As in title, in terms of group theory (I'm not familiar with category-theoretic terms), question comes from algebraic topology but seems to be of general interest. (Other questions on MSE touch on the topic but I haven't found a direct answer).  What is the difference between the two? Is one special case of the other?
122375
I'm trying to proof that the following diagram in the category of groups with $i_1$ and $i_2$ being inclusions is a pushout iff $G$ is the free product with amalgamation (up to isomorphism). It should be sufficient to proof $"\Leftarrow"$ since the pushout is unique up to isomorphism. How does the proof of $"\Leftarrow"$ go? \begin{array}{ccc}  G_0 &amp; \xrightarrow{i_1} &amp; G_1 \\ \downarrow_{i_2} &amp;  &amp; \downarrow_{\varphi_1}\\ G_2 &amp; \xrightarrow{\varphi_2} &amp; G \end{array} I tried defining the unique homomorphism of the pushout by $$\phi((g_1*...*g_n)N) = h_i(g_1)*...*h_i(g_n)$$ but couldn't proof it's well-definedness.
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What would be your recommendation for the text which could be useful for someone starting with lattice theory? Suggestion for both books and online materials/lecture notes are welcome.    This was asked as a part of  and I did not like that it was combination of two different questions. (One of them on geometric modular lattices and the other one about text recommendation.)  I don't expect the user to post a follow-up question, since he was not seen at MSE for some time. I think that this could be a useful for some users here, so I've decided to post it myself.
61983
In , one learns two definitions of free groups associating with sets.        Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where \begin{equation} \operatorname{Obj}(\mathcal{C})=\{A\xrightarrow{g}G\}, \end{equation}where the codomain $G$ are groups, and \begin{equation} \operatorname{Hom}(\mathcal{C})=\{\text{Commutative Diagrams } A\xrightarrow{g_1}G_1\xrightarrow{\phi}G_2\xleftarrow{g_2}A\}  \end{equation} where $\phi$ are group homomorphisms.   Also $F(A)$ has the concrete construction with elements being non-redundant words with alphabet $A$, and the group multiplication being juxtaposition and reduction.      But what good are free groups? Why are they useful?  says they are useful in topology, but does not explain why explicity.   Can someone give some examples that an undergraduate can understand? I am asking about examples that can show the usefulness of this abstract construction. So I guess the identification of $\mathbb{Z}=F({a})$ does not really count.
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I was hoping for an answer in general terms avoiding things like holonomy, Chern classes, Kahler manifolds, fibre bundles and terms of similar ilk. Simply, what are the compelling reasons for restricting the landscape to admittedly bizarre Calabi-Yau manifolds? I have  but haven't read it yet, nor, obviously,  :)
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What is the difference between free groups and free modules/vector/algebras spaces, or in other words what does free mean in algebra?   I have seen two uses of the term free:   something of infinite order that is generate by a linear independent set called a basis. I have seen this applied to (Groups/abelian groups/vector spaces/modules/algebras) an algebraic structure in which there is no defined relation between elements. I have seen this apllied to vector spaces/modules-used in tensor product definition.   Thanks
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$\mathbb R^*$ refers to $\mathbb R$ without zero.  Please explain the statement "$\mathbb R$  is not a group under multiplication, it is a group under addition." in the . Basically:   Why is multiplicative group $(\mathbb R^*, ×)$ a group? Why is $(\mathbb R, ×)$ not group? Why are $(\mathbb R^*, +)$ and $(\mathbb R, +)$ groups?
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I've seen there are few questions similar, but I haven't seen anyone so precise or with a good answer.   I'd like to understand the reason why we ask in the definition of a manifold the existence of a countable basis. Does anybody has an example of what can go wrong with an uncountable basis? When does the problem arise? Does it arise when we want to differentiate something or does it arise before? Thank You
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At the moment I'm taking an algebraic topology course. In our current exercise we have to prove the following: Consider any topological space $X$ and any two maps $f, g : X \to S^n$, such that $f(x) \neq -g(x)$. It follows that $f$ is homotopic to $g$.  My idea is to use the mapping cylinders $M_f, M_g$ to construct a homotopy via $$M := M_f \amalg M_g/ (x, 0)_f \sim (x,0)_g$$ But this doesn't seem to work. In addition I don't know how $f(x) \neq -g(x)$ is going to help me (I can see how this could help if $n=1$, but I have no clue how this would work for $n&gt;1$).  I would appreciate any hints towards a solution.  Thanks in advance.
