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presentation axiom in nLab

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The basis of it all

 Set theory

set theory

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foundational axioms

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Idea

In the foundations of mathematics, it's interesting to consider the axiom that the Category of Sets Has Enough Projectives; in short: CoSHEP (pronounced /ko:-shep/). This is more commonly known as the presentation axiom: PAx. It is a weak form of the axiom of choice.

Statement

In elementary terms, CoSHEP states

Axiom (CoSHEP)

For every set AA, there exists a set PP and a surjection P→AP \to A, such that every surjection X↠PX \twoheadrightarrow P has a section.

This should be read in view of the definition of projective objects:

Definition

An object PP in a category CC is (externally) projective iff the hom-functor C(P,−):C→SetC(P, -): C \to Set takes epis to epis. This is the same as saying: given an epi p:B→Ap: B \to A and a map f:P→Af: P \to A, there exists a lift g:P→Bg: P \to B in the sense that f=p∘gf = p \circ g.

Accordingly, in a topos the CoSHEP axiom says equivalently

Borrowing from the philosophy of constructivism, we may also call this a complete presentation.

Consequences

The existence of sufficiently many projective presentations plays a central role in homological algebra as a means to construct projective resolutions of objects. Tradtionally one often uses the axiom of choice to prove that categories of modules have enough projectives, on the grounds that the free modules are projective.

But the weaker assumption of CoSHEP is already sufficient for this purpose: while not every free module will be projective, one can still use CoSHEP to find a projective presentation for every free module (and thus for every module). This is discussed in more detail here.

Note that we normally assume (3) for the category of sets, which is true in any (constructively) well-pointed pretopos and true internally in any pretopos whatsoever, so one normally says that DC and CC simply follow from the existence of enough projectives (CoSHEP). Equivalently, internal DC and internal CC follow from internal CoSHEP.

Proof

Condition 1 easily implies 2. Condition 2 says precisely that the natural numbers object ℕ\mathbb{N} is externally projective, and since 11 is a retract of ℕ\mathbb{N}, it is projective under condition 2, so 2 implies 3. It remains to show 3 implies 1.

Let XX be inhabited, so there exists an entire relation given by a jointly monic span

1←epiU→fX,1 \stackrel{epi}{\leftarrow} U \stackrel{f}{\to} X,

and similarly let

X←epiπ 1R→π 2XX \stackrel{epi \pi_1}{\leftarrow} R \stackrel{\pi_2}{\to} X

be an entire binary relation. Let p:P→Xp: P \to X be a projective cover. Since 11 is assumed projective, the cover U→1U \to 1 admits a section σ:1→U\sigma: 1 \to U, and the element fσ:1→Xf \sigma: 1 \to X lifts through pp to an element x 0:1→Px_0: 1 \to P. Next, in the diagram below, pp lifts through the epi π 1\pi_1 to a map q:P→Rq: P \to R, and then π 2q\pi_2 q lifts through pp to a map ϕ\phi (since PP is projective):

P →ϕ P ↙p ↓q ↓p X ←π 1 R →π 2 X\array{ & & P & \stackrel{\phi}{\to} & P \\ & \swarrow p & \downarrow q & & \downarrow p \\ X & \underset{\pi_1}{\leftarrow} & R & \underset{\pi_2}{\to} & X }

By the universal property of ℕ\mathbb{N} (see recursion), there exists a unique map h:ℕ→Ph: \mathbb{N} \to P rendering commutative the diagram

1 →0 ℕ →s ℕ id↓ ↓h ↓h 1 →x 0 P →ϕ P ↙p ↓q ↓p X ←π 1 R →π 2 X\array{ 1 & \stackrel{0}{\to} & \mathbb{N} & \stackrel{s}{\to} & \mathbb{N} \\ id \downarrow & & \downarrow h & & \downarrow h \\ 1 & \underset{x_0}{\to} & P & \underset{\phi}{\to} & P \\ & \swarrow p & \downarrow q & & \downarrow p \\ X & \underset{\pi_1}{\leftarrow} & R & \underset{\pi_2}{\to} & X }

Clearly ⟨ph,phs⟩:ℕ→X×X\langle p h, p h s \rangle : \mathbb{N} \to X \times X factors through ⟨π 1,π 2⟩:R→X×X\langle \pi_1, \pi_2 \rangle : R \to X \times X, i.e., ∀ n:ℕ(ph(n),ph(n+1))∈R\forall_{n: \mathbb{N}} (p h(n), p h(n+1)) \in R, thus proving that dependent choice holds under CoSHEP.

Example

A topos in which CoSHEP holds but 11 is not projective is Set CSet^C, where CC is the category with three objects and exactly two non-identity arrows a→b←ca \to b \leftarrow c. For if U:C→SetU: C \to Set is a functor with U(a)={a 0}U(a) = \{a_0\}, U(b)={b 0,b 1}U(b) = \{b_0, b_1\}, and U(c)={c 0}U(c) = \{c_0\}, with U(a→b)(a 0)=b 0U(a \to b)(a_0) = b_0 and U(c→b)(c 0)=b 1U(c \to b)(c_0) = b_1, then the map U→1U \to 1 is epi but has no section, so 11 is not projective. On the other hand, as noted below, every presheaf topos satisfies CoSHEP, assuming that SetSet itself does.

