making a cone universal in a faithful way

[Dimitri Ara <dimitri.ara@gmail.com>]

Dear List,

Has the following elementary problem been already studied?

Let C be a category, I a small category, F : I -> C a functor and
alpha : F => c a cocone (c is an object of C). When does there exist a
category D and a faithful functor G : C -> D taking alpha to a universal
cocone?

For example, if I is the empty category, the question becomes "when can
you make c an initial object in a faithful way?". If I is the final
category, then the cocone alpha amounts to a morphism f : F(*) -> c and the
question becomes "when can you make f an isomorphism in a faithful way?".

There are two obvious necessary conditions.
1) Let f,g : c -> d be two morphisms of C. If f alpha_i = g alpha_i for
every i in I, then we should have f = g.
2) Let f : a -> F(i) and g : a -> F(j) be two morphisms of C such that
alpha_i f = alpha_j g. Then for every cocone beta : F => d, we should
have beta_i f = beta_j g.

In the case of the empty category, the first condition means that for
every object d there is at most one arrow c -> d and the second condition
is void. In the case of the final category, the first condition means
that f is an epi and the second that f is a mono. It is not hard to prove
that in both cases, theses conditions are sufficient.

Question: are they sufficient in the general case?

Regards,
--
Dimitri


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