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5

You have $0\le_B 1$, but $g(0)\not\le_A g(1)$. So $g$ is not an arrow in $\mathbf{Poset}$. More generally, $(U,\le_A)$ is totally disordered, so any map $U\to V$ ($V$ with any order relation) is monotone, because there's no pair that can falsify it. It's quite easy to see that if $A=(U,\le_A)$ and $B=(V,\le_B)$ are isomorphic in $\mathbf{Poset}$ and $A$ is ...


5

$[R^{op}, \text{Ab}]$ is the category of right $R$-modules, which is equivalently the category of left $R^{op}$-modules. The reason to prefer taking right modules here is the same reason why presheaves are contravariant functors and not covariant functors: it's so that the Yoneda embedding, which in this case is $R \to [R^{op}, \text{Ab}]$, is covariant. In ...


4

Yes, $\mathcal{V}$ is cocomplete and this is well-known. Probably the easiest and most constructive proof describes the colimit by generators and relations. One just "adds" the generators and the relations and mods out the relations forced by the transition maps. For example, the colimit of $\mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{2} \dotsc$ in ...


4

I can't think of any direct disadvantages other than the complication of the definition. But then, engineers, analysts and applied mathematicians aren't usually bothering with exponentiability of spaces anyway-although mathematical physicists might, in trying to formalize some of the "spaces" that arise in physics.


3

I think the main idea is that of a "category which is adequate and convenient for all purposes of topology" as I wrote in the Introduction of my first, 1963, paper, available from here. That is, the idea of looking at the properties of the category of the objects one is studying is a useful one. In analysis, one may need something different. See the book ...


3

As I said in a comment I wouldn't call this construction the $\pi_1(\mathcal C,c)$. First of all your construction distingushes between an $n$-tuple of composable arrows in $\mathcal C$ and their composite that in topological case is not considered: this problem arise because you consider $n$-tuples of composable arrows, i.e. a functor from $n$ in $\mathcal ...


2

This is almost never satisfied since the endomorphisms (automorphisms) of $X$ resp. $Y$ commute when extended to $X \otimes Y$. It is more reasonable to ask if $\mathrm{End}(X \otimes Y) = \mathrm{End}(X) \times \mathrm{End}(Y)$ holds. For example, this holds in the free monoidal category on a category. It fails in most symmetric resp. braided monoidal ...


2

"I have seen direct proofs of the general statement [...] but I am not looking that.": Since you haven't specified what a direct proof or a category theoretic proof is for you, I am not sure if the following proof will satisfy your requirements. If not, please tell me why and we will try to find something else. Proof 1. Recall that in general $M \otimes_R ...


2

Yes: Suppose $\{M_i\}_{i\in I}$ is a generating set of Noetherian objects in the given locally Noetherian Grothendieck category ${\mathscr A}$. Then for any nonzero $X\in{\mathscr A}$ there exists some $i\in I$ and a non-zero morphism $\varphi: M_i\to X$. The image of this morphism is a nonzero Noetherian subobject of $X$. Even more: $X$ is the direct ...


2

The only dualisable object in a cartesian monoidal category is the terminal object. Thus, a cartesian compact closed category must be trivial. Indeed, suppose $A$ has a dual $A^*$. Since the unit object is terminal, the counit $\epsilon : A \times A^* \to 1$ is forced. Consider the unit $\eta : 1 \to A^* \times A$. This decomposes into components as a ...


2

Expanding on the comment of Adeel, you have to exploit the following easy fact about orthogonal classes of arrows in categories: Let $F\dashv G$ be two adjoint functors between categories $\mathcal{C}\leftrightarrows\mathcal{D}$; then $Ff\perp g$ in the category $\mathcal D$ (i.e., $Ff$ has the LLP with respect to $g$ in $\cal D$) if and only if $f\perp Gg$ ...


2

In fact, $\mathcal{V}$ is even locally finitely presentable. (I assume your operations are finitary.) It is a standard result that the following are equivalent for a category $\mathcal{C}$: $\mathcal{C}$ is locally finitely presentable. $\mathcal{C}$ is finitely accessible and cocomplete. $\mathcal{C}$ is finitely accessible and complete. Now, it is ...


2

The conditions $\sigma\circ\psi_{G}=\varphi_{G}$ and $\sigma\circ\psi_{H}=\varphi_{H}$ are determining for $\sigma$. This because: ...


1

Let $P$ be a nonzero projective in the category. Consider its injective envelope $D$ in the category of abelian groups, which is a torsion divisible module. Thus $D$ is a direct sum of Prüfer groups and so there is a nonzero morphism $f\colon P\to G$ where $G$ is a Prüfer group, say $G=\mathbb{Z}(p^\infty)$, for some $p$. Let $G_0=\{0\}\subset G_1\subset ...


1

Consider the two morphisms \begin{align} i&\colon G\to G\times H, &&i(x)=(x,1)\\ j&\colon H\to G\times H, &&j(y)=(1,y) \end{align} (what you called $\psi_G$ and $\psi_H$). If $\alpha\colon G\to K$ and $\beta\colon H\to K$ are morphisms of abelian groups, then you can define $$ \sigma\colon G\times H\to K $$ by $$ ...


1

You are right: the implication (i) => (ii) holds in general, while the converse is equivalent to the axiom of choice. See some relevant discussion on the nLab.


1

Let $\mathbf{A}$ be an abelian (or Grothendieck) category, and consider the injective model structure on the category of cochain complexes $\mathrm{Ch}(\mathbf{A})$. Then the derived category $\mathcal{D}(\mathbf{A})$ is the homotopy category $\mathrm{Ho}(\mathrm{Ch}(\mathbf{A}))$ by inverting the weak equivalences.



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