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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Non trivial colimit for rings in a finite diagram
I’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a ...
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Sequence of direct summands of modules over a ring
Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules:
$$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$
I claim that ...
- 16.1k
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(Co)Products are bifunctors, but are general (co)limits also functors?
In a category with all products or coproducts, the (co)product operation can be understood as a bifunctor. More generally let $\mathcal{C}$ be a category with all limits of shape $D$, where for ...
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Subtleties in commuting colimits
For context, I am reading Weibel's k-book and I am trying to express the homology of $BS^{-1}S$, the group completion of the classifying space of a symmetric monoidal category, as a colimit. In ...
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Is a canonical morphism from the wedge sum to the product monic? A section?
Let $C$ be a category with a terminal object $\ast$. For two pointed objects, define their wedge sum $c\vee d$ as the pushout
$$
\require{AMScd}
\begin{CD}
\ast @>>c_0> c \\
@VVd_0V@VV\...
- 7,021
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Colimits of full subdiagrams vs topological subspaces
Let $F:J \rightarrow \mathrm{Top}$ be a diagram in the category of topological spaces and $X:=\mathrm{colim} F$ be its colimit. For each $a \in \mathrm{ob}(J)$, denote by $\imath_a:F(a)\rightarrow X$ ...
- 151
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$\mathcal{O}(U)$ as a projective limit of Hilbert spaces
It is well-known that the space of holomorphic functions $\mathcal{O}(U)$ (with the standard topology of compact-uniform convergence) on an open set $U \subset \mathbb{C}$ is a projective limit of ...
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Is there a direct limit in the category of rings for hypercomplex numbers [closed]
I recently learned about the concept limits in categories. From R we can construct C the H etc... by iterating the Cayley-Dickson construction.
My question is: Can we construct a (non-associative)ring ...
- 27
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Showing that the diagonal functor $\Delta:\mathbb{C} \to \mathbb{C} \times \mathbb{C}$ having a right adjoint implies $\mathbb{C}$ having products.
I started brushing up on my understanding of adjunctions and came across this well-known fact (rephrased in my own words):
Let $\mathbb{C}$ be a category, and let $\Delta:\mathbb{C} \to \mathbb{C} \...
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Inverse limit of a quotient space (simple question)
Setup:
I have a tower of abelian groups $\hspace{1em} \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0$.
There are similar towers for $B_i, C_i$, and $D_i$.
There ...
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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\...
- 3,047
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Direct and inverse limit after deleting some groups
Suppose we have a preordered set $(I,\le)$ and a sequence of abelian groups $\{G_i\}$ with a homomorphism $\alpha_{i,j}:G_i\rightarrow G_j$ if $i\le j$ in $I$.
Let $G$ be the direct limit of this ...
- 3,047
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Cech cohomology on infinite open cover commutes with colimit on Noetherian space? (Exercise 5.2.6 in Qing Liu's book)
This is Exercise 5.2.6 in Qing Liu's book Algebraic Geometry and Arithmetic Curve. In part (b), I can show (b) if the open covering has only finitely many open subsets, since the the colimit of the ...
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Direct limit of a system: computation
I am considering a homomorphism from $\mathbb{Z}_4\rightarrow \mathbb{Z}_6$ given by $\bar{1}\mapsto \bar{3}$. My question is:
What is the direct limit of this system in the category of abelian ...
- 3,047
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A locally $\kappa$-presentable category is also locally $\lambda$-presentable for $\lambda>\kappa$? (Typo?)
In Riehl's Category Theory in Context, Sect. 4.6, we find the following:
Definition 4.6.16. Let $\kappa$ be a regular cardinal.¹ A locally small category $\mathsf{C}$ is locally $\kappa$-presentable ...
