Given categories $X$ and $Y$ and a strong functor $$D:X^{op}\times Y\to Cat$$ we can of course build the oplax colimit $$\mathrm{colim}^{oplax}_{X^{op}\times Y}D$$ via the usua …

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-equval …

This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation. Recall that every space (or ∞-grou …

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 …

In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is level …

Given a topological space $X$, we consider the following category $\mathsf{CW}_{X\to}$. The objects are finite CW complexes $Y$ equipped with a continuous map $X\to Y$. The morph …

In order to test the monadicity of a functor, there is a precise monadicity theorem (PM) as well as a crude monadicity theorem (CM), see the nlab. In CM, the forgetful functor shou …

In Wikipedia's article on Kan extensions 1, in the view of Kan extensions as colimits, I am confused about the notation: $(Lan_F X)(b) = \varinjlim_{f:Fa \to b} X(a)$. Wikipedia s …

I'm in need of a condition that is analogous to the "finality" condition in the following lemma: Lemma: A functor $F\colon A\to B$ is final if and only if for any functor $x\colon …

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the approp …

I wanted to ask the following question, Suppose $\mathbf{M}$ a cof generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ i …

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 t …

Background on the Adèles The Adèles $\mathbb{A}_K$ of a number field or function field $K$ are defined as a restricted product of the complete local fields $K_\nu$, where $\nu$ ra …

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 …

It is a fairly special property for the Yoneda embedding $A \hookrightarrow \mathcal{P}A$ of a category to have a left adjoint defined everywhere (this happens just when $A$ is tot …