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Feb
10
comment Explain “homotopy” to me
Feel free to ask if you need additional clarifications.
Feb
7
comment Can we define structures like groups or monoids in the context of pure category theory?
@JoshChen did you mean the commutative version?
Feb
5
comment Name for categories with a certain property on coproducts
@ZhenLin if that's the case, well, it seems really hard to me that there are any interesting examples where the induced mappings between the hom-sets $\hom(Y,\bigoplus_{i \in I} X_i)$ and $\prod_{i \in I}\hom(Y,X_i)$ are injective.......
Feb
5
comment Name for categories with a certain property on coproducts
What are the projections maps? In a generic category with coproducts there is no reason why you should have natural mappings from $\bigoplus_{i \in I} X_i \to X_i$. Or am I missing something?
Feb
2
comment Type theoretic proof that $\lnot (A \lor B) \Rightarrow \lnot A \land \lnot B$
Well in my opinion in this case one can use chat in an asynchronous way, it isn't really important to have an instant answer.
Feb
1
comment Type theoretic proof that $\lnot (A \lor B) \Rightarrow \lnot A \land \lnot B$
By the way if you want to discuss this arguements I would be very interested, since I'm studying this subject too. Maybe via chat?! ;)
Feb
1
comment Type theoretic proof that $\lnot (A \lor B) \Rightarrow \lnot A \land \lnot B$
Indeed, I supposed so.
Feb
1
comment Example of Spherical Element (Simplicial Homotopy)
AAAAAAAH Sorry I forgot you have degeracy maps in $\Delta$, so stupid by me. Again apologize for the confusion.
Feb
1
comment Example of Spherical Element (Simplicial Homotopy)
@NajibIdrissi is the simplicial set with only one $0$-simplex and no simplex in higher dimension. Clearly this cannot be a the terminal simplicial set.
Feb
1
comment Example of Spherical Element (Simplicial Homotopy)
@NajibIdrissi if you like you could think them as two different kind of slice-categories one is the canonical $1/\mathbf{sSet}$, where $1$ is the terminal presheaf, the other one is $S^{0}/\mathbf{sSet}$, where $S^{0}$ represent the simplicial set where $S^{0}_0$ is a singleton and each $S^0_n$ is empty for $n > 0$.
Feb
1
comment Example of Spherical Element (Simplicial Homotopy)
I see, let me be more clear about my doubt. The problem lies in the notion of point you want to use: by point do you mean an element of $X_0$ (that is a $0$-simplex) or a morphism from $1$ (the terminal simplicial set) into $X$?
Feb
1
comment Example of Spherical Element (Simplicial Homotopy)
Could you point out your definition of pointed simplicial set? There are at least two that come to my mind.
Feb
1
comment Prove that a group G is finitely generated if and only if there is a surjective homomorphism $F(\{1,..,n \}) \to G$
@Jxt921 Ok. I wanted to give an hint.... but unfortunately I couldn't find any so I provided a solution. The point is that it seems quite straightforward once you know the properties of homomorphisms and the concrete characterization of free groups......
Feb
1
comment Prove that a group G is finitely generated if and only if there is a surjective homomorphism $F(\{1,..,n \}) \to G$
I don't understand, are you looking for a solution to your problem or do you want an hint?
Jan
31
comment Defining natural transformations based on generalized elements?
As an additional remark, yoneda also implies that the natural isomorphism $\langle -,-\rangle$ determines completely the structure of the product on $A \times B$.
Jan
28
comment If $F : \mathbf{C} \to \mathbf{D}$ is an equivalence, does $\alpha_{GD} = G(\beta_D)$ hold in general?
Finally if you look carefully to the counterexample above that exactly what it does: it takes two unrelated natural transformation between the identity functors, i.e. two elements of the set $\mathbb C^{\mathbb C}[1_{\mathbb C},1_{\mathbb C}]$. To prove that the functor $1_{\mathbb C}$ is self-equivalent we just need to provide an element of that set, this could be the identity but it doesn't have to be it.
Jan
28
comment If $F : \mathbf{C} \to \mathbf{D}$ is an equivalence, does $\alpha_{GD} = G(\beta_D)$ hold in general?
I think that the point is that while we can choose many different, unrelated, proofs of the fact that $F$ and $G$ are equivalences, if we want that our proofs also .... prove that the functors are adjoint we need to choose them more carefully.
Jan
28
comment If $F : \mathbf{C} \to \mathbf{D}$ is an equivalence, does $\alpha_{GD} = G(\beta_D)$ hold in general?
@StudentType think of the natural isos as proofs of the equivalences between the functors. An equivalence of categories provides two functors between the categories with two proofs that these functor are equivalences of categories. In an adjoint equivalence we need to require something more, for instance we want that the two proofs satisfy the triangle identities.
Jan
27
comment If $F : \mathbf{C} \to \mathbf{D}$ is an equivalence, does $\alpha_{GD} = G(\beta_D)$ hold in general?
By the way, it is true that if $\langle F,G,\alpha,\beta \rangle$ is an adjoint equivalence then $\alpha_{G(D)}=G(\beta_D)$ for every $D \in \mathbf D$, that's a consequence of the triangle identities.
Jan
24
comment Is the axiom $g1 = g$ essential for a group action
Just a remark: how would you conclude that $(\omega\cdot g)\cdot g^{-1}=\omega$? Usually to prove that one uses the axiom $\omega \cdot 1=\omega$.....