The usual way to prove snake lemma is by diagram chasing, which requires you to work on the elements.   But snake lemma still holds on abelian categories.   Could someone provide a version of proof for snake lemma by chasing arrows?
25509
Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A&#39; \mathrel{\overset{f&#39;}{\to}} B&#39; \mathrel{\overset{g&#39;}{\to}} C&#39; \to 0 $$ and morphisms $a : A \to A&#39;, b : B \to B&#39;, c : C \to C&#39;$ making the obvious diagram commute. The snake lemma states that there is then an exact sequence $$0 \to \ker a \to \ker b \to \ker c \to \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c \to 0$$ where the morphisms between the kernels are induced by $f$ and $g$ while the maps between the cokernels are induced by $f&#39;$ and $g&#39;$.  It is not hard to show that the morphisms induced by $f, g, f&#39;, g&#39;$ exist, are unique, and that the sequence is exact at $\ker a, \ker b, \operatorname{coker} b, \operatorname{coker} c$. With the use of a somewhat large diagram shown , we can even construct the connecting morphism $d : \ker c \to \operatorname{coker} a$. However, I'm stuck showing exactness at $\ker c$ and $\operatorname{coker} a$. I thought Freyd might have had an element-free proof in his book, but it turns out he proves it by diagram chasing and invoking the Mitchell embedding theorem [pp. 98–99]. Is there a direct proof?
61307
Since $T_p G$ is isomorphic to $T_e G$ for all $p\in G$, it makes sense that each vector in $T_p G$ can be identified with a vector in $T_e G$. Hence, to make the map from $TG$ one to one, we must identify each vector in $T_e G$ with a point in $G$, but I'm having trouble saying this precisely.  Thanks so much!
49783
So I'm asked to prove that $$S_{3}\cong D_{3}$$ where $D_3$ is the dihedral group of order $6$. I know how to exhibit the isomorphism and verify every one of the $6^{2}$ pairs, but that seems so long and tedious, I'm not sure my fingers can withstand the brute force. Is there a better way?
76794
Prove that structure $(Q/{0}; =;*)$ has infinite number of automorphisms I think it's about we don't have $&lt;$ or $&gt;$, so we don't have strict order in structure, so we can do whatever we want? but i dont know how to say it more formal/
108082
are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, Gabriel gives  of quivers of finite type using just five Dynkin diagrams.  Although these are both deep and surprising, but I am not sure why quivers deserve so much attention. The only potential application I can think of (although highly unlikely to be true) that they might be useful to answer certain questions in category theory since the notion of quivers are similar to categories, and a representation is very much like a functor from a quiver to some $\mathcal{k}$-$\operatorname{vect}$.   So I wonder whether someone can give a hint why quivers deserve so much attention? Do they naturally show up in problems? And do representations of quivers really help to solve these problems?
1529
Problem statement: I need to find out if $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$ and explain my answer. I already know that it is isomorphic, so now all I need to do is to prove it.   I assume that first we need to look where it sends $s$ and $t$ and it must send them to elements that satisfy the same relations() And then need to show that those two are conjugation by some element in $D_4$(?)  Any help is appreciated. I was hoping for a duplicate post, but couldn't find it.  Thank you!
75042
This is from Qing Liu's Algebraic Geometry and Arithmetic Curves (page 89). When he proves (c), he first prove the result when $K/k$ is finite, and then says that the general case immediately follows from the finite case and lemma 2.6 (posted below). I have no clue of how the lemma can be use to reduce the general case to the finite case.   Moreover, there is a remark pictured below that also uses the lemma 2.6 and once again I'm confused about how to use it. How can I complete these details?
91930
I'm currently reading Goldblatt's Topoi - The categorial analysis of logic and at some point he lets one verification to the reader and I must admit that I'm stuck.  He's just introduced $\mathbf{M}$-sets where $\mathbf{M}$ is a monoid, and he's showing that the category of $\mathbf{M}$-sets and multiplication-preserving maps is a topos.  He also introduced the concept of a left ideal and called $L_M$ the set of left ideals of $M$, and his claim is that $L_M$ together with the multiplication $m\cdot B = \{n\mid nm \in B\}$; and with the true arrow $true : \{0\} \to L_M$, $true(0) = M$ is a subobject classifier in this category.   Then he considers a monic arrow $k: X\to Y$ and argues that the character $\chi : Y\to L_M$ associated with this subobject is defined by $\chi(y) = \{n \mid n\cdot y \in X\}$.   This is where I'm stuck : I've managed to show that this does make the associated square a pullback square, and that any other arrow $g:Y\to L_M$ that makes it a pullback square has $g(x) = M$ for $x\in X$. I've also shown that it has $\chi(y) \subset g(y)$ for all $y\in Y$, but I can't prove that $\chi$ is the only one that works...  Any indications?
