6.3.8 2-Categorical Monomorphisms in Rel

In this section we characterise (for now, some of) the 2-categorical monomorphisms in Rel, following Chapter 11: Types of Morphisms in Bicategories, Section 11.1.

Let R:A|B be a relation.

  1. 1. Representably Faithful Morphisms in Rel. Every morphism of Rel is a representably faithful morphism.
  2. 2. Representably Full Morphisms in Rel. The following conditions are equivalent:
    1. (a) The morphism R:A|B is a representably full morphism.
    2. (b) For each pair of relations S,T:X|A, the following condition is satisfied:
      • If RSRT, then ST.
    3. (c) The functor
      R:(P(A),)(P(B),)

      is full.

    4. (d) For each U,VP(A), if R(U)R(V), then UV.
    5. (e) The functor
      R!:(P(A),)(P(B),)

      is full.

    6. (f) For each U,VP(A), if R!(U)R!(V), then UV.
  3. 3. Representably Fully Faithful Morphisms in Rel. Every representaly full morphism in Rel is a representably fully faithful morphism.

Item 1: Representably Faithful Morphisms in Rel
The relation R is a representably faithful morphism in Rel iff, for each XObj(Rel), the functor
R:Rel(X,A)Rel(X,B)

is faithful, i.e. iff the morphism

R|S,T:HomRel(X,A)(S,T)HomRel(X,B)(RS,RT)

is injective for each S,TObj(Rel(X,A)). However, HomRel(X,A)(S,T) is either empty or a singleton, in either case of which the map R|S,T is necessarily injective.

Item 2: Representably Full Morphisms in Rel
We claim Item (a), Item (b), Item (c), Item (d), Item (e), and Item (f) are indeed equivalent:
  • Item (a)Item (b): This is simply a matter of unwinding definitions: The relation R is a representably full morphism in Rel iff, for each XObj(Rel), the functor
    R:Rel(X,A)Rel(X,B)

    is full, i.e. iff the morphism

    R|S,T:HomRel(X,A)(S,T)HomRel(X,B)(RS,RT)

    is surjective for each S,TObj(Rel(X,A)), i.e. iff, whenever RSRT, we also have ST.

  • Item (c)Item (d): This is also simply a matter of unwinding definitions: The functor

    R:(P(A),)(P(B),)

    is full iff, for each U,VP(A), the morphism

    R|U,V:HomP(A)(U,V)HomP(B)(R(U),R(V))

    is surjective, i.e. iff whenever R(U)R(V), we also necessarily have UV.

  • Item (e)Item (f): This is once again simply a matter of unwinding definitions, and proceeds exactly in the same way as in the proof of the equivalence between Item (c) and Item (d) given above.
  • Item (d)Item (f): Suppose that the following condition is true:
    • For each U,VP(A), if R(U)R(V), then UV.
    We need to show that the condition
    • For each U,VP(A), if R!(U)R!(V), then UV.
    is also true. We proceed step by step:
    1. (c) Suppose we have U,VP(A) with R!(U)R!(V).
    2. (d) By Chapter 7: Constructions With Relations, Item 7 of Proposition 7.4.4.1.3, we have

      R!(U)=BR(AU),R!(V)=BR(AV).

    3. (e) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 we have R(AV)R(AU).
    4. (f) By assumption, we then have AVAU.
    5. (g) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 again, we have UV.
  • Item (f)Item (d): Suppose that the following condition is true:
    • For each U,VP(A), if R!(U)R!(V), then UV.
    We need to show that the condition
    • For each U,VP(A), if R(U)R(V), then UV.
    is also true. We proceed step by step:
    1. (c) Suppose we have U,VP(A) with R(U)R(V).
    2. (d) By Chapter 7: Constructions With Relations, Item 7 of Proposition 7.4.1.1.3, we have

      R(U)=BR!(AU),R(V)=BR!(AV).

    3. (e) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 we have R!(AV)R!(AU).
    4. (f) By assumption, we then have AVAU.
    5. (g) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 again, we have UV.
  • Item (b)Item (d): Consider the diagram

    and suppose that RSRT. Note that, by assumption, given a diagram of the form

    if R(U)=RURV=R(V), then UV. In particular, for each xX, we may consider the diagram

    for which we have RS[x]RT[x], implying that we have

    S(x)=S[x]T[x]=T(x)

    for each xX, implying ST.

  • Item (d)Item (b): Let U,VP(A) and consider the diagram

    By , we have

    R(U)=RU,R(V)=RV.

    Now, if R(U)R(V), i.e. RURV, then UV by assumption.

, Fully Faithful Monomorphisms in Rel: This follows from Item 1 and Item 2.

Item 2 of Proposition 6.3.8.1.1 gives a characterisation of the representably full morphisms in Rel.

Are there other nice characterisations of these?

This question also appears as [MO 467527].


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