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Questions tagged [limits-colimits]

For questions about categorical limits and colimits, including questions about (co)limits of general diagrams, questions about specific special kinds of (co)limits such as (co)products or (co)equalizers, and questions about generalizations such as weighted (co)limits and (co)ends.

The notion of limit and the dual notion of colimit are important in category theory.

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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of the guys in my office, and despite a very shady…
Asaf Karagila
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Category-theoretic limit related to topological limit?

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…
Rafael Mrden
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The "magic diagram" is cartesian

I am trying to solve an exercise from Vakil's lecture notes on algebraic geometry, namely, I want to show that $\require{AMScd}$ \begin{CD} X_1\times_Y X_2 @>>> X_1\times_Z X_2\\ @V V V @VV V\\ Y @>>> Y \times_Z Y \end{CD} is a cartesian…
Paul
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What are the product and coproduct in the category of topological groups?

What are the product and coproduct in the category of topological groups? I know the limits in the categories of groups, abelian groups and topological spaces, and was wondering about the same thing.
faridrb
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On limits, schemes and Spec functor

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…
iago
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Right adjoints preserve limits

In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an adjunction, then if $D:I\to \mathcal{D}$ is a diagram that…
Bruno Stonek
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Examples of a categories without products

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category of fields. But I cannot formally prove why…
ChesterX
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Compact subset in colimit of spaces

I found at the beginning of tom Dieck's Book the following (non proved) result Suppose $X$ is the colimit of the sequence $$ X_1 \subset X_2 \subset X_3 \subset \cdots $$ Suppose points in $X_i$ are closed. Then each compact subset $K$ of $X$ is…
Luigi M
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Calculating (co)limits of ringed spaces in $\mathbf{Top}$

Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces. There are forgetful functors $$ U_{\mathbf{LRS}}: \mathbf{LRS} \to \mathbf{RS} $$ (the…
user8463524
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Atiyah Macdonald – Exercise 2.15 (direct limit)

Atiyah-Macdonald book constructs the direct limit of a directed system $(M_i,\mu_{ij})$, (where $i\in I$, a directed set, and $i\leq j$) of $A$-modules as the quotient $C/D$, where $C=\bigoplus_{i\in I} M_i$, and $D$ is the submodule generated by…
gradstudent
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Do products preserve colimits in the category of locales?

Does the functor $X\times-:\mathbf{Loc}\to\mathbf{Loc}$ preserve small colimits for all locales $X$? The reason that I'm interested in this question is that the same property fails in the category of topological spaces. If products preserve…
Oscar Cunningham
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What really is a colimit of sets?

This is probably a question I should have asked myself a bit earlier. For some reason I always thought I knew the answer so I did not bother, but now that I actually need to use it (I am studying the $+$ construction on presheaves) I realize I am…
user121314
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Choosing products in categories appearing in practice

Is there any category $\mathcal{C}$ appearing in practice* which has binary products but it is not possible to choose them in a canonical way? By dualization, what about binary coproducts? A bit more formal, it is required that $\mathsf{ZF}$ (plus…
Martin Brandenburg
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Do Wikipedia, nLab and several books give a wrong definition of categorical limits?

It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and write some errata emails. A standard definition of a…
joriki
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Can the fundamental group and homology of the line with two origins be computed as a direct limit?

Let $X$ be the line with two origins, the result of identifying two lines except their origins. Let $X_n$ be the result of identifying two lines except their intervals $(-\frac{1}{n},\frac{1}{n})$. $X_n$ is a Hausdorff space that exists in…
183orbco3
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