58678
I'm currently self-studying "Categories and Sheaves"by Schapira and Kashiwara, and I've been stuck on problem 1.19 all day today, so I was hoping that someone could help me out.  Let $\mathcal{C}$, $\mathcal{C&#39;}$ be categories and $L_v:\mathcal{C} \rightarrow \mathcal{C&#39;}$, $R_v:\mathcal{C&#39;} \rightarrow \mathcal{C}$ be functors such that $(L_v,R_v)$ is a pair of adjoint functors (v=1,2). Let $\epsilon_v: id_\mathcal{C} \rightarrow R_v L_v$ and  $\eta_v: L_v \circ R_v  \rightarrow id_{\mathcal{C&#39;}}$ be the adjunction morphisms. Prove that the two maps $\lambda$,$\mu$ : $$Hom_{Fct(\mathcal{C},\mathcal{C&#39;})}(L_1,L_2) \rightleftarrows^{\lambda}_{\mu} Hom_{Fct(\mathcal{C&#39;},\mathcal{C})}(R_2,R_1)$$ given by : $$\lambda(\varphi):R_2 \rightarrow^{\epsilon_1 \circ R_2} R_1 \circ L_1 \circ R_2 \rightarrow^{R_1 \circ \varphi \circ R_2} R_1 \circ L_2 \circ R_2 \rightarrow^{R_1 \circ \eta_2} R_1$$ for $\varphi \in Hom_{Fct(\mathcal{C},\mathcal{C&#39;})}(L_1,L_2)$ $$\mu(\psi) : L_1 \rightarrow^{L_1 \circ \epsilon_2} L_1 \circ R_2 \circ L_2 \rightarrow^{L_1 \circ \psi \circ L_2} L_1 \circ R_1 \circ L_2 \rightarrow^{\eta_1 \circ L_2} L_2$$ for $\psi \in Hom_{Fct(\mathcal{C}&#39;,\mathcal{C})}(R_2,R_1)$ are inverse to eachother.  So far, I've been trying to play around with the zig-zag identity, but it doesn't seem to lead me anywhere. Any help will be greatly appreciated!
8357
Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you.
110385
I know that there are many proofs regarding this. However, the book i'm using seems to suggest another way to do it without giving an answer. What i mean by the another way is some other proofs that do not use the fact that elementary row operation can be expressed by multiplying elementary matrices. The book says that the lemma need to be proved only when the size of identity matrix is 2 by 2. Does it have something to do with mathematical induction? That is, did the author want us to find out both induction hypothesis and induction step?
30949
I have a question about the proof of the following:     (Lemma 37) Assume $\mathbf V = \mathbf L$, and let $\kappa$ be a cardinal. Then   $\mathcal P (\kappa ) \subseteq L_{\kappa^+}$.   Assume we have proved the following:      (Claim 38) There exists a finite subset $T$ of $ZF + (\mathbf V =  \mathbf L)$ such that whenever $x$ is transitive and $\langle x, \overline{\in} \rangle \models T$ then there exists $\gamma \in  \mathbf{ON}$ such that $x = L_\gamma$.      (Lemma 35) Let $\alpha \in \mathbf{ON}$ and $x, a_0, \dots , a_{k-1}  \in L_\alpha$ and let $\varphi(z, a_0, \dots ,a_{k-1})$ be a formula   of $L_S$. Then there exists a $\beta &gt; \alpha$ such that for every $z  \in x$: $$ \langle L_\beta , \overline{\in} \rangle \models \varphi  [z, a_0 , \dots, a_{k-1}]  \hspace{0.5cm} \text{ iff } \hspace{0.5cm}  \mathbf L \models \varphi [z, a_0 , \dots, a_{k-1}] $$     The proof in the book starts as follows: Suppose $\mathbf V = \mathbf L$ holds, let $\kappa$ be as in the assumptions of lemma 37 and suppose $\kappa \in \mathcal P ( \kappa )$. Fix an ordinal $\alpha &gt; \kappa$ such that $y \in L_\alpha$. Since the theory $T$ of claim 38 is finite we can form the conjunction $\varphi$ of all sentences in $T$.   Question: But what is $T$? I know that it's the theory from claim 38 but it's not clear to me how we can use an unknown theory to prove lemma 37. And why do we take the conjunction over $T$?  Then the proof proceeds: By lemma 35,    Question: How does $\gamma &lt; \kappa^+$ follow from $(2)$?  Many thanks for your help.
