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The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those statem …

For functors $E,F,E',F':X\to Y$ I would like to say that transformations $\tau:E\to F$ and $\tau':E'\to F'$ are isomorphic if they are isomorpic in the arrow category of functors $X\to Y$, that is, t …

So I wonder which terminology can I use and which "precautions" should be taken so that my simple arguments concerning functor categories will be mathematically correct. … Do you know any text that introduces "the most conventional" terminology for these matters? I would not like to mention universes, and I don't like terms like "small set". …

I am interested in those objects in the ("topological") stable homotopy category $SH$(I call them spectra) whose homology (with integral coefficients; should I call it singular or stable, or $H\mathbb …

Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex $D$ of coherent sheaves. Then (if I understand these matters and the answer to the question Characterization of …

How do people call an additive functor from a triangulated category $C$ to an abelian one that converts distinguished triangles into long exact sequences. Does one usually call a covariant functor o …

I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider th …

Does there exist any term for finite groups with no non-trivial homomorphisms into $\mathbb{Z}/p\mathbb{Z}$ for a fixed prime $p$, or any term related to this property (so that I could write "finite …

In our text we have several statements of the following sort: for a certain morphism $f:X\to Y$ of varieties over an (algebraically closed) field of characteristic $p$ and some $c>0$ the corresponding …

Are these properties of fields and schemes related to any existing terminology? …

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with respe …

Let $X$ be an equidimensional variety of dimension $n$ over a field that can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ (for a large enough $N$; we consid …

A cellular tower in SH or in a "more general stable homotopy category" is a chain of morphisms $\dots X^{(n)}\stackrel{g^n}{\to} X^{(n+1)}\to \dots$ along with some more data and conditions; one of th …

In my preprint I propose to call $X/S$ quasi-smooth if $X$ can be embedded into a smooth $X'/S$. Does this sound fine? Upd. So, smoothly embeddable is better? Is it ok to call a morphism smoothly emb …