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Jul
2
awarded  Curious
Apr
26
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?
Apr
26
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 $\eta_c=\langle 1_c,1_c\rangle$?
Apr
25
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?
Apr
25
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?
Apr
25
asked A very simple question on adjunctions that I simply can't look at correctly.
Apr
23
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)?
Apr
22
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?)
Apr
22
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 $f(c)$?
Apr
22
asked Given an arrow from a category to a product of categories, can we evaluate it on an object using its unique decomposition?
Apr
22
answered What is the relationship between the second isomorphism theorem and the third one in group theory?
Apr
20
accepted Proving naturality of an isomorphism in MacLane's CWM.
Apr
20
comment Proving naturality of an isomorphism in MacLane's CWM.
Thanks! So, is my attempt wrong?
Apr
19
asked Proving naturality of an isomorphism in MacLane's CWM.
Apr
11
accepted Query on a simple exercise involving representations of functors.
Apr
11
comment Query on a simple exercise involving representations of functors.
Yes, of course. How stupid of me. Thank you very much!
Apr
11
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.
Apr
11
awarded  Commentator
Apr
11
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 $\text{Hom}_{\mathcal{D}}(-,h)$?
Apr
11
asked Query on a simple exercise involving representations of functors.