surjection (changes) in nLab

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Definition

A function ff (of sets) from AA to BB is surjective if, given any element yy of BB, y=f(x)y = f(x) for some xx. A surjective function is also called onto or a surjection.

Surjections are the same as epimorphisms in the category of sets and effective epimorphisms in the (infinity,1)-category of sets.

A bijection is a function that is both surjective and injective.

The surjection preorder

In classical set theory, one writes |B|≤ *|A||B| \leq^* |A| to mean that either there is a surjection A→BA \to B or B=∅B=\empty. The relation ≤ *\leq^* is a preorder on the class of all sets, and its restriction to inhabited sets is the preorder reflection of the category Surj inhSurj_{inh} of inhabited sets and surjections.

To make this definition less piecemeal and more constructive, one can define |B|≤ *|A||B| \leq^* |A| to mean BB is a subquotient of AA, in other words one has a surjection B′→BB' \to B and an injection B′→AB' \to A. For BB inhabited and a subquotient of AA, excluded middle implies there is a surjection from AA to BB, so in classical mathematics this coincides with the piecemeal definition.

Contrast with the notation |B|≤|A||B| \leq |A| if there is an injection B→AB\to A. Since subobjects are subquotients (e.g. we can take B′=BB'=B above), |B|≤|A||B| \leq |A| implies |B|≤ *|A||B| \leq^* |A|. (If we wanted to prove this using the piecemeal definition, we would require excluded middle.)

Axioms of choice

The axiom of choice states precisely that every surjection in the category of sets has a section. Thus in this setting one has: |B|≤ *|A||B| \leq^* |A| implies |B|≤|A||B| \leq |A|, and so |B|≤ *|A||B| \leq^* |A| iff |B|≤|A||B| \leq |A| assuming AC. Some authors who doubt the axiom of choice use the term ‘onto’ for a surjection as defined above and reserve ‘surjective’ for the stronger notion of a function with a section (a split epimorphism).

The axiom WISC has an equivalent statement (that works in any Boolean topos) due to François Dorais phrased almost entirely in terms of surjections (or epimorphisms):

For every set XX there is a set YY such that for every surjection q:Z→Xq\colon Z \to X there is a function s:Y→Zs\colon Y \to Z such that q∘s:Y→Xq\circ s\colon Y\to X is a surjection.

One can view this as really a statement about the Grothendieck fibration over Set with fibre over XX the full subcategory of Set/XSet/X on the surjections: every fibre has a weakly initial object.

In other categories

Since an element aa in a set AA in the category of sets is just a global element a:1→Aa:1\rightarrow A, one could define surjections in any category 𝒞\mathcal{C} with a terminal object 11:

Just because a morphism is a surjection in a concrete category does not mean that the morphism in the underlying category of sets is a surjection; equivalently, the forgetful functor from any category 𝒞\mathcal{C} to Set does not preserve surjections.

Well-pointedness

The fact that surjections are epimorphisms in Set is a result of the fact that Set is well-pointed. This could be generalised to any category with a terminal separator 11.

Axiom of choice

The axiom of choice for surjections in Set is the following statement:

• Every surjection is a split epimorphism.

This axiom could be defined in every category with a terminal object, and could be contrasted with the axiom of choice for epimorphisms, as not every epimorphism is an surjection, nor is every surjection an epimorphism.

In dependent type theory

In dependent type theory, surjections are defined in the same way as they are in set theory: a function f:A→Bf:A \to B between types AA and BB is a surjection if for all terms b:Bb:B the fiber of ff over bb is inhabited.

isSurjective(f)≔∏ b:B‖fiber(f,b)‖isSurjective(f) \coloneqq \prod_{b:B} \Vert fiber(f, b) \Vert

However, unlike set theory, surjections are not necessarily the same as epimorphisms in dependent type theory. For example, the unique function from the boolean domain 𝟚\mathbb{2} to the unit type is a surjection, but it is not a epimorphism if the type theory has the circle type with its large recursion principle. In fact, surjections and epimorphisms correspond in dependent type theory if and only if an axiom of set truncation holds in the type theory.

Instead, surjections in dependent type theory correspond to the effective epimorphisms in set theory and category theory. This is because in dependent type theory, types in general represent $\infty$-groupoids and form an (infinity,1)-category, and so the relevant notion is that of an effective epimorphism in an (infinity,1)-category.

• injection

• epimorphism

• subquotient

• monotone function

• equivariant function

References

For surjections in dependent type theory, see section 15.2 of:

• Egbert Rijke, Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (arXiv:2212.11082)

or section 4.6 of:

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

see also:

• Egbert Rijke, The join construction, 26 Jan 2017, (arXiv:1701.07538)