CoSHEP also implies several weaker forms of choice, such as the axiom of multiple choice and WISC. In weakly predicative mathematics, it can be combined with the existence of function sets to show the subset collection axiom.

Variants of the presentation axiom

 Split surjections

In the absence of the full axiom of choice, there are actually two notions of surjections: surjections, and split surjections. Hence, we get four possible different statements of the presentation axiom, depending on whether one uses surjections or split surjections:

  1. For every set AA, there exists a set PP and a surjection P→AP \to A, such that every surjection X↠PX \twoheadrightarrow P has a section.

  2. For every set AA, there exists a set PP and a split surjection P→AP \to A, such that every surjection X↠PX \twoheadrightarrow P has a section.

  3. For every set AA, there exists a set PP and a surjection P→AP \to A, such that every split surjection X↠PX \twoheadrightarrow P has a section.

  4. For every set AA, there exists a set PP and a split surjection P→AP \to A, such that every split surjection X↠PX \twoheadrightarrow P has a section.

By definition of split surjection, every split surjection has a section, so the third and fourth versions of the presentation axioms are simply:

These two versions of the presentation axiom are always true: take the set PP to be AA and the (split) surjection P→AP \to A to be the identity function on AA.

This leaves the first and second version of the presentation axiom as non-trivial axioms that one can add to the foundations. In general, the version using a split surjection into AA is stronger than the version using a surjection into AA, because not every surjection is a split surjection unless the full axiom of choice holds. And then the usual axiom of choice is simply the presentation axiom where the surjection is required to be a bijection.

External and internal versions

In addition, every set theory has an internal logic defined on its subsingletons:

Then there also exist internal versions of the presentation axiom, which states that the presentation axiom is true when expressed internally in the set theory, that a particular subsingleton defined using the set theoretic operations above is a singleton.

BHK interpretation of the presentation axiom

In addition, there are two different ways to interpret predicate logic in the internal logic of a set theory:

This means that we get even more versions of the presentation axiom in set theory, depending on where one uses the traditional interpretation of predicate logic and where one uses the BHK interpretation of predicate logic in the internal logic:

  • For surjections, whether the fibers of the function f:P→Af:P \to A are internally inhabited or pointed

  • For split surjections, whether given a function f:P→Af:P \to A there exists a section g:A→Pg:A \to P, or whether one has a section-retraction pair (f,g):(P→A)×(A→P)(f, g):(P \to A) \times (A \to P)

All of these combine together to form a huge number of possible axioms of the presentation axiom, ranging from the provable to stronger than the usual presentation axiom.

In dependent type theory

In dependent type theory, not all types are sets, and sets (and other types) are usually elements of types called universes, so there are even more versions of the presentation axiom, depending on

  • Whether given a set A:UA:U there exists a set P:UP:U or whether one has a pair of sets (A,P):U×U(A, P):U \times U,

  • Whether one uses sets or types in defining the presentation axiom.

In a topos

When working in the internal logic of a topos, the “internal” meaning of CoSHEP is “every object is covered by an internally projective object.” (Compare with the internal axiom of choice: every object is internally projective.) As regards foundational axioms for toposes (in the sort of sense that the axiom of choice is regarded as “foundational”), the internal version of the presentation axiom should be taken as the default version.

Proposition

Suppose that 11 is (externally) projective in EE. Then EE satisfies PAx whenever it satisfies internal PAx.

Internal PAx does not follows from external PAx; see Counterexample 5.3. However, if every object is projective (AC), then every object is internally projective (IAC).

A stronger version of PAx may be worth considering. Say that an object is stably projective if its pullback to any slice category is projective. Then stably projective objects are internally projective (proof?). Similarly, if we say that a topos EE satisfies stable PAx if every object is covered by a stably projective object, then a topos satisfies internal PAx if it satisfies stable PAx.

Example

Every presheaf topos Set C opSet^{C^{op}} has enough projectives, since any coproduct of representables is projective. If in addition CC has binary products, then by this result, Set C opSet^{C^{op}} validates internal PAx.

Counterexample

However, not every presheaf topos validates internal PAx, even though every presheaf topos validates external PAx. As an example, let CC be the category where Ob(C)Ob(C) is the disjoint sum ℕ∪{a,b}\mathbb{N} \cup \{a, b\}, and preordered by taking the reflexive transitive closure of relations n≤n+1n \leq n+1, n≤an \leq a, n≤bn \leq b. The claim is that neither C(−,a)C(-, a), nor any presheaf PP that maps epimorphically onto C(−,a)C(-, a), can be internally projective. Indeed, consider the presheaf FF defined by F(a)=F(b)=∅F(a) = F(b) = \emptyset and F(n)=[n,∞)F(n) = [n,\infty), with F(n+1→n)F(n+1 \to n) the evident inclusion. Let GG be the support of FF, so that we have an epi e:F→Ge: F \to G.