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I am trying to read the book  by Anderson and Fuller.  On  it is stated that the category of Groups is not a subcategory of a category of Sets, some explanations are given there. Can any one please give me some more details about? especially why $mor_G((G,\circ),(H,\circ))\not\subset Map((G,\circ),(H,\circ))$ ?
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This has been asked about here: , and  said      Personally I'd say $M$ wasn't a valid manifold with boundary because the $\{2\}$ doesn't have a neighborhood with any structure like an open ball/half-ball.    This is actually an exercise from An Introduction to Manifolds by Loring W. Tu and is not mentioned in an .        I have spent almost 2 hours thinking about an exercise that looked like it would take only 15 minutes and even tried pasting lemma (it's a good thing Professor Tu has solutions unlike ): The result of all that thinking is that I don't think $[0,1) \cup \{2\}$, $(\varepsilon,1) \cup \{2\}$ or $\{2\}$ is homeomorphic to any open subset of $\mathscr H^1$ or $\mathscr L^1$. I was able to show $\{0\} \subseteq \partial M$ and $(0,1)\subseteq M^0$, but I don't quite know where $2$ belongs. I believe $M$ is not locally $\mathscr H^1$. Also, I have double checked: I believe "manifold boundary" was defined for manifolds with boundary, so this isn't some trick where "manifold boundary" is actually defined for a Hausdorff and second countable space that need not be locally $\mathscr H^n$.     To generalize,     Is a half-open interval and a point not in the interval's closure a manifold with boundary?
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I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly  arbitrary. I'm not sure what exactly it is about the axioms that motivated them defining groups.  My textbook asks you to list 'common features' of vector spaces and then later defines a set of axioms for vector spaces under addition, scalar multiplication and both, noting that the axioms under addition form an Abelian group. So are groups just a generalisation of vector spaces under any binary operation?  My main problem is that the book notes that 'the axioms may seem rather arbitrary' and links groups with vector spaces but doesn't elaborate.   When introducing groups, you are tasked to complete Cayley tables for the symmetries of an equilateral triangle and square.  Then, similarly to the delivery of vector spaces, notes that the tables have common properties (Closure, Identity, inverse and association) and defines a group as a set of elements under a binary operation that has these features.  So what's so important about these 4 properties? For example, if 1 or 2 the properties were excluded form the  axioms, or we added an extra few properties as axioms how would that cripple the effectiveness of groups?  Are the group axioms ever difficult to work with or do they always work, forgive the crude Littlewood analogy, like a mathematical skeleton key?  What is it about these 4 properties that make groups such a powerful tool in mathematics and physics?  My best guess is that a group is the best way to express our sense of  symmetry and what is symmetric mathematically, but I would prefer some elaboration.
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I have heard that some issues in group theory prevent classifying all manifolds upto homotopy using the fundamental group invariant. Does anyone know what are those issues?  Thanks, K.
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Given any abstract group $ G $, how much is known about which types of topological/Lie group structures it might have?  Any abstract group $ G $ will have the structure of a discrete topological group (since generally, any set can be given the discrete topology), but there are groups that have no smooth structure. An example of this from Wikipedia is the group $ \mathbb{Q} $ with the subspace topology inherited from $ \mathbb{R} $. Which groups can occur as Lie groups? Are there specific families of groups that are known to have no smooth structure?  Similarly, how much can we know about the possible topologies on an abstract group $ G $? For example, which types of abstract groups admit a nontrivial (i.e., not the usual compact case) group structure?  In particular, I am curious about the extent to which properties of the abstract group determine properties of any associated topology/smooth structure.  Does anyone have any good references or a succinct answer for this question?
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sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and problems, and I want it to be the sort of book in which you always have an idea about what the next results are going to be. Something I really liked about Dummit and Foote is I understood what was going on the first time I read it, but when I read it a second time I felt a lot better than the first time (I'm looking forward to a third).  Sorry if this just sounds like a lost of demands, I just wanted to give some information about what I was looking for.  Thank you kindly  Regards.
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I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks?  More particularly, is it worth investing the time to study these formalisms?
63937
Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group structure of G) is equal, in $\pi_1(G,e)$, to the loop $s_1*s_2$ where this product is given by first going around $s_1$ and then $s_2$ (i.e., do we have $[s_1s_2] = [s_1*s_2]$)? If yes, what is the proof for this?
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Allen Hatcher seems impossible and this is set as the course text?  So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online.   Any good intro to Algebraic topology books?  I can find a big lists of Algebraic geometry books on here. On a very old thread on Maths overflow someone recommended that a person should read James Munkres Topology first, then you should read Allen Hatcher book.   It just seems like Rudin's book but crammed with ten times more material.