# 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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Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits. In general representables preserve limits, but the hypothesis ...
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### Understanding a product of objects as a limit of a discrete diagram

I have read that a product of objects $\{A_i\}_{i\in I}$ in a category $\mathcal{A}$ can be defined as a limit of a discrete diagram i.e. a diagram $D:\mathcal{I}\to\mathcal{A}$ where the only ...
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### Compute $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$

I know that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})=\prod_p({\widehat{\mathbb{Z}_p}/\mathbb{Z}})$. But the following steps took me a long time to figure out what was wrong with ...
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### Definition of limit by nLab

Let $d:\mathbf I^{\text{op}}\to \mathbf{Set}$ be a functor, with $\bf I$ a small category. Set $t:\mathbf I^{\text{op}}\to \mathbf{Set}$ the (constant) functor sending every object to the singleton. I ...
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### Applications of bounded cocomplete semilattices

Call an ordered set a poset when the ordering is transitive and antisymmetric. Call a poset bounded when it has a top and a bottom element (i.e. a greatest and least element). Call a poset cocomplete ...
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### Existence of Limits in Category theory in Roman's book

I am self-reading An Introduction To The Language Of Category by Roman and having some questions about the set-up for proving the existence of limits. In the book, the hom-classes are assumed to be ...
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### Counterexample: existence of categorical product of family does not imply existence of categorical product of subfamily [duplicate]

Assumptions/context: Let's say we have some objects $X_i$ in some category, with $i$ belonging to some index set $\mathbf{I}$, such that there exists some object, call it $X_{\mathbf{I}}$, in the ...
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### Are singleton sets nullary products or unary products in the category of sets? (Or are they both nullary and unary simultaneously?)

I understand why terminal objects are "nullary products" in any category, and I understand why singleton sets are terminal objects in the category of sets. So I understand why singleton sets ...
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### Simpler description of pushout in $\mathsf{Set}$ and $\mathsf{Top}$ when one of the two maps is injective

It is known that the pushout in $\mathsf{Top}$ has same underlying set as the pushout in $\mathsf{Set}$ (this is because the forgetful functor $\mathsf{Top}\to\mathsf{Set}$ is a left adjoint). ...
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### Forgetful functor from the slice category creates limits

I am reading "Category theory in context" and I'm having difficulties with proposition 3.3.8, top of page 92, pdf. Theorem: The forgetful functor $F : c/C \to C$ strictly creates all limits ...
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### What's going on in this notation for the projective limit in Serre?

$\newcommand{\Z}{\mathbf Z}\newcommand{\Q}{\mathbf Q}$I am currently reading Serre's A Course in Arithmetic, and in Chapter 2 where he introduces the $p$-adics, he mentions the projective limit. My ...
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Really quite a silly question, but I'm confused about the notation around limits in category theory. Given a diagram of shape $I$, say $F$, I would write the limit of $F$ in some category as $\... 2 votes 1 answer 39 views ### Products in quotient category Suppose$\mathcal{C}$is a category,$\sim$is a congruence relation on$\mathcal{C}$, and$[-]: \mathcal{C} \to \mathcal{C}/{\sim}$is the quotient map. I'm able to prove that if$\mathbf{0}$is ... 3 votes 0 answers 38 views ### Does the forgetful functor from the category of models of a cartesian theory preserve and create limits? It is well-known that a monadic functor preserves and creates limits, which in particular shows that any algebraic category, or equivalently, any category of models of an algebraic theory, has limits, ... 2 votes 0 answers 48 views ### Notation: Interpolation between functors in extension/lifting problems with simple categories In chapter 6 on homotopy (co-)limits of Jeffrey Strom's Modern Classical Homotopy Theory he gives the following definition on p. 156 (I'm explaining the terminology at the bottom): [Let$\mathscr{C}$... 0 votes 2 answers 103 views ### Prove the forgetful functor$U: Grp \to Set$preserves limits Let$F: A \to B$be a functor. F preserves limits iff any for any diagram$ D: I \to A$s.t.$\operatorname{lim}_I D $exists,$\operatorname{lim}_I FD$exists and$\operatorname{lim}_I FD \cong F( \... 1 vote
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### Why is the limit the product, but not the coproduct?

On page 489 of Algebra, Paolo gives an example of a limit: Let $I$ be the discrete category with 2 objects with only identity morphisms, let $\alpha$ be a functor from $I$ to any category $C$, and let ...
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### Inverse limit and crossed product

Let $p$ be a prime and $H$ be a uniform, pro-$p$ group. Then the Iwasawa algebra $\mathbb{F}_p[[H]]$ can be seen as the $I$-adic completion of the group algebra $\mathbb{F}_p[H]$ for $I$ the ...
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### Do ample sheaves descend along limits / noetherian approximation?

Suppose $X \to S$ is a proper (projective) morphism of schemes, $S$ is quasi-compact and quasi-separated, and $\mathcal L$ is a relatively ample sheaf on $X$. By stacks, tag 01ZA, the scheme $S$ is a ...
### Proving $\operatorname{dir.lim}(M_i\otimes_{A_i}N_i)\cong \operatorname{dir.lim} M_i\otimes_{\operatorname{dir.lim} A_i}\operatorname{dir.lim} N_i$
If $\left ((G_i)_{i\in I},(\varphi_{ij})_{i\le j}\right )$ is an inductive system of abelian groups then its limit $G=\varinjlim G_i$ is the abelian group whose underlying set is the quotient of the ...