Mathematics Stack Exchange
Mathematics Stack Exchange

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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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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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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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 another variable $x$, may be written as a …
Tim
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Is every commutative ring a limit of noetherian rings?

Let $\mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of commutative rings with one. Let $A$ be in $\mathsf{CRing}$. Question 1. Is there a functor from a small category to…
Pierre-Yves Gaillard
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Any criteria for a category to have all connected limits?

There are many theorems in category theory that give criteria for the existence of a large class of limits in terms of the existence of a few special, more manageable types of limits. For instance: A category has all (small) limits if it has…
Tomo
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When do coproducts map canonically to products?

I have noticed that in certain categories (e.g. $k$-vector spaces, pointed sets, ...), given an indexed family $\{x_i\}_i$ of objects, we always have a canonical map $$\bigsqcup_ix_i\longrightarrow\prod_ix_i.$$ I was wondering what conditions do we…
Daniel Robert-Nicoud
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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 limits and for each object $A$ in $C$, $A…
Conan Wong
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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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