6.3.8 2-Categorical Monomorphisms in
In this section we characterise (for now, some of) the -categorical monomorphisms in , following Chapter 11: Types of Morphisms in Bicategories, Section 11.1.
Let be a relation.
- 1.
Representably Faithful Morphisms in . Every morphism of is a representably faithful morphism.
- 2.
Representably Full Morphisms in . The following conditions are equivalent:
- (a)
The morphism is a representably full morphism.
- (b)
For each pair of relations , the following condition is satisfied:
- (c)
The functor
is full.
- (d)
For each , if , then .
- (e)
The functor
is full.
- (f)
For each , if , then .
- 3.
Representably Fully Faithful Morphisms in . Every representaly full morphism in is a representably fully faithful morphism.
Item 1: Representably Faithful Morphisms in
The relation is a representably faithful morphism in iff, for each , the functor
is faithful, i.e. iff the morphism
is injective for each . However, is either empty or a singleton, in either case of which the map is necessarily injective.
Item 2: Representably Full Morphisms in
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 is a representably full morphism in iff, for each , the functor
is full, i.e. iff the morphism
is surjective for each , i.e. iff, whenever , we also have .
- Item (c)Item (d): This is also simply a matter of unwinding definitions: The functor
is full iff, for each , the morphism
is surjective, i.e. iff whenever , we also necessarily have .
- 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: We need to show that the condition is also true. We proceed step by step:
- (c) Suppose we have with .
- (d) By Chapter 7: Constructions With Relations, Item 7 of Proposition 7.4.4.1.3, we have
- (e) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 we have .
- (f) By assumption, we then have .
- (g) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 again, we have .
- Item (f)Item (d): Suppose that the following condition is true: We need to show that the condition is also true. We proceed step by step:
- (c) Suppose we have with .
- (d) By Chapter 7: Constructions With Relations, Item 7 of Proposition 7.4.1.1.3, we have
- (e) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 we have .
- (f) By assumption, we then have .
- (g) By Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.10.1.2 again, we have .
- Item (b)Item (d): Consider the diagram
and suppose that . Note that, by assumption, given a diagram of the form
if , then . In particular, for each , we may consider the diagram
for which we have , implying that we have
for each , implying .
- Item (d)Item (b): Let and consider the diagram
By
, we have
Now, if , i.e. , then by assumption.
, Fully Faithful Monomorphisms in : 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 .
Are there other nice characterisations of these?
This question also appears as [MO 467527
].