Timeline for Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

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Oct 2, 2018 at 9:32	comment	added	მამუკა ჯიბლაძე		@NajibIdrissi Thank you, this is certainly clearer than what I wrote. What I wanted to make more clear is that while replacing quotients by gluing in paths ensures that "projectivity is not damaged" (since we avoid non-split quotients), I don't understand how replacing pairs $(x,y)$ satisfying $f(x)=g(y)$ with paths between $f(x)$ and $g(y)$ ensures that "injectivity is not damaged"
Oct 2, 2018 at 9:09	comment	added	Najib Idrissi		I'm not sure what you are looking for. The way I view it is that for a quotient, instead of killing the relation $x = y$, you add an "path" between $x$ and $y$ (either a literal edge, or a new generator with $dz = x - y$). Similarly in a fibre, instead of only looking at couples where $f(x) = g(y)$, you instead look at couples $(x,y)$ together with a fixed "path" between $f(x) = g(y)$ (either a literal path, or an element such that $dz = f(x) - g(y)$). In both cases it's about adding paths... And I guess this is because I think in terms of 1-categories.
Oct 2, 2018 at 8:03	history	asked	მამუკა ჯიბლაძე	CC BY-SA 4.0