objects such that commutes with certain colimits
As there are already at least five distinct notions of finite set in constructive mathematics, so there must be at least five distinct notions of finite object internal to a topos. Additionally, the definitions may also be interpreted in an ‘external’ sense, giving even further notions. Only some are mentioned below.
Also beware that in category theory the term ‘finite object’ is also used in a much more general sense to mean a compact object. Similar finiteness meaning may also be attributed to dualizable objects in monoidal categories and to perfect complexes (of abelian sheaves) in geometry.
The internal version of a “finite set” is an object such that “ is a finite cardinal” is true in the internal logic. This is equivalent to the following
An object is locally isomorphic to a finite cardinal, if there is an epimorphism and a generalized element such that over . Equivalently, there is a such that is a finite cardinal in the slice topos .
An internally finitely indexed object is an object is which is locally a quotient of a finite cardinal, hence such that there is an epimorphism , a finite cardinal in the slice topos and an epimorphism .
An “internally finitely indexed” object is generally called a Kuratowski-finite object or -finite object for short, and an “internally subfinitely indexed” one is called a -finite object.
Since it is still provable in the internal logic that any decidable finitely indexed set is finite, the “internally finite” objects (those that are locally isomorphic to a finite cardinal, as above) can be characterized as the decidable -finite objects.
The following lists closure properties of K-finite objects, def. 1.
A coproduct is K-finite precisely if both summands are.
A product of two K-finite objects is K-finite.
This appears in (Johnstone) as lemma D5.4.4, corollary D5.4.5, pro. 5.4.8.
The full subcategory of finite cardinals in any topos is again a topos, and it is Boolean. Its subobject classifier is , which in the ambient topos is the classifier only of decidable subobjects. This means that classically valid arguments, including all of finitary combinatorics, can generally be applied easily to finite cardinals, as long as we always interpret “subset” to mean “decidable subset.”
The category of -finite objects is a topos if and only if every -finite object is decidable, and the category of -finite objects is a topos if (but not only if) the subobject classifier is -finite.
The first statement appears as (Johnstone, theorem 5.4.18).
The full subcategory can be regarded as the “stack completion” of the topos of finite cardinals.
An object is K-finite precisely if the étale geometric morphism
In any Boolean topos, all four internal notions coincide. In a well-pointed topos, each internal notion coincides with its external notion. Therefore, in a well-pointed Boolean topos, including the topos Set as usually conceived, all notions of finiteness coincide.
In a presheaf topos , the finite cardinals are the finite-set-valued functors which are constant on each connected component. In particular, if is a group, then the topos of finite cardinals is equivalent to FinSet.
By contrast, the -finite objects in are the finite-set-valued functors each of whose transition functions is surjective, and the decidable K-finite objects are the finite-set-valued functors each of whose transition functions is bijective.
In particular, if is a groupoid, the topos of decidable -finite objects is equivalent to . Since the topos of presheaves on a groupoid is Boolean, this gives an example of a Boolean topos in which the finite cardinals (“externally finite objects”) and the (decidable) -finite objects (“internally finite objects”) fail to coincide.
In the category of sheaves over a topological space, the decidable K-finite objects are those that are “locally finite;” i.e. there is an open cover of such that over each open in the cover, the sheaf is a locally constant function to . These are essentially the same as covering spaces of with finite fibres.
|geometry||monoidal category theory||category theory|
|perfect module||(fully-)dualizable object||compact object|
finite cardinal objects are discussed in section D5.2, Kuratowski finite objects in section D5.4
O. Acuña-Ortega, Fred Linton, Finiteness and decidability: I Springer Lecture Notes in Mathematics, (1979), Volume 753/1979, 80-100, DOI: 10.1007/BFb0061813
B. P. Chisala, M.-M. Mawanda, Counting Measure for Kuratowski Finite Parts and Decidability , Cah.Top.Géom.Diff.Cat. XXXII 4 (1991) pp.345-353. (pdf)
T. Streicher, P. Freyd, F. Linton, P.Johnstone, W. Lawvere, catlist discussion ‘finiteness in toposes’, January 1997. (link)
C. Kuratowski, Sur la notion d’ensemble fini , Fund. Math. 1 (1920) pp.129-131. (pdf)