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### Constructing pointwise Kan extensions as adjoints to some functor

Background
I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because ...

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### (Co-)Limits and fibrations of DG-Categories?

First of all, let me see if i got the 1-categorical version right:
Let $\mathcal F:C\to Cat $ be a
(pseudo-) functor. The 2-colimit
$\mathrm{colim}_C\mathcal F$ is then
given by the grothendieck
...

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### Reference request: colimits of locally presentable categories

Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small ...

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### Colimit of an etale diagram of schemes

It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More ...

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### Composition of Cat-valued distributors - compatible with grothendieck construction?

Let $C$ be a category and $F\in[C^{op}, Cat]$ be a strong functor.
(1) There are functors
$$hom_C(c',c)\times F(c)\to F(c').$$
(2) The grothendieck construction gives a 2-equvalence
$$\int_C: ...

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### The inductive and projective limits of compact Hausdorff topological groups

Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit ...

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### Direct Limits and Limits of Nets

A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed ...

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### how many ways can an algebra be a weighted colimit of free algebras?

For a given weight $W : \mathcal{S}^{op} \to \mathcal{V}$ and diagram $D : \mathcal{S} \to \mathcal{A}$, the weighted colimit is an object $W \cdot D$ together with an isomorphism
$$\mathcal{A}(W\cdot ...