# Questions tagged [limits-and-colimits]

For questions on limits and colimts in the sense of category theory, and related notions.

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### 2(Probably quite simple) Math Questions [closed]

https://i.stack.imgur.com/LTJXB.jpg I need help with these two questions.I can't try anything because i'm not good at math.Thanks for the help.
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### Primitive ideals of inductive limits of $C^*$-algebras

I am trying to understand ideals of direct limits in the category of $C^{\ast}$-algebras. Let $(A_n,f_n)$ be a direct sequence of $C^{\ast}$-algebras and let $I$ be a primitive (modular) ideal of ...
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### Reference for homotopy colimit = total complex

I'm looking for a reference for the following fact: take a simplicial chain complex $X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
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### Can filtered colimits be computed in the homotopy category?

For $\mathcal{S}$ the $(\infty,1)$-category of spaces its homotopy category $h\mathcal{S}$ does not have pushouts or pullbacks. Even if it does, they won't always agree with the (homotopy) pushouts or ...
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### $L^{\infty}$ as colimit

I recently read a result (in Jarchow's book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following. Let $\mu$ be a ...
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### Gluing together dense subset of Projective Limit in $Ban_1$

Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
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### $L^1_{\mu}$ as limit

Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Does there exist a countable set of finite measures $\{\mu_n\}_{n \in \mathbb{N}}$ on $(X,\Sigma)$ such that $L^1_{\mu}(\Sigma)$ can be ...
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### Diagonal of a diagram of codescent objects

Given the following diagram in a $2$-category, in which squares of the same "type" commute, where each column and each row is a strong codescent diagram (Edit: it should be reflexive as well), is ...
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### Can homotopy colimits recover cohomology sheaves?

The question is basically the one outlined in the title. Let $\mathcal{T}$ be a triangulated category containing infinite direct sums (e.g. $D_{qc}(X)$ for some separated, finite type over a field $k$,...
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### Existence of homotopy limits and colimits in model categories

I am not an expert, thus I apologize if my question is very naive. Let $\mathsf{M}$ be a model category (I do not assume any functoriality on the factorization), Q1. Is there a reference where it ...
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### Can $L^1_{loc}$ be represented as colimit?

Let $L^1_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1_{m_K}$ denote the space ...
### Limit of balls in $L^p$
Setup: Let $\mu$ be a measure on a measurable space $(X,\Sigma)$, such that for every $p ,q\in [1,\infty)$, $L^p_{\mu}(\Sigma)\subseteq L^q_{\mu}(\Sigma)$ if $p\geq q$. Furthermore, the inclusions ...