mathcal{U}^{\perp\perp}$.

We refer to the elements of $\mathcal{U}$ as finitary, and the elements of $\mathcal{U}^\perp$ as cofinitary.

3. Properties

• All finite subsets of $X$ belong to $\mathcal{U}^\perp$ for any $\mathcal{U}$. Thus, in a finiteness space all finite subsets are both finitary and cofinitary.

• Any set of the form $\mathcal{U}^\perp$ 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 $\mathcal{U}^\perp$ is closed under finite unions. Thus, any finite union of finitary subsets is finitary, and any finite union of cofinitary subsets is cofinitary.

4. Examples

• Taking $\mathcal{U} =$ all finite subsets of $X$ gives a minimal finiteness structure on $X$.

• Dually, taking $\mathcal{U}=$ all subsets of $X$ gives a maximal finiteness structure.

• In fact, if $(X,\mathcal{U})$ is a finiteness space, so is $(X,\mathcal{U}^\perp)$, 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 $X$ are finitary, then all cofinitary subsets must be finite, hence $X$ is maximal. Thus, to obtain finiteness spaces other than the minimal or maximal ones, we must invoke some other structure on $X$.

• Let $X$ be a poset and $\mathcal{U}$ the set of all subsets of $X$ 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 $X$ be a linearly ordered set and let $\mathcal{U}$ the set of all left-finite subsets, i.e. those $u$ such that $u\cap \{ x \mid x \le y\}$ is finite for all $y$. Since this is of the form $\mathcal{V}^\perp$ for $\mathcal{V}$ the set of subsets $\{x \mid x\le y\}$, it is a finiteness structure.

5. Categories of finiteness spaces

A relation or morphism of finiteness spaces is a relation $R$ from $X$ to $Y$ such that

1. If $u\subseteq X$ is finitary, then $R[u] = \{ y\in Y \mid \exists x\in u, x R y \}$ is finitary.
2. If $v\subseteq Y$ is cofinitary, then $R^{-1}[v] = \{ x\in X \mid \exists y\in v, x R y \}$ is cofinitary.

An equivalent way to say this is that

• If $u\subseteq X$ is finitary and $v\subseteq Y$ is cofinitary, then the set $\{ (x,y) \mid x\in u \wedge y\in v \wedge x R y \} \subseteq X \times Y$ has finite images under both projections $X\times Y\to X$ and $X\times Y\to Y$.

Additionally, in the presence of condition (1), condition (2) is equivalent to its special case when $v$ is a singleton.

Let $FinSp_{rel}$ denote the category of finiteness spaces and relations, and $FinSp_{par}$ and $FinSp_{fun}$ its wide subcategories in which the morphisms are restricted to be partial functions or functions, respectively.

Note that a partial function lies in $FinSp_{par}$ 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 $u\subseteq X$ and cofinitary $v\subseteq Y$, we have a surjection $u\cap f^{-1}(v) \to f(u) \cap v$ so that finiteness of the former implies finiteness of the latter.

Monoidal structures

• All three categories are symmetric monoidal and the inclusions $FinSp_{fun} \hookrightarrow FinSp_{par} \hookrightarrow FinSp_{rel}$ are strict symmetric monoidal.
• $FinSp_{rel}$ is star-autonomous (see Ehrhard).
• $FinSp_{par}$ is closed (see BCJS).

Initial and terminal structures

Let $Set_{par}$ denote the category of sets and partial functions. Note that $Set_{par}$ is equivalent to the category $Set_*$ of pointed sets, and hence in particular is complete and cocomplete.

The following theorem says that $FinSp_{par}$ and $FinSp_{fun}$ are almost topological concrete categories over $Set_{par}$ and $Set$ respectively.

Theorem. The (faithful) forgetful functors $U_{par}:FinSp_{par} \to Set_{par}$ and $U_{fun}:FinSp_{fun} \to Set$ have initial lifts for all (possibly large) sources $\{ f_i : X \rightharpoonup U(S_i) \}$ having the property that for all $x\in X$ there exists an $i$ such that $x\in dom(f_i)$.

Proof. The functor $U_{fun}$ is simply the pullback $U_{par}$ along the inclusion $Set \hookrightarrow Set_{par}$, so it suffices to consider the former, which we denote simply $U$.

