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… effective (and its dual)
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I think maybe a lot of these can be settled by an argument similar to this one for Top: Take the quotient of the underlying congruence in Set and put the quotient topology on it. Then |
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| Say a cocongruence $j_1, j_2 : X \rightrightarrows E$ in a category with binary copowers is <i>regular</i> if the epimorphism $(j_1; j_2) : X + X \twoheadrightarrow E$ is in fact a regular epimorphism. | ||
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| A category <i>has effective regular congruences</i> if it has binary copowers, and every regular cocongruence in the category is effective (see <a href="/category-property/effective cocongruences">here</a> for definition). |
At the moment, this is very much in draft status: I have 24 undecided categories for "has effective regular congruences" and 21 undecided categories for "has effective regular cocongruences".
The immediate purpose of this PR is to get the last preliminary property in place before I can introduce the property of "Grothendieck quasitopos" along with the Giraud-like theorem for such quasitopoi (Johnstone C2.2.13). (Which is why I was formulating this one before the related property of "has quotients of regular congruences".)