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Alex Chen

• @ThomasRead Yes, thanks for pointing this out. Continuing with Puzzle 18, if $$f_{\ast}$$ has a left-adjoint $$g_{\ast}$$, then $$f$$ must be surjective. From before, we know that if $$f_{\ast}$$ has a left-adjoint $$g_{\ast}$$, then $$f$$ must be…
• I think there is another example of a poset (Puzzle 4) from Owen Biesel's puzzles in the comments of Lecture 17 (OB1 to OB4): the set of galois connections $$\{ (f, g) : f \dashv g \}$$ between two posets $$A$$ and $$B$$ forms a poset. A galois c…
• I could contribute how I understand Puzzle TR1, though I'm not sure how terse it is. We know that $$g(b)$$ is an upper bound for the set $$A_b = \{a \in A : f(a) \leq_B b\}$$. We want to show that $$g(b)$$ is a least upper bound for $$A_b$$. It is…
• Following Dan Schmidt's post 5, it seems that if $$f_{\ast}$$ has a left adjoint $$g_{\ast}$$, then $$f$$ must be injective. Otherwise, if $$f$$ is not injective, there are elements $$a_1, a_2 \in A, a_1 \neq a_2$$ such that $$f(a_1) = b = f(a_2)$$.…
• Hi, I'm confused about the proof of proposition 1.88, where they show that if $$Q$$ is a preordered set with all meets, then any monotone map $$g: Q \to P$$ that preserves meets is a right adjoint. They do this by constructing a left adjoint \(f: P …