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This follows from the Yoneda lemma - probably my favourite way of thinking about this fact is via coends in the way described here by Todd Trimble, which I think makes it quite clear what is going o …

One can prove that for any non-zero ring $R$ the category $R$-Mod$^{op}$ is not a category of modules. Indeed any category of modules is Grothendieck abelian i.e., has exact filtered colimits and a ge …

I don't know the paper well enough but the notion of plethory in J. Borger, B. Wieland, Plethystic algebra, Advances in Mathematics 194/2 (2005), pp 246-283 (which is available from Borger's website) …

It is the minimal separator in the sense that it corepresents the forgetful functor Grp $\rightarrow$ Set, but this uses sets so probably isn't what you are after. In fact it is the same as the statem …

To answer the first question provided one has, as you say, (small) products and equalizers the notion of sheaf makes sense as one has the right diagram corresponding to any cover. But we can just say …

There are still strictly speaking elements floating around in the following since we are using indexing sets but maybe it is better? Consider for a set $S$ and an abelian group $A$ the isomorphisms $$ …

Dan Murfet has some notes on foundations for category theory which can be found here. They contain an introduction to Grothendieck universes as well as some references for learning about NBG class th …

Suppose that $F\colon C\to D$ is a functor. Then there are many situations in which thinking of finding left and right adjoints to $F$ as solving approximation problems is very good intuition. So thes …

The adjoint functor theorem as stated here and the special adjoint functor theorem (which can also both be found in Mac Lane) are both very handy for showing the existence of adjoint functors. First …

One can take the category of modules over a Laurent polynomial ring in one variable $\textrm{Mod}\;k[t,t^{-1}]$ and think of $k[t,t^{-1}]$ as the group algebra of $\mathbb{Z}$. The corresponding cocom …

I assume you are working in some fixed abelian category $\mathcal{A}$. It is not true in general that every short exact sequence in $\mathcal{A}$ will split. The problem is that although you can pick …

At least if one takes labeled graphs (LGrphs) and labeled digraphs the functor you suggest, say D, is right adjoint to the forgetful functor which I'll call U. There is a canonical natural transformat …

A counterexample which is non-trivial is given in Chapter 6 of Neeman's book Triangulated Categories. The category in question is the full subcategory of additive functors Cat(S^{op}, Ab) where S sati …

So the fact that you had a hard time thinking of cosplit sequences of groups and the last question got me thinking (along the lines of Joel's comment actually)... what I came up with is probably stand …

I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation th …