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If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory:

  • never perform quotients, add structure instead;
  • never require subobjects, take fibres instead.

Although having encountered instances of both many times, I must confess that I understand the first one much better than the second one. I understand conceptual reasons behind the "no-quotients-please" principle well enough: whenever you have (non-split) quotients, you get non-projective objects, and then you have to resolve them by projectives anyway. Instead of this you may enrich your structure so that these resolutions are "already there". Algebraically this means avoiding any identities/relations at the expense of having richer algebraic structure: identities for algebraic structures are replaced by "higher level" structures, and everything "remains projective" all the way up.

With the "no-kernels-please" principle I can understand the conceptual reason (which is straightforwardly dual) but not much more. Conceptually it seems I now want to achieve that all injective resolutions are "already there" and that in fact everything "remains injective". However I don't understand at all what corresponds to this principle on the (maybe co-?) algebraic side. What is the dual of the "replace identities by higher order structure" slogan? How does it ensure keeping everything injective?

What I understand even less is that upon stabilization there seems to be a tendency for projectives and injectives to coincide, i. e. to get something "Frobeniusish". Again conceptually this is sort of trivial - if I want to follow both principles simultaneously, then everything will remain projective and injective at the same time. But, while obviously a Frobenius situation is even closer to the "ideal" vector spaces situation, I have no idea what does correspond to it on the algebraic side. Somehow identities must become the same as "co-identities" but I have no idea what they might be.

Could somebody help me out? Is there an algebraic model where incorporation of both these principles has a clear purely algebraic interpretation? It would be also OK with me if, say, in fact I need both algebra and coalgebra.

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    $\begingroup$ 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. $\endgroup$
    – Najib Idrissi
    Commented Oct 2, 2018 at 9:09
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    $\begingroup$ @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" $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 2, 2018 at 9:32

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