# Terminology generalizing “quasi-isomorphism”

Background: In homological algebra, a quasi-isomorphism of chain complexes is a chain map$\phi:(C,d) \to (C',d')$ so that the induced map on homology $\phi_\ast:H_\ast(C,d) \to H_\ast(C',d')$ is an isomorphism.

This is clearly a special instance of the following much more general phenomenon: given a functor $\mathcal{F}:\mathcal{C} \to \mathcal{D}$, between categories $\mathcal{C}$ and $\mathcal{D}$, a morphism in $\mathcal{C}$ is taken to an isomorphism in $\mathcal{D}$ by $\mathcal{F}$.

What is the standard terminology for a morphism of $\mathcal{C}$ so that its image under $\mathcal{F}$ is a (mono, epi, iso) morphism?

I couldn't find a term for this in MacLane's book, but I'm sorry if this is standard stuff.

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I don't think this is completely standard, so if I were going to use it I would explain it first, but a natural possibility is "$\mathcal{F}$-isomorphism" (-monomorphism, -epimorphism).

Quasi-isomorphisms are in fact sometimes called $H_*$-isomorphisms, and similarly maps inducing isomorphisms on homotopy groups are sometimes called $\pi_*$-isomorphisms (the main example of this usage that I can think of is for symmetric spectra, where $\pi_*$-isomorphisms are not the same as stable equivalences and thus need their own name).

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There is no standard terminology that I know. But if I had to create one, there are precedents: I would steal from the language of homotopy theory.

A model category is a category equipped with a model structure: three special classes of morphisms, called weak equivalences, cofibrations, and fibrations (satisfying some axioms). There is a general construction called the "homotopy category" of a model category $C$: this is in some appropriate sense the universal category with a functor from $C$ which turns weak equivalences into isomorphisms.

Thus I'd be comfortable calling morphisms which are sent to isomorphisms equivalences. Quasi-isomorphisms are in fact the weak equivalences for a model structure on the category of chain complexes, so this example is a fortunate one.

Sometimes, for some particular examples of model structures, the cofibrations are defined to be the morphisms which are sent to monomorphisms by some natural forgetful functor. Similarly, sometimes the fibrations are the morphisms sent to epimorphisms.

Thus I'd be vaguely comfortable using the same terminology for this situation. However, of course, one should be up front about this, to avoid confusing those readers who know about model categories already and might imagine that you are building one.

Of course, it may be that in the example you're considering, there really is a model category structure somewhere...

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Or at the very least, 'weak equivalences'... –  David Roberts Jul 11 '12 at 23:23