94492
I read a section of a book and it made mention of the set of rationals not being a $G_\delta$. However, it gave no proof. I read on  about using contradiction, but it made use of the Baire category theorem, which is unfamiliar to me. I was wondering if anyone could offer me a different proof; perhaps using the fact that the complement of $G_\delta$ is $F_\sigma$.  Thanks.
91140
The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit worried about the (*) line. It looks a bit awkward, but is it too bad? Any suggestions?    Theorem: If $\varphi : G \rightarrow H$ is a homomorphism of groups, then $ker \varphi \trianglelefteq G$ and $G/ker \varphi \cong \varphi (G)$.  Proof: Let $\varphi:G \rightarrow H$ be a group homomorphism. If $g \in G$ and $h \in ker \varphi$, then we have that $ghg^{-1} \in ker \varphi$ because $$\varphi (ghg^{-1})= \varphi (g) \varphi (h) \varphi (g^{-1})= \varphi (g) \varphi (g^{-1}) = \varphi (gg^{-1}) = \varphi (1) =1,$$ so that $ker \varphi \trianglelefteq G$. Now, define $\phi: G/ker \varphi \rightarrow \varphi (G)$ by $$\phi (g(ker \varphi))=\varphi(g),$$ for each $g(ker \varphi) \in G/ker \varphi$, for some $g \in G$. This is well defined because if $g' \in g(ker \varphi),$ then $$\phi (g'(ker \varphi))= \varphi(g')= \varphi(g)= \phi (g(ker \varphi)).$$ Also, this is a group isomorphism because for each $\varphi (h) \in \varphi (G),$ for some $h \in G$, we have $$(*) \ \phi^{-1}\{\varphi(h)\}= \phi^{-1} \varphi[g(ker \varphi)]= \phi^{-1} \varphi(\pi^{-1}\{g(ker \varphi)\})= \{g(ker \varphi)\},$$ a set with a single element of $G/ker \varphi$, so that it is a bijection, and for every $g(ker \varphi), g'(ker \varphi) \in G/ker \varphi$, for some $g,g' \in G$, we have $$\phi [g(ker \varphi) g'(ker \varphi)]=\phi [(gg')(ker \varphi)]=\varphi(gg')=\varphi(g) \varphi(g')=\phi [g(ker \varphi)] \phi [g'(ker \varphi)],$$ so that it is a group homomorphism.
114726
In the context of representation theory, direct graphs —known there as quivers— appear all the time. Here are  a few samples (from the book of Assem, Simson andSkowroński) First, a couple of gabriel quivers:  These are in general directed graphs which allow parallel arrows (that is, the graph need not be simple) and loops. Next, the beautiful (!) A-R quiver of $E_6$ (which is missing a few dashed lines denoting the translation functor, but still)  I am 100% sure people working in the subject have come up with Tikz styles, tricks and whatnot to draw these things, but googling does not come up with much. In particular (and amazingly!) there is no AR quiver at   Do you have any relevant resources for this?
44209
I am writing another exam in Algebra this week and this time the main topic is automorphism. I was again going through the example exercises and exams from previous years and this problem is giving me a hard time to understand:  Prove that for group of automorphisms $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})$ holds  $\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$ where $\mathbf{Z_{n}^{*}}=\{1 \leq k \leq q | \gcd(k,n)=1\} $.  My main issue with this problem is that it seems very "general" and I don't really know how to address it with any of the techniques we have used.  Could you please show me some ways this could be approached? I appreciate your help.
59193
I'm wondering whether the category whose objects are short exact sequences of abelian groups, and whose morphisms are commutative diagrams of such short exact sequences, is cocomplete. Working naively, it seems you can get coproducts by taking them componentwise. However, for coequalizers, I think we are not so lucky. Consider the two short exact sequences $0 \rightarrow 0 \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0$ and $0 \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0 \rightarrow 0$ with maps between integers being the identity. Consider the morphism from the first sequence to the second which is the identity on the middle map between the integers and obviously zero elsewhere. Then taking cokernels componentwise would give $0 \rightarrow \mathbb{Z} \rightarrow 0 \rightarrow 0 \rightarrow 0$, which clearly cannot be exact.   So the obvious candidate for cokernels is not the correct one, but perhaps there is another not-so-obvious choice. I'm wondering how I might go about showing whether or not there are coequalizers.