Let $X$ be a set and $\{ f_i : X \rightharpoonup U(S_i) \}$ a $U$-structured source satisfying the given condition, where each $(S_i,\mathcal{V}_i)$ is a finiteness space. Let $\mathcal{V} = \big(\bigcup_i f_i^{-1}(\mathcal{V}_i^\perp)\big)^\perp$, making $X$ a finiteness space for which $u\subseteq X$ is finitary if and only if $u \cap f_i^{-1}(v)$ is finite for all cofinitary $v\subseteq S_i$. Thus each $f_i$ becomes a morphism in $FinSp_{par}$.

For universality, suppose given a finiteness space $T$ and a partial function $g:U(T)\rightharpoonup X$ such that each composite $f_i\circ g$ is a morphism in $FinSp_{par}$. The latter means that $(f_i\circ g)^{-1}(v)$ is cofinitary for all cofinitary $v\subseteq S_i$, which is to say $u \cap g^{-1}(f_i^{-1}(v))$ is finite for all finitary $u\subseteq T$. It follows that $g(u) \cap f_i^{-1}(v)$ is finite (being a surjective image of $u \cap g^{-1}(f_i^{-1}(v))$), hence $g(u)$ is finitary, i.e. condition (1) in the definition of morphism holds.

Thus it remains to show that for any $x\in X$ the preimage $g^{-1}(x)$ is cofinitary in $T$. By assumption, there exists an $i$ such that $x\in dom(f_i)$, which is to say $x\in f_i^{-1}(f_i(x))$. But then $g^{-1}(x) \subseteq g^{-1}(f_i^{-1}(f_i(x)))$, hence is cofinitary.  ▮

In the case of $U_{fun}$, the extra condition reduces to the statement that if $X$ is inhabited then so is the set of indices $i$. This condition is necessary: empty $U_{fun}$-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, $FinSp_{fun}$ does not have a terminal object, though it does have all inhabited limits, and the forgetful functor $FinSp_{fun} \to Set$ is a Grothendieck fibration.

A similar argument shows that $FinSp_{par}$ has all inhabited limits. (The above is essentially the argument given for this in BCJS, unraveled into an explicit construction of limits in $Set_{par}$ in terms of the equivalence to $Set_*$.) But $FinSp_{par}$ also has a terminal object, namely the empty set with its unique finiteness structure, so it is a complete category.

In fact, $FinSp_{par}$ is also a cocomplete category, as shown in BCJS. But its coequalizers are not preserved by $U_{par}$, and in particular not constructed as final lifts of $U_{par}$-structured sinks.

6. Matrices

Let $R$ be any semiring; we can define a category $FMat(R)$ as follows:

• Its objects are finiteness spaces.
• Its morphisms $X\to Y$ are functions $M:X\times Y \to R$ whose support $\{ (x,y) \mid M(x,y) \neq 0 \}$ is a morphism $X\to Y$ in $FinSp_{rel}$. We write this morphism in $FinSp_{rel}$ as $supp(M):X\to Y$.

The composite of $M:X\to Y$ and $N:Y\to Z$ is defined by

$$(N M)(x,z) = \sum_{M(x,y)\neq 0 \neq N(y,z)} N(y,z) \cdot M(x,y).$$

To see that this is well-defined, first note that by assumption, for any $x$ the set $\{ y \mid M(x,y)\neq 0 \}$ is finitary, and for any $z$ the set $\{ y \mid N(y,z)\neq 0 \}$ is cofinitary. Thus their intersection is finite, so the above sum is finite and thus makes sense. Now to check that $N M$ is indeed a morphism $X\to Y$ in $FMat(R)$, note that if $(N M)(x,z)\neq 0$ it must be that there exists a $y$ such thath $M(x,y)\neq 0 \neq N(y,z)$, and therefore $(supp(N) \circ supp(M))(x,z)$ holds. In other words, $supp(N M) \subseteq supp(N) \circ supp(M)$, and therefore $supp(N M)$ is also a morphism in $FinSp_{rel}$.

It is straightforward to check associativity and unitality, so we have a category $FMat(R)$. We note the following properties:

• $FMat(R)$ is enriched over abelian monoids (abelian groups if $R$ is a ring), and indeed over $R$-modules.

• $FMat(R)$ 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 $R= \{\bot,\top\}$ the set of truth values, we have $FMat(R) = FinSp_{rel}$.

