Add category Mono of injective set functions and commutative squares#266
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dschepler wants to merge 2 commits into
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Add category Mono of injective set functions and commutative squares#266dschepler wants to merge 2 commits into
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ScriptRaccoon
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Jul 5, 2026
| For this proof we will work in the equivalent category of pairs $(X, X')$ where $X' \subseteq X$. Thus, suppose we have a coreflexive corelation $p : (X \sqcup X, X' \sqcup X') \twoheadrightarrow (E, E')$ with coreflexivity morphism $r : (E, E') \to (X, X')$. From the assumption that $p$ is an epimorphism, we have that $p : X \sqcup X \to E$ is surjective. Since $\Set$ is co-Malcev, it follows that $E \simeq X \sqcup_Y X$ for some subset $Y \subseteq X$. It remains to show that $E' = i_1(X') \cup i_2(X') \subseteq X \sqcup_Y X$. Certainly, since we have a morphism $(X \sqcup_Y X, i_1(X') \cup i_2(X')) \to (E, E')$ induced by $p$, we must have $i_1(X') \cup i_2(X') \subseteq E'$. On the other hand, any element of $E'$ is equal to either $i_1(x)$ or $i_2(x)$ for $x \in X$. In the first case, we must have $r(i_1(x)) = x \in X'$, so $i_1(x) \in i_1(X')$; and similarly for the second case. | ||
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| - property: effective cocongruences | ||
| proof: 'See the proof that $\Mono$ is co-Malcev: It shows that in fact any coreflexive corelation is equivalent to an effective cocongruence $X \sqcup X \to X \sqcup_Y X$.' |
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Do you think it's worth adding a property "every reflexive relation is effective" to the database? (Not in this PR of course.)
I think we have seen this a couple of times already (Haus, ...).
| @@ -0,0 +1,158 @@ | |||
| id: Mono | |||
| name: category of sets with a distinguished subset | |||
| notation: $\Mono$ | |||
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I really like the decision to change the presentation of the category. But now, the notation is a bit off, don't you think?
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Do you also want to add the functors to the database which appear in this discussion? For example, the functor Mono ---> Set which maps (X,X') to X.
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The purpose here is primarily to provide a simple example of a quasitopos which is neither an elementary topos nor thin. (Of course, a lot of the manual proofs here will go away once we can add a "Grothendieck quasitopos" property; this category is equivalent to the$\lnot\lnot$ -separated presheaves on the walking morphism category, where the $\lnot\lnot$ topology is generated by the single morphism $0 \to 1$ forming a covering sieve of $1$ .)