The objects C(−,a),C(−,b)C(-, a), C(-, b), and GG are subterminal and G≅C(−,a)∩C(−,b)≅C(−,a)×C(−,b)G \cong C(-, a) \cap C(-, b) \cong C(-, a) \times C(-, b). The set F C(−,a)(b)F^{C(-, a)}(b) is empty because there is no C(−,a)×C(−,b)→FC(-, a) \times C(-, b) \to F (it would give a section G→FG \to F of e:F→Ge: F \to G, but none exists), whereas G C(−,a)(b)G^{C(-, a)}(b) is inhabited by C(−,a)×C(−,b)≅GC(-, a) \times C(-, b) \cong G. For any PP covering C(−,a)C(-, a), the set F P(b)F^P(b) is empty (because any section s:C(−,a)→Ps: C(-, a) \to P of P→C(−,a)P \to C(-, a) induces a function F P(b)→F C(−,a)(b)=0F^P(b) \to F^{C(-, a)}(b) = 0), and the set G P(b)G^P(b) is inhabited (the map P→C(−,a)P \to C(-, a) induces a map 1≅G C(−,a)(b)→G P(b)1 \cong G^{C(-, a)}(b) \to G^P(b)). Thus the map e P:F P→G Pe^P \colon F^P \to G^P cannot be epic.

Counterexample

Any topos that violates countable choice, of which there are plenty, must also violate internal PAx.

Example

An interesting example of a topos that has enough projectives and satisfies internal CoSHEP (at least, assuming the axiom of choice in Set), although it violates the full (internal) axiom of choice, is the effective topos, and more generally any realizability topos. The reason for this is quite similar to the intuitive justification for CoSHEP given above. Technically, it results from the fact that realizability toposes are exact completions; an explanation is given in this remark.

For a Grothendieck topos example, the sheaves on the interval [0,1][0,1] with its usual topology give a topos in which the internal axiom of countable choice fails, hence internal PAx must also fail.

In categories which are not topoi

Example

According to Peter Scholze in this comment on the nCafé, an example of a category that satisfies external CoSHEP is the category of condensed sets, assuming that Set satisfies the axiom of choice. The category of condensed sets do not form a topos, only an infinitary pretopos.

However, internal CoSHEP fails in condensed sets.

Further properties

Since Set is (essentially regardless of foundations) an exact category, if it has enough projectives then it must be the free exact category PSet ex/lexPSet_{ex/lex} generated by its subcategory PSetPSet of projective objects. By the construction of the ex/lex completion PSet ex/lexPSet_{ex/lex}, it follows that every set is the quotient of some “pseudo-equivalence relation” in PSetPSet; i.e., the result of imposing an equality relation on some completely presented set. See SEAR+ε for an application of this idea.

Proof

Since CoSHEP implies WISC, and WISC has this implication (a result of van den Berg).

 In higher category theory

In higher category theory, there are different versions of CoSHEP:

The difference between these versions of CoSHEP is that sets cover.

Justification

Although perhaps not well known in the literature of constructive mathematics, the CoSHEP axiom may be justified by the sort of reasoning usually accepted to justify the axioms of countable choice and dependent choice, which it implies, by Proposition above.

To be explicit, every set AA should have a ‘completely presented’ set of ‘canonical’ elements, that is elements given directly as they are constructed without regard for the equality relation imposed upon them. For canonical elements, equality is identity, so the BHK interpretation of logic justifies the axiom of choice for a completely presented set. This set is PP, and AA is obtained from it as a quotient by the relation of equality on AA. This argument can be made precise in some forms of type theory, which thus justify CoSHEP, much as they are widely known to justify dependent choice.

References

When Peter Aczel was developing CZFCZF (a constructive predicative version of ZFC), he considered this axiom, under the name of the presentation axiom, but ultimately rejected it.

  • Peter Aczel, The type theoretic interpretation of constructive set theory. Logic Colloquium ‘77 (Proc. Conf., Wroclaw, 1977), pp. 55–66, Stud. Logic Foundations Math., 96, North-Holland, Amsterdam-New York, 1978. doi:10.1016/S0049-237X(08)71989-X

The presentation axiom was, however, adopted by Erik Palmgren in CETCSCETCS (a constructive predicative version of ETCS):

Its relationship to some other weak axioms of choice is studied in

  • Michael Rathjen, Choice principles in constructive and classical set theories, In: Z. Chatzidakis, P. Koepke, & W. Pohlers (Eds.), Logic Colloquium ‘02 (Lecture Notes in Logic, pp. 299-326) (2006). Cambridge: Cambridge University Press. doi:10.1017/9781316755723.014. author pdf.

Last revised on February 23, 2024 at 23:23:51. See the history of this page for a list of all contributions to it.