Niels.Remb05
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 Jul2 awarded Curious Apr26 comment A very simple question on adjunctions that I simply can't look at correctly. I mean, the fact that $a\times b$ is a limit, says that we have an iso $\hom(\Delta c,(a,b))\cong (c,a\times b)$, natural in $c$ and $(a,b)$. This satisfies the requirements for an adjunction between $\Delta$ and $\times$. But by the data I just gave, I have no clue about what the bijection actually looks like. What I was trying to ask is, can you recover the fact that the isomorphism is the mapping $(f,g)\mapsto \langle f,g\rangle$ just from this data? This is what I was trying to do in my first reply, where I proved that $\phi(f,g)=(f\times g)\circ\eta_c$. Now why is this of the above form? Apr26 comment A very simple question on adjunctions that I simply can't look at correctly. Hmm...But what if I didn't know either what the mapping is, or what the unit is. And suppose I just know, because of the definition of the product, that we must have an adjunction given by the bijection on the hom-sets as you mention it in the first line of your last comment. Then shouldn't I be able to recover that the mapping is $(f,g)\mapsto \langle f,g\rangle$ or that the unit is $\eta_c=\langle 1_c,1_c\rangle$? Apr25 comment A very simple question on adjunctions that I simply can't look at correctly. ... but this uses the fact that $\eta_c=\langle 1_c,1_c\rangle$, which is what I want to prove, doesn't it?...I was trying to prove that form of the adjunct, in order to prove that $\eta_c=\langle 1_c,1_c\rangle$...Isn't this your argument? Apr25 comment A very simple question on adjunctions that I simply can't look at correctly. Hmm, thanks! I actually tried to find the explicit form of the adjunct, and, if $\phi$ is the bijection of the adjunction, and $f:c\to a$, $g:c\to b$, I found via naturality, that $$\phi(f,g)=(f\times g)\circ \eta_c$$ I struggle a bit to see why this is precisely $\langle f,g\rangle$. Am I doing something wrong? Apr25 asked A very simple question on adjunctions that I simply can't look at correctly. Apr23 comment Given an arrow from a category to a product of categories, can we evaluate it on an object using its unique decomposition? Hmm...But $\delta_c$ is defined to be the unit of the adjunction $\Delta \dashv \times$. So shouldn't the letters in the diagram be small (not capitals)? Apr22 comment Given an arrow from a category to a product of categories, can we evaluate it on an object using its unique decomposition? Arghh I have been trying to do Exercise 3 of Chapter IV.1 in MacLane's CWM. I got very confused by trying to make sense of all the capital $C$ in the diagram. Shouldn't they be small $c$ everywhere?!...(Oh, I just noticed that maybe you don't have a copy of CWM?) Apr22 comment Given an arrow from a category to a product of categories, can we evaluate it on an object using its unique decomposition? Say $\pi_{\mathcal{D}}\circ f=F$ and $\pi_{\mathcal{E}}\circ f=G$. What I mean is, if we know what $F(c)$ and $G(c)$ are, can we say something about $f(c)$? Apr22 asked Given an arrow from a category to a product of categories, can we evaluate it on an object using its unique decomposition? Apr22 answered What is the relationship between the second isomorphism theorem and the third one in group theory? Apr20 accepted Proving naturality of an isomorphism in MacLane's CWM. Apr20 comment Proving naturality of an isomorphism in MacLane's CWM. Thanks! So, is my attempt wrong? Apr19 asked Proving naturality of an isomorphism in MacLane's CWM. Apr11 accepted Query on a simple exercise involving representations of functors. Apr11 comment Query on a simple exercise involving representations of functors. Yes, of course. How stupid of me. Thank you very much! Apr11 comment Derivatives on Functors The functor of immersions can be viewed as the "first derivative" of the functor of embeddings of a manifond into another one. This sentence makes sense by a technique called "Calculus of Functors", first developed by Goodwillie. Apr11 awarded Commentator Apr11 comment Query on a simple exercise involving representations of functors. Ok, thanks! But I am afraid I don't see exactly how Yoneda proves the statement you mention in your second paragraph: I can see that $$\text{Hom}_{[\mathcal{D}^{op},\mathbf{Set}]}(\text{Hom}_{\mathcal{D}}(-,r), \text{Hom}_{\mathcal{D}}(-,r'))\cong \text{Hom}_{\mathcal{D}}(r,r')$$ by Yoneda, but why is every natural transformation coming from the left-hand side, of the form $\text{Hom}_{\mathcal{D}}(-,h)$? Apr11 asked Query on a simple exercise involving representations of functors.