In all the examples we have seen that coproduct for the comonoids have been monomorphisms.
The categorical theory of relations and quantizations
This is equivalent to assuming that the map r is a monomorphism.
The categorical theory of relations and quantizations
The fact that h is a monomorphism in C follows from the universality as i did for πA δ,γ in proposition 36.
The categorical theory of relations and quantizations
Then the morphism δ l : δ −→ a ⊗A δ is a monomorphism.
The categorical theory of relations and quantizations
But al = δA = ar and δA is a monomorphism so we have lA a .
The categorical theory of relations and quantizations
Let Q be the universal localization of R with respect to the set Σ of monomorphisms between ﬁnitely generated pro jective right R-modules with ﬁnite dimensional cokernels.
$K_0$ of purely infinite simple regular rings
Since T and therefore N ′ is isomorphic to a direct sum of Pr¨ufer modules, there is a monomorphism f : S → N ′ with S a simple ob ject of t.
Infinite dimensional representations of canonical algebras
Since g ′ : T → T is a split monomorphism, and T ∈ ω0, we see that L ∈ ω0, thus belongs to ω .
Infinite dimensional representations of canonical algebras
Using Ext1 (ω, ω ) = 0, we see that g is a split monomorphism.
Infinite dimensional representations of canonical algebras
Since δ(P ) = −1, we conclude that the restriction of f to P must be a monomorphism, using that the kernel Ker f and the image f (P ) are submodules of G.
Infinite dimensional representations of canonical algebras
This shows that there are many monomorphisms P → S [∞].
Infinite dimensional representations of canonical algebras
This shows that both f and g are monomorphisms.
Infinite dimensional representations of canonical algebras
And we obtain a characterization of the modules in D as follows: The modules in D are the cokernels of monomorphisms in ω .
Infinite dimensional representations of canonical algebras
For a module M in q, we need a monomorphism of this kind, and M will be the cokernel.
Infinite dimensional representations of canonical algebras
Any monomorphism X ′ → Y ′ with X ′ ∈ ω ′ 0 splits.
Infinite dimensional representations of canonical algebras
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