• The proof above that $supp(N M) \subseteq supp(N) \circ supp(M)$, plus a simpler argument in the unary case, shows that for any $R$ we have an identity-on-objects colax functor $supp: FMat(R) \to FinSp_{rel}$. (For general $R$, $FMat(R)$ is not a 2-category, but it is when $R$ is a distributive lattice, including the case of $FinSp_{rel}$; thus this makes sense.)

• For general $R$, the category $FMat(R)$ is star-autonomous, and the functor $supp: FMat(R) \to FinSp_{rel}$ preserves this structure strictly.

• Any morphism $H : X\to Y$ in $FinSp_{rel}$ induces a morphism $\hat{H}:X\to Y$ in $Mat(R)$ by setting $\hat{H}(x,y) = 1$ if $H(x,y)$ and $0$ otherwise. However, this does not define a functor $FinSp_{rel} \to FMat(R)$ since morphisms of the form $\hat{H}$ need not be closed under composition, as the composite of two such might involve a finite sum of copies of $1$. There are two cases when this does work: when $1$ is an additive idempotent in $R$ (such as when $R$ is a distributive lattice), and when we restrict the domain to $FinSp_{par}$. In particular, we have a strict symmetric monoidal functor $\chi : FinSp_{par} \to FMat(R)$ for any $R$ that is a partial section of $supp$.

7. Linearization

Let $(X,\mathcal{U})$ be a finiteness space and $A$ an abelian group (or more generally an abelian monoid), and define

$$A \langle X\rangle = \{ f : X\to A \mid supp(f) \in \mathcal{U} \}$$

where $supp(f) = \{ x\in X \mid f(x)\neq 0 \}$ is the support of $f$. Then $A\langle X\rangle$ is an abelian group called the linearization of $X$ (with coefficients in $A$).

Note that if $A=\{\bot,\top\}$, then $A\langle X\rangle$ is just the set $\mathcal{U}$ of finitary subsets of $X$.

It is shown in BCJS that if $X$ is a partial finiteness monoid (i.e. a monoid in $FinSp_{par}$) and $R$ a ring, then $R\langle X\rangle$ is also a ring with

$$(f\cdot g)(m) = \sum_{(m_1,m_2)\in X_m(f,g)} f(m_1)\cdot g(m_2)$$

where $X_m(f,g)$ is the set of pairs $(m_1,m_2)\in supp(f) \times supp(g)$ such that $m_1 \cdot m_2 = m$. In fact $R$ need only be a semiring (in which case so is $R\langle X\rangle$).

If $X$ is merely a monoid in $FinSp_{rel}$, then the above multiplication is still defined, but may not be associative or unital. This can be explained more abstractly as follows. We have $R\langle X\rangle = FMat(R)(1,X)$, where $1$ is the one-element finiteness space (the monoidal unit, which is terminal in $FinSp_{fun}$ but not in any of the other categories of finitenes spaces). Thus, $R\langle X\rangle$ inherits a multiplicative monoid structure as soon as $X$ is a monoid object in $FMat(R)$. As noted above, we have a monoidal functor $\chi : FinSp_{par} \to FMat(R)$, which therefore preserves monoids; but no such functor whose domain is $FinSp_{rel}$.

Examples

• If $X$ is a minimal finiteness space, then the linearization $A\langle X\rangle$ is then the copower of $A$ by the set $X$ in Ab. Any monoid (or partial monoid) $M$ is a finiteness monoid with the minimal finiteness structure, and when $R$ is a ring the resulting ring structure on $R\langle M\rangle$ is induced on the basis elements by the multiplication of $M$. When $M$ is a group $G$, this is the group algebra $R[G]$.

• If $X$ 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 $X$ with coefficients in $R$ (see BCJS). Thus, this includes rings of formal power series, formal Laurent series?, polynomials, Laurent polynomials, and Hahn series.

• If $X$ 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.

8. Related pages

• Finiteness spaces were defined by analogy to coherence spaces in which the property “is finite” is replaced by “is a subsingleton”. Thus, both are special cases of arity spaces for two different sets of arities.

9. References

• T. Ehrhard, Finiteness spaces, Mathematical Structures in Computer Science 15, pp. 615–646 (2005)

• Richard Blute, Robin Cockett, Pierre-Alain Jacqmin, and Philip Scott, Finiteness spaces and generalized power series, Electronic Notes in Theoretical Computer Science, 341 (2018), 5-22, doi, arXiv:1805.09836