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.
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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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 $X_i$ ...
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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_{\...
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3
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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 ...
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Example of a functor which preserves all small limits but has no left adjoint
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 ...
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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 ...
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Filtered colimits commute with forgetful functors
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 ...
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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 $...
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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 $(-\frac{1}{n},\frac{1}{n})$. $...
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Meaning of pullback
I was wondering if the following two
meanings of pullback are related and
how:
In terms of Precomposition with a function:
a function $f$ of a variable $y$, where $y$
itself is a function of ...
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