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.
925
questions
9
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191
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Fundamental groupoid of a filtered union
Let $X$ be a topological space and let $(X_i)_{i\in I}$ be a filtered family of subspaces. Let $X =\bigcup_{i \in I} X^°_i$ be the union of the interiors of the $X_i$. I want to prove the following ...
3
votes
1
answer
142
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Writing $\operatorname{Spec}\mathbb{Z}$ as an inverse limit of finite $T_0$-spaces
I want to show that $\operatorname{Spec}\mathbb{Z}$ can be written as an inverse limit of finite $T_0$-spaces. First off, $\operatorname{Spec}\mathbb{Z} = \{(0), (2),(3),(5),...\}$, so the closed sets ...
1
vote
1
answer
80
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Why is a short exact sequence the same as a bicartesian square?
Let $N \rightarrowtail G \twoheadrightarrow H$ be a sequence of group homomorphisms, with maps $\alpha: N \rightarrowtail G$ a monomorphism and $\beta: G \twoheadrightarrow H$ an epimorphism. I am ...
- 313
2
votes
0
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50
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Proving that the functor $\mathscr{B} \rightarrow 1$ is limit-preserving. Need help constructing the diagram.
I am newbie to Category Theory and I just read about limit preserving functors and I was trying to prove that the functor $\mathscr{B} \rightarrow 1$ preserves limits. The book I am following is ...
- 315
3
votes
1
answer
136
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Finite sets are compact objects in the category $\mathrm {\mathbf{Sets}}$
Object $C$ of category $\mathcal C$ is called compact if
$$\mathrm{Hom} (X, \mathrm{colim} _I Y_i) \cong \mathrm{colim} _I \mathrm{Hom} (X, Y_i) $$
for every filtered colimit.
I want to prove that ...
- 332
0
votes
0
answers
54
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Every object of category $\mathcal{C}$ is compact in $Ind(\mathcal{C})$
Let $\mathrm{Ind}(\mathcal{C})$ be the Ind-completion. We can define it in two different (but similar) ways: as filtered colimits of representable presheaves and as the category of diagrams over ...
- 332
1
vote
1
answer
43
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Condition that limit of a diagram $F$ exists if $\alpha: Mor(-,L) \to \lim Mor(X,F(i))$
Suppose $F : I \to \mathcal C$ is a diagram for a small category $I$ and a category $\mathcal C$. Suppose also that there is an object $L$ of $\mathcal C$ such that for all $X \in Obj(\mathcal C)$ ...
- 313
4
votes
1
answer
100
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Subcategory of functor category is complete
Let $D$ be a complete and cocomplete symmetric monoidal closed category (a Bénabou cosmos). Let $P$ be the permutation category. Consider the substitution product $\circ$ on the functor category $[P^{...
- 1,769
2
votes
1
answer
53
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Reference for construction of coproduct of Boolean algebras
I was recently trying to understand how coproducts of Boolean algebras work, since I need them for my research. I came across a StackExchange question with a great constructive answer from "Math ...
- 21
1
vote
1
answer
70
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Alternate definition of limits in category theory
I was reading about limits and colimits from May's A Concise Course in Algebraic Topology (Chapter 2, Sec 6). He defines the limits essentially like so:
Let $\mathsf I$, $\mathsf C$ be categories and ...
- 4,119
3
votes
1
answer
151
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Is every short exact sequence the direct limit of short exact sequences between finitely presented modules?
Let $R$ be a unitary, associative ring. It is well-known that every $R$-module is the direct limit (filtered colimit) of finitely presented $R$-modules. Is it also true that every short exact sequence ...
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4
votes
1
answer
144
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Automorphism group of the functor $P:G\text{-FinSet} \to \text{Set}$
I am trying to understand the proof of the following theorem from Qiaochu Yuan's answer
Theorem: Let $G$ be a group and let $P : G\text{-FinSet} \to \text{Set}$ be the forgetful functor from the ...
- 601
0
votes
0
answers
61
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Show the category of Hausdorff spaces has Pushouts and Coequalizers [duplicate]
Let $\mathbf{Haus}$ be the category of Hausdorff spaces. I'm required to show that $\mathbf{Haus}$ has Pushouts and Coequalizers.
I don't know if I'm even close, but here is what I've tried:
I tried ...
- 821
2
votes
0
answers
65
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Definition of a direct limit over a directed set in the category of $R$-modules. [duplicate]
Definition: Let $(I,\leq)$ be a directed set. Let $(\{M_i\},\{f_{ij}\})$ be a direct system of $R$-modules. We define the injective limit to be the disjoint union of the $M_i$ modulo a certain ...
- 2,673
1
vote
1
answer
70
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Completion with respect to the topology defined by the subgroups of finite index
Let $M$ be an toplogical abelian group.
I heard we can define $\hat{M}$ by completion of $M$ with respect to the topology defined by the subgroups of finite index.
For example, if $M=\Bbb{Z}$, then ...
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