Giorgio Mossa
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 May 4 comment Natural transformation is a mono iff the components are. @magma that's not what I meant. The first part of the proof show that one of the two implication does hold for every category $\mathbf D$ as codomain of functors, i.e. it holds also for $\mathbf {Set}$. The second part shows how the other implication holds for $\mathbf{Set}$ valued functors, indeed if I remember well this second implication doesn't hold for a generic category $\mathbf D$. May 4 revised Natural transformation is a mono iff the components are. Fixed a grammar error May 3 answered Natural transformation is a mono iff the components are. Apr 29 comment Does Gödel's Completeness Theorem still hold even if the set of variables is finite? @MauroALLEGRANZA I agree that probably Manin assume the hypothesis of infinite set of variables. Nonetheless my observation is that the proof of completeness theorem doesn't seems requiring the infinity of the set of variables in order to be proven. Apr 28 comment Does Gödel's Completeness Theorem still hold even if the set of variables is finite? @Amr Following your argument: if you have to prove an $L$-formula somewhere in your proof you are gonna eliminate the $10$-th variable by an application of a generalization and a particularization (application of modus ponens to formulas of the form $\forall x A(x) \rightarrow A(t)$ for $t$ a term). My conjecture is that by replacing in your proof every occurence of the variable with the term is used in the particolarization you should still be able to have a valid proof. Anyway I could be wrong, that's why I said that I suppose that should work :) Apr 28 answered Does Gödel's Completeness Theorem still hold even if the set of variables is finite? Apr 17 awarded Nice Answer Apr 11 comment Query on a simple exercise involving representations of functors. @Niels.Remb05 take a look to the edit and see if that solves all your doubts. Apr 11 revised Query on a simple exercise involving representations of functors. Added specifications Apr 11 comment Query on a simple exercise involving representations of functors. It follows by the definition of the yoneda isomorphism. In some minutes I'm gonna add that to the answer. :) Apr 11 answered Query on a simple exercise involving representations of functors. Apr 10 revised When is $\langle x,y\rangle$ equal to $\langle x\rangle\langle xy\rangle$? edited title Apr 9 answered Preservation of Limit by Hom: Naturality Question. Apr 8 comment Do we lose everything, if the natural transformations in a monad are exactly inverse? @NiftyKitty95 $\eta_{TA} \circ \mu_A$ while being well typed is usually false for monads. Consider $T$ the free monoid monad then for every set $X$ we have $T(X)$ the free-monoid. Now $\mu_X((x),(y))=(x,y)$ where $x,y \in X$ and $\eta_{TX}\circ \mu_X ((x),(y))=((x,y)) \ne ((x),(y))$ to $\eta_{TX} \circ \mu_X \ne 1_{TTX}$. Mar 31 answered Showing hom-sets are disjoint in a morphism category Mar 31 comment Spec($A$) is connected if $A$ is local @WLOG Right I supposed that it was well known notation: $D(I) = Spec(R) \setminus V(I)$ a standard open set. Mar 31 answered Spec($A$) is connected if $A$ is local Mar 31 revised Spec($A$) is connected if $A$ is local Corrected a typo. Mar 24 comment What is category theory? @AsafKaragila I would glad to hear your opinion about it :) (maybe in a chat?) Mar 24 comment What is category theory? @Number9 "is category theory widely accepted?" the answer is "it depends, what do you mean for category theory being accepted?"