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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.

915 questions
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Proving $\varprojlim S_i \ \cong \ \varprojlim S_j $ where $J \subseteq I$ cofinal.

Let $I$ be a directed poset and $J \subseteq I$ cofinal. Let $(S_i)_{i \in I}$ be a collection of sets, with a collection of maps $$\{ f_{ij} \ : \ S_i \rightarrow S_j \ | \ i,j \in I \text{ and } i \geq j \}$$ that satisfies the usual commutation…
Koenraad van Duin
  • 3,156
3
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Surjecting onto inverse limit of a subsystem

Let $(I, \le)$ be a poset and $\{ A_i \}$ a collection of sets indexed by $I$, such that the projections $\varprojlim A_i \to A_i$ are surjective for all $i \in I$. Is it true that the natural map $$\varprojlim_{i \in I} A_i \to \varprojlim_{j \in…
Mr. Chip
  • 5,009
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About definition of direct limit

In the definition of direct limit of abelian groups (or modules or ...), one takes abelian groups $G_i$ ($i\in I)$ with a morphism $f_{ij}:G_i\rightarrow G_j$ with following conditions: for every $i\in I$, $f_{ii}:G_i\rightarrow G_i$ is…
Maths Rahul
  • 2,877