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There are important functors like this associated with coalgebras or corings. E.g., let $C$ be a coassociative coalgebra with counit over a field and $N$ be a right $C$-comodule. Then the cotensor p …

One general rule that unites some of the examples above is that if you have two categories whose objects are sets endowed with some structure, and there is an equivalence between these two categories …

This short answer is a complement to Simon's detailed answer. In Simon's answer he considers arbitrary $\kappa$-accessible categories $A$ (in the first theorem) and locally $\kappa$-presentable ones …

In MacLane's "Categories for the Working Mathematician", in the end of section VII.1 there is an argument due to Isbell proving that one cannot strictify a monoidal category by replacing it with its s …

There is a natural functor with such property in the theory of coalgebras and co/contramodules over them. Given a (coassociative, counital) coalgebra $C$ over a field $k$, a left comodule $M$ over $C …

Let $\mathcal C$ and $\mathcal D$ be two categories, and let $F\colon\mathcal C\longrightarrow \mathcal D$ and $G\colon\mathcal D\longrightarrow\mathcal C$ be two functors, with $F$ left adjoint to $G …

The following pair of examples follows the idea of Jeremy Rickard suggested in a comment on Math Stack Exchange under the link. Inverting the arrows, it suffices to construct an example of abelian ca …

Another and probably more natural interpretation of the sentence in the Wikipedia article may be called "localizing an adjunction to an equivalence". Let $\mathcal C$ and $\mathcal D$ be two categori …

The right definition is: take the free associative (tensor) algebra generated by $V$; divide out the ideal generated by the elements $xy-(-1)^{|x||y|}yx$ for all homogeneous $x$, $y\in V$ and $z^2=0$ …

For any diagram $B_i$ and an object $A$ in a category, there are natural maps of sets: colim Hom($A,B_i) \to$ Hom($A$, colim $B_i$) colim Hom($B_i,A) \to$ Hom(lim $B_i, A$) These maps need not be …

Yes, there exists such a treatment by Deligne, see "Cohomologie a supports propres", SGA4, Tome 3, Lect. Notes Math. 305, subsections 1.2.1-1.2.2. Basically, what one needs is that for any object X i …

I came to the following question when thinking about the (infinitely generated) tilting-cotilting correspondence, where it appears to be relevant. Does there exist a locally presentable abelian ca …

Let $J^\bullet$ be an acyclic complex of injective objects in an abelian category $\mathcal A$. Consider its finite subquotient complexes of canonical truncation $0\to Z^m\to J^m\to J^{m+1}\to \dotsb …

Two morphisms of groups are isomorphic as functors between the related categories if and only if they differ by an inner automorphism of the target group. The choice of a particular isomorphism of su …

The category of CDG-rings (curved DG-rings) does not have an initial object. (A CDG-ring B = (B,d,h) is a graded ring B endowed with an odd derivation d of degree 1 and an element h of degree 2 such …