Add extremal (co)generating sets and single extremal (co)generators (WIP)#280
Add extremal (co)generating sets and single extremal (co)generators (WIP)#280dschepler wants to merge 5 commits into
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FYI #283 means that the files have changed their location, so a (trivial) rebasing is required. |
Co-authored-by: Script Raccoon <scriptraccoon@gmail.com>
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This can be refactored after #288 has been merged.
EDIT: it is merged now
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| _Proof._ This is a straight forward generalization of [this result](/category-implication/generator_via_coproduct). We remark that the assumption about $S$ implies that each inclusion $G \to U$ has a left inverse. Now let $f,g : A \rightrightarrows B$ be two morphisms with $f h = g h$ for all $h : U \to A$. If $G \in S$, any morphism $G \to A$ extends to $U$ by our preliminary remark. Thus, $fh = gh$ holds for all $h : G \to A$ and $G \in S$. Since $S$ is a generating set, this implies $f = g$. <span class="qed">$\square$</span> | ||
| _Proof._ We remark that the assumption about $S$ implies that each inclusion $G \to U$ has a left inverse. Now let $f,g : A \rightrightarrows B$ be two morphisms with $f h = g h$ for all $h : U \to A$. If $G \in S$, any morphism $G \to A$ extends to $U$ by our preliminary remark. Thus, $fh = gh$ holds for all $h : G \to A$ and $G \in S$. Since $S$ is a generating set, this implies $f = g$. |
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To adjust the notation to your proof:
| _Proof._ We remark that the assumption about $S$ implies that each inclusion $G \to U$ has a left inverse. Now let $f,g : A \rightrightarrows B$ be two morphisms with $f h = g h$ for all $h : U \to A$. If $G \in S$, any morphism $G \to A$ extends to $U$ by our preliminary remark. Thus, $fh = gh$ holds for all $h : G \to A$ and $G \in S$. Since $S$ is a generating set, this implies $f = g$. | |
| _Proof._ We remark that the assumption on $S$ implies that each coprojection $i_G : G \to U$ has a left inverse. Now let $f,g : A \rightrightarrows B$ be two morphisms with $f \circ \bar a = g \circ \bar a$ for all $\bar a : U \to A$. If $G \in S$, any morphism $G \to A$ extends to $U$ by our preliminary remark. Thus, $f \circ a = g \circ a$ holds for all morphisms $a : G \to A$ with $G \in S$. Since $S$ is a generating set, this implies $f = g$. |
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| title: Construction of Generators | ||
| description: How to construct a generator from a generating set | ||
| author: Martin Brandenburg |
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| authors: | |
| - Martin Brandenburg | |
| - Daniel Schepler |
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| - property: generator | ||
| proof: The $0$-dimensional one-point manifold is a generator since it represents the forgetful functor $\Top \to \Set$. | ||
| - property: extremal generator |
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I think one can actually prove that R is an extremal generator.
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I guess the idea is that tangent vectors at a point are equivalent to certain equivalence classes of curves going through the point; so if f induces a bijection of those curves... I'll see about formalizing that argument.
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actually, it is dense
I remember (long ago) being fascinated by the fact that Man is hence a full subcategory of the category of M-sets, where M = End(R), and I wondered if we can describe the image.
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| - property: cogenerator | ||
| proof: 'The manifold $\IR$ is a cogenerator, since for every smooth manifold $M$ and points $p \neq q$ in $M$ there is a smooth function $f : M \to \IR$ with $f(p) = 1$ and $f(q) = 0$ (John Lee, Introduction to Smooth Manifolds, Prop. 2.25).' | ||
| Now, since the tangent spaces of $M$ and $N$ are defined in terms of certain evaluations of smooth functions on $M$ and $N$, and since $f$ induces a bijection between these smooth functions, we can conclude that $f$ induces isomorphisms between tangent spaces of $M$ and $N$. Thus, $f$ is a diffeomorphism. |
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Please make this part more precise.
| @@ -25,8 +25,8 @@ satisfied_properties: | |||
| - property: generator | |||
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This is now redundant.
Actually, the redundancy script yields several redundancies. But maybe first rebase.
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Remark: many proofs here show something stronger, or are close to that, namely that the set or object is a dense subcategory. This property is not here yet, but maybe we should add it soon. Not in this PR perhaps because it is already big, unless you think it clarifies the proofs much better. We can also have separate PRs then for the missing proofs. Btw, should I stop looking at this PR as long is it is in draft mode? |
No, comments and suggestions in the mean time are very useful. Incidentally, I've looked at the remaining unsettled cases, and they seem very tricky. I can maybe make more detailed comments later once I'm done with work for the day. (The exception is extremal cogenerator for Sp which seems like it should be doable if I could get my head around it better. Maybe something like the set of all quotients of So I should probably be able to take the PR out of draft status soon, and then we can decide on what questions to submit to MO and/or which categories we're OK with leaving unknown for the moment. |
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My primary motivation in the short term is to get "has an extremal generating set" into the database for use in entering one of the equivalent conditions in the Giraud-type theorem for Grothendieck quasitopoi. The property is, of course, of broader interest.
Current status:
extremal generator: 18 unknown
extremal generating set: 2 unknown
extremal cogenerator: 26 unknown
extremal cogenerating set: 16 unknown
Blocker: for Top, currently unknown whether it has a single extremal cogenerator.