# 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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### Coequalizer in the category of modules

I am trying to prove that the category of modules is cocomplete. It suffices to show that it has all coequalizers and coproducts. It's relatively easy to show that all coproducts exist, and I am left ...
• 3,790
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### Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory?

I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around. (1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$...
• 1,632
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### Spec of an infinite intersection of ideals, Spec of a colimit

This comes from the study of Krull's Intersection Theorem, and deriving a geometric meaning. Let $I \subset R$ be an ideal of a commutative ring (we shall see the case when $R$ is Noetherian). ...
• 447
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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 ...
• 239k
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### Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives

Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result. I have no idea on ...
• 3,790
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### Interpretation of closure in inverse limit

Can one interpret the closure of a set inside an inverse limit as the closure of its individual components? I have not been able to find a source confirming or denying this claim. I have only been ...
1 vote
133 views

### Does profinite completion preserve injectivity?

Let $G$ be an abelian group. Let $\widehat{G}$ be a profinite completion of $G$. Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are ...
• 6,266
1 vote
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### Confusion about colimits in the category $\mathbf{Set}$

It is well known that $\mathbf{Set}$ is an $\aleph_0$-accessible category, but I'm very inexperienced and I'm not sure how to prove it in detail. In particular, I need to find a set $\Omega$ of ...
• 33
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### Difference between different definitions of diagram in a category

I'm currently reading the book "Topoi: The Categorial Analysis of Logic" by Robert Goldblatt, and in chapter 3.11, in order to define limits and co-limits he defines a diagram in a category ...
• 4,952
1 vote
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### Non trivial colimit for rings in a finite diagram

I’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a ...
33 views

### Sequence of direct summands of modules over a ring

Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules: $$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$ I claim that ...
• 16.1k
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### (Co)Products are bifunctors, but are general (co)limits also functors?

In a category with all products or coproducts, the (co)product operation can be understood as a bifunctor. More generally let $\mathcal{C}$ be a category with all limits of shape $D$, where for ...
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### Subtleties in commuting colimits

For context, I am reading Weibel's k-book and I am trying to express the homology of $BS^{-1}S$, the group completion of the classifying space of a symmetric monoidal category, as a colimit. In ...
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