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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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 ...
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Ind-objects with "full support"
Now cross-posted to MO.
Let $C$ be a small category. Let's say a presheaf $P\colon C^{\mathrm{op}}\to \mathsf{Set}$ has "full support" if $P(X)\neq \varnothing$ for all objects $X$. We ...
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Why are split coequalizers "contractible"?
In the book Toposes, Triples and Theories by Barr and Wells, the authors define a contractible coequalizer (elsewhere known as a split coequalizer) to be a commutative diagram:
$A \rightrightarrows_{d^...
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Proof of Theorem 3.4.12 in Emily Riehl's "Category Theory in Context"
I have questions about the proof of Theorem 3.4.12 in Emily Riehl's Category Theory in Context.
The theorem states that the colimit of a small diagram $F\colon \mathsf J \to\mathsf C$ can be expressed ...
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How to understand the effect of adjoint functors?
I have a good grasp of all different definitions/interpretations of adjoint functors, but still do not know have to interpret the left or right adjoint of a give functor, when it exist. It would be a ...
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Coends and adjunctions
I was reading Fosco Loregian's paper This is the co/end, my only co/friend, and here's something that I don't understand in an exercise.
The exercise is to prove that given $F: C\to D, U: D\to C$ ...
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Colimits for gluing schemes and the functor of points 1
Closely related questions been asked several times in different forms on here but I feel like none really spell out what's going on. I have been looking more at glueing schemes, and particularly ...
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Proving étale maps have discrete fibers by abstract nonsense?
I'm trying to prove that an étale map of spaces has discrete fibers. The first diagram I drew is:
$$\require{AMScd} \begin{CD} f^\ast\coprod_i \left\{ x \right\} @>>> f^\ast \coprod_i U_i @&...
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Surjectivity of a map on inverse limits
I found the following statement in P. Gabriel's thesis:
Lemma. Let $(I, \leq)$ be a directed poset, and $(M_i, \mu_{ji}:M_j\rightarrow M_i)_{j\geq i},$ $(N_i, \nu_{ji}:N_j\rightarrow N_i)_{j\geq i}$...
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Example of surjective inverse system where projections from limit are not surjective
I am reading "Profinite Groups" by Ribes and Zalesskii and on page 9 it says that the projections of the nonempty inverse limit of a surjective inverse system are not necessarily surjective, ...
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Direct limit of sequences induced by fusing together copies of $\mathbb{Z}$
Let $A=l^\infty(\mathbb Z,\mathbb Z)$ be the abelian group of bounded sequences $\mathbb Z\to\mathbb Z$. Define a homomorphism $f\colon A\to A$ by
$$f(a)(n)=a(2n)+a(2n+1),$$
for $a\in A$ and $n\in\...
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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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Sheaf Axioms and Limits - Intuition
The question is based on the following problem from Vakil's notes in Algebraic Geometry:
2.2.C. The identity and gluability axioms (of sheaves) may be interpreted as saying that $\mathcal{F}(\cup_i ...
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Limits as initial objects
I am relatively new to category theory and was wondering about the following problem:
Can I consider a limit as an initial object in some categories?
Let $\mathscr{C}$ be a category and $\mathbf{J}$ a ...
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Surjection on limits
Suppose we have a map of diagrams $X \to Y$ of shape $D$ in the category of sets. Suppose further this is an objectwise surjection. That is, $X_d \to Y_d$ is a surjection for all $d \in D$. Are there ...
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