One of the features of the new user profile is the next badge section. As can be seen below, my next badge on meta is the Informed badge. But meta does not have an Informed badge; see the list of badges on meta. When I click on the box, a pop-up appears with the option to "Go get it". When I f...

my question / request is about some badges that still can not be followed in the "badges" box of the profile. For instance pundit and taxonomist between others. Would it be possible to add them in the future? I have been reading previous questions at Meta about how to follow badges and in the p...

See the below image: This is in the user's profile, in the "Activity" tab. The above part is in meta profile, and the below part is in the main site profile. In meta, there is the option to track the next tag badge, while in the main site there is not this option. Is there a way to track the...

I propose creating limits-colimits for limits and colimits in the sense of category theory. (Perhaps it might be useful to create also colimits and categorical-limits and add them as synonyms.) We have many questions about limits and colimits in category-theory tag. Some of them are tagged as li...

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For $n \geq m$, let $\varphi_{m,n}$ denote the canonical ring homomorphism $R / I^n \to R / I^m$. Let $J = \cap_n I^n$. Then $R / J$ is going to end up being the $I$-adic completion of $R$, coming equipped with the obvious natura...

Suppose we have three objects $A,B,C$ of an (abelian) category $\mathbf{C}$ and a short exact sequence $ 0\to A \to B \to C \to 0 $ such that $B$ is a central extension of $C$ by $A$ ($im(A\to B)\subset ker(B\to C)$) 1.) What is a pullback of this central extension? 2.) What is the minimum ...

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We have the 'product like description' $(v,w)\in V\oplus W$ 2.) We have $v+w\in V\oplus W$ Is there...

In many cases, filtered colimits commute with forgetful functors to $\mathbf{Set}$, for example with $\mathbf{CRing} \to \mathbf{Set}$ or $R-\mathbf{Mod} \to \mathbf{Set}$. Is there a slick way of showing this? I am mainly interested in this because you use this fact for the computation of stalk...

Suppose we have three directed sequences of $C^*$-algebras, say $(A_n,\varphi_n)$,$(B_n,\psi_n)$ and $(C_n,\theta_n)$ and $*$-homomorphisms $\alpha_n:A_n\rightarrow C_n$ and $\beta_n:B_n\rightarrow C_n$, then we can take the pullback $A_n\times_{C_n}B_n$ for all $n\in\mathbb{N}$ and can also take...

Given the direct system $$\mathbb{Z}^2 \xrightarrow{A} \mathbb{Z}^2 \xrightarrow{A}\mathbb{Z}^2 \xrightarrow{A}\cdots$$ with $$A = \begin{pmatrix} 1 & 1 \\ 2 & 0 \end{pmatrix},$$ the direct limit should be $\mathbb{Z} \oplus \mathbb{Z}[\frac{1}{2}]$. How is this done? Well, the eigenvalues of $A...

Let $X, Y_1, Y_2, \cdots$ be a sequence of topological vector spaces, and let $f_n : X \to Y_n$ be a sequence of continuous linear maps. Define the product space $\mathcal Y_N := Y_1 \times \cdots \times Y_N$, and let $\mathcal Y_\infty := \prod_n Y_n$ denote the cartesian product (equipped with...

By way of background to this question, I am interested in the properties of direct limits. They are usually defined in terms that assume there is an underlying directed poset, but according to category theory, direct limits do in fact exist for general diagrams that are not directed. I am inter...

The General Adjoint Functor Theorem (Category Theory) states that for a locally small and complete category $D$, a functor $G\colon D \to C$ has a left adjoint if and only if $G$ preserves all small limits and for each object $A$ in $C$, $A \downarrow G$) has a weakly initial set. Could someone ...

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? With more generality and summarizing, with which generality there exist limits and colimits in Schem...

I find it is hard to understand the "create limit." (You can find it in Mac Lane's Categories for the working mathematician, P112; there it defines: "A functor $V:A→x$ creates limits for a functor $F:J→A$.") Here are my questions: (1) Why does it use "a functor $F:J→A$," and how does it match w...

Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a category-theoretic setting of some non-trivial topological space such that these different concepts o...

And, lo, the contadiction tag did battle with the mighty Trogdor, but it was a foregone conclusion. Though the humble tag did parry and thrust with all its might, the armor-like scales resisted all attacks. The dragon-man picked up the tag with its arm, and with a single great breath of flame the...