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Let $\mathcal{C}$ be a category and $\mathcal{G}=(G,\delta, \epsilon)$ be a comonad on $\mathcal{C}$. Here $G: \mathcal{C}\to \mathcal{C}$ is a functor, $\delta: G\to G^2$ and $\epsilon: G\to id_{\mat …

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and $F$ and $G: \mathcal{C}\to \mathcal{D}$ be two functors. Suppose $F$ and $G$ have right adjoints $F^{\wedge}$ and $G^{\wedge}: \mathcal{D}\to …

I read on this n-lab page that a fully faithful functor $F: C\to D$ reflects all limits and colimits by the universal property. On the other hand, I think a fully faithful functor does not always pres …

Let $\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{G …

In Chapter 15 Section 15.2.1 of Quantum Groups and Noncommutative Geometry, 2nd edition, the authors raised a question: can we reconstruct a finite group $G$ from its category of finite dimensional co …

Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but morph …

Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category. Now we consider dgCat as a 2-category, which objects small dg-categories, $1$- …

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under formin …

Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under i …

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We coul …

Let $\mathcal{C}$ be a category. The pro-category pro-$\mathcal{C}$ is defined as (see this nLab page) follows: its objects are diagrams $F: D\to \mathcal{C}$ where $D$ is a small cofiltered category. …

One of the remarkable results in Toen's paper is the existence of internal homs of dg-categories. Actually for two dg-categories $A$ and $B$, there exists a dg-category $RHom(A,B)$ such that for any d …

Let $A$ and $B$ be differential graded algebras over a field $k$ and $D(A)$ and $D(B)$ be the derived categories of differential graded modules. Let $N$ be an $A$-$B$ bimodule. Then $(-)\otimes^{\mat …

Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums. A triangula …

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined $$ RHom(C …