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.
1,038 questions
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Show that $\varprojlim \mathbb{Z}/n!\mathbb{Z}\cong\prod\limits_{p \text{ prime}}\mathbb{Z}_p$ by giving a direct isomorphism
A exercise from my number theory course asks me to show that $$\lim_\leftarrow\mathbb{Z}/n!\mathbb{Z}\cong\prod\limits_{p \text{ prime}}\mathbb{Z}_p$$ as rings, by giving some explicit isomorphism. I ...
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Local presentability of the category of normed vector spaces
I am wondering if the category $\mathbf{NormVect}$ of normed vector spaces with linear contractions is locally presentable. Likewise, the category $\mathbf{SemiNormVect}$ of semi-normed vector spaces ...
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universal property of the coproduct of abelian groups
universal property of the coproduct of abelian groups
Suppose for every pair of indices $i, j$ with $i \leq j$ there is a map $\rho_{ij} : A_i \to A_j$ such that the following hold:
i. $\rho_{jk} \...
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Non-cocomplete topos with a geometric morphism to the topos of sets
Is there an elementary topos $\mathcal{E}$ such that there exists a geometric morphism $\mathcal{E} \to \mathbf{Set}$ (or equivalently, the functor $\mathcal{E}(1, -)$ has a left adjoint, which is ...
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Example of a finite category without sequential colimits?
I assume that there must be a finite category which does not have all sequential colimits. What is an explicit example?
Context. I have been writing down some proofs for the existence of directed ...
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What is the conceptual reason that $B : \mathbf{Mon} \to \mathbf{Cat}$ preserves pushouts?
The delooping functor $B : \mathbf{Mon} \to \mathbf{Cat}$ has a left adjoint (MSE/574745), hence preserves all limits. It does not preserve coproducts, quite drastically, since the category $B(M) + B(...
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Formal definition for a canonical morphism [duplicate]
I'm relatively new to the concept of categories and there always seems to be a similar reasoning gap in my understanding in almost every proof - suppose there are "canonical" morphisms ...
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Is the category $\mathbf{Pos}$ of partial orders coregular?
This question deals with regular categories. The category $\mathbf{Pos}$ of partial orders is not regular (see Example 3.14 in the linked nlab article). But I wonder if it is coregular. By definition, ...
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Homogeneous generators for inverse limit of graded modules
$\newcommand{\tensor}{\otimes}$
I would like to present a proof for the following lemma, that is an extension to the graded case of a
theorem (II, 9.3A) that is stated in Hartshorne, Algebraic ...
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When is the pushout along a surjective epimorphism along a split monomorphism a surjective epimorphism?
I was doing a problem and I came upon the following query.
I have a limit sketch $\mathbb{T}$, an epimorphism of models $f:X\to Y$ in the category $Set$ that is also surjective-on-components and a ...
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Does inductive limit topology with cofinal chain satisfy the subspace property?
Let $\{ (E_\alpha, \tau_\alpha) \}_{\alpha\in A}$ be a family of locally convex spaces, where each $E_\alpha$ is a vector subspace of a vector space $E$ over $\mathbb K.$ Suppose further that the $E_\...
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Nuclear maps and direct limits
I am looking for references on a nuclearity result. Let's consider a direct system of topological vector spaces $(E_i)_{i \in \mathbb{N}}$, whose morphisms are all nuclear maps. Moreover, let's define ...
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Does existence of inductive topology follow from lattice property?
Let $X$ be a set and $\frak T$ the system of all topologies on $X$. It is well known that $\frak T$ is a complete lattice under the "finer than" relation. On page 5 of Schaefer & Wolff's ...
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Local base for inductive limit topology
The following is based on Schaefer & Wolff's Topological Vector Spaces, 2nd edition, §II.6. Let $E$ be a vector space, $\{E_\alpha\}_{\alpha\in A}$ be locally convex spaces (not necessarily ...
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Explicit description of left adjoints in pushout diagram
Suppose that there is a diagram $A_1 \overset{G_1}{\rightarrow} A_0 \overset{G_2}{\leftarrow} A_2$ of right adjoints of presentable $\infty$-categories, and let $A_3 = A_1 \times_{A_0} A_2$ denote the ...
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