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$\DeclareMathOperator{\Hom}{Hom}\newcommand{\amod}{\mathscr{A}\text{-}{\bf Mod}}\newcommand{\scrA}{\mathscr{A}}\newcommand{\scrE}{\mathscr{E}}\newcommand{\Ab}{\mathbf{Ab}}\DeclareMathOperator{\Lex}{\m …

The category of Banach spaces and contractions (over the reals or any other complete normed field, I think) is an example of an $\aleph_{1}$-presentable category which is not $\aleph_{0}$-presentable. …

If $X$ is a left bounded complex, you may take a quasi-isomorphism $X \to I$ where $I$ is a left bounded complex of injectives. Such a complex exists and can be found explicitly in many books on homol …

Note: I'm only addressing David's question how one one can "add cokernels" to an additive category and how one can use this in order to embed an additive category into an abelian one, even in a univer …

Part 1: The "cheeky" answer is: huge. There is a left adjoint to the forgetful functor $LCTVS \to Vect$ (in particular there is a left adjoint to the forgetful functor $LCTVS \to Sets$): Equip a vecto …

Here are some examples that are less of an algebraic nature (but all seem to be subsumed by Qiaochu's observation in that they are "weakly initial" or "weakly terminal" objects in appropriate categori …

The answer is yes, at least if you believe Thomason-Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, which David Ben-Zvi already mentioned. I quote from Appendix B.3 (p. 40 …

You essentially answered your first question yourself: the ground field is a (contractive) retract of any nonzero Banach space by Hahn-Banach. If there were a non-zero projective Banach space in your …

This is too long for a comment only, therefore I write it into an answer (all categories are assumed to be additive). This will not answer your question but maybe it will provide some insight. Let me …