# 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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### Colimit of categories don't preserve equivalences [duplicate]

I have a question about the following problem: Find two diagrams of shape J in Cat $F,G:J\rightarrow Cat$ and a natural transformation $\eta:F\rightarrow G$ such that $\eta_i:F(i)\rightarrow G(i)$ is ...
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### Intuition behind cokernel pair

I'm trying to find some insights about cokernel pairs. I understand the definition and I know how to calculate them in concrete categories, but while the kernel pairs seem to have a very sensible ...
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### Inverse limit and direct limit of an infinite sequence of sets

Let $S_1 \subset \cdots \subset S_n \subset \cdots$ be an infinite sequence of finite sets. We can define both inverse limit and direct limit of this sequence. What is the difference between them? ...
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### Inductive limit in category of (small) categories

Recently I attended a talk where the speaker mentioned about inductive limit in the category of (small) categories. I have never seen anything like this so I was wondering how does one constructs ...
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### Is the inclusion into a coproduct an equalizer?

Let $\mathcal{C}$ be a category with finite limits and binary coproducts. Let $1$ denote the terminal object, and let $\langle\rangle_A : A \rightarrow 1$ denote the unique map from $A$ to $1$. If we ...
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### Statement in Riehl's book of the result "a diagram of functors has a limit if it has objectwise limit"

$\def\A{\mathsf{A}} \def\C{\mathsf{C}} \def\ob{\operatorname{ob}} \def\ev{\operatorname{ev}} \def\J{\mathsf{J}}$ I am having a little trouble understanding the last sentence of this result in p. 93 ...
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### the hom-functor is left exact: an equivalent condition

I would like to understand the equivalence of the 2 facts that hom is left exact in the sense that it carries short exact sequences to left exact sequences and that it preserves all small existing ...
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### any $\lambda-$pure morphism in a $\lambda$-accessible category is a monomorphism

I'm trying to understand the proof of the proposition 2.29 in the book Locally presentable and accessible categories which is also given in the snippet below. I've got stuck in the -2nd paragraph ...
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