# 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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### 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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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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### Products In the Categary of Skew-Cocommutative Coalgebras Are Skew Tensor Products

Let $A = \bigoplus^n_{i=0} A_i,B = \bigoplus^b_{i=0} B_i$ be a graded modules over the same commutative ring $R$ . The twisting isomotphism $\tau_{A,B} : A \otimes B \to B \otimes A$ is defined on ...
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