What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from …

What is the standard or best reference for the adjointnes of bar and cobar constructions?

Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijectio …

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 …

I recently learned about two-variable adjunctions and multi-variable adjunctions, and about adjunctions in 2-categories. The nCatLab page on two-variable adjunctions talks about h …

In answering this MO question, the issue was raised of characterizing when a given endofunctor $R:C\to C$ has the form $U\circ F$ where $F:C\to D$ is left adjoint to $U:D\to C$, i. …

It was asked here whether every functor is a composition of adjoint functors. The answer is no, because all adjoint functors induce homotopy equivalences on the nerve, and we can c …

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 am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming th …

Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair? In my mind …

Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X) …

Let $F: \mathscr{C}\to \mathscr{C}^{op}$, with an adjoint $G$, and $\eta: 1_\mathscr{C} \Rightarrow G\circ F $ and $\varepsilon: F\circ G\Rightarrow 1_{\mathscr{C}^{op}}$ with …

Because adjoint functors are just cool, and knowing that a pair of functors is an adjoint pair gives you a bunch of information from generalized abstract nonsense, I often find mys …

Hi, assuming we have $\mathbb{K}$-linear categories and functors $F:D\to C$ and $G:C\to D$ which are adjoint $F\dashv G$, then there exist unit and counit functorial morphisms $\va …

I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arr …