# Categorical Fibrations

## Main Question or Discussion Point

I wasn't sure where to put this question. Sorry if it's in the wrong place.

I'm trying to read a paper by Hovey on model categories, so I turned to his book, Model Categories. Given a category, he gives the definition in terms certain morphisms called weak equivalences, fibrations and cofibrations.

A weak a equivalence is a morphism that becomes an iso when passed to the homotopy category. And I know the topological defintions of (co)fibrations.

Is there a categorical definition of fibration and cofibration? Or should I take them as undefined terms subject the axioms of a model category? I've even thought about constructing the category of CW-complexes, letting my weak equivalences be weak homotopies, and fibrations and cofibrations be the topological versions. Then try to see what the (co)fibrations do categorically speaking.

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matt grime
Homework Helper
Hmm. It's a tricky one to answer. Basically, you have some concrete honest to goodness category of something - spectra, modules for a ring, sheaves, etc, and then you pass to some triangulated stucture, often. You'd not be remiss in thinking of a cofibration as an injection, and a fibration as a surjection, though really we mean monomorphism, and epimorhpism, and to be honest what we actually mean is probably an inflation and a deflation. But, no, these are topological ideas. A 'generic' (co)fibration, is something that behaves like a (co)fibration in some analogous category. But this is a guess because I don't have the Hovey/Palmeri/Strickland stuff on me (they're in my office). See the link with stable module categories for a better idea.

Hmm. It's a tricky one to answer. Basically, you have some concrete honest to goodness category of something - spectra, modules for a ring, sheaves, etc, and then you pass to some triangulated stucture, often. You'd not be remiss in thinking of a cofibration as an injection, and a fibration as a surjection, though really we mean monomorphism, and epimorhpism, and to be honest what we actually mean is probably an inflation and a deflation. But, no, these are topological ideas. A 'generic' (co)fibration, is something that behaves like a (co)fibration in some analogous category. But this is a guess because I don't have the Hovey/Palmeri/Strickland stuff on me (they're in my office). See the link with stable module categories for a better idea.
I'd noticed that the arrows for fibrations and cofibrations are drawn like those for injections and surjections, so I, at first, thought that they would correspond to epi and monomorphisms. But as I read further, I realized that this wasn't the case.

I've briefly checked Hovey and Quillen's Homotopical Algebra, but they seem to use the terms without defining them.

Drawing some diagrams, (topological) fibrations look like projectives in module theory, so I wondered if there was a connection. Also, the pictures resemble the ones we draw for projective covers. I'm going to see if I can find a connection between covering spaces with fibrations and projective covers. It may be coincidence, but it sounds like a nice way to spend the afternoon.

Also, the whole point of this is to read Hovey's Model Category Structures on Chain Complexes of Sheaves. I'm not made it much past the abstract because I was trying to understand what a model category actually is. Do you think it's the case that I don't need a categorical definitions of (co)fibrations? I can just take them as classes of arrows that satisfy certain conditions.

ETA: I don't see the link on stable module categories.

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matt grime
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Stable module categories are stable homotopy categories, just in the language of algebra, not topology.

The notion of surjection doesn't make sense in topology in quite the same way. You know how to construct the cone of a map of topological spaces f:X-->Y? Now look at homotopy categories of complexes over an abelian category. Loot at the mapping telescope, or the mapping telescope conjecture, for example.

matt grime
Homework Helper
if you'd just googled fibration you'd've found the categorical definition, btw:

http://en.wikipedia.org/wiki/Fibration

but they ought to be thought of as the topological notions of injections and surjections. Or more rigorously, perhaps, you should think of short exact sequences:

X-->Y-->Z

the first being a replacement for cofibration, the second for fibration. A map can be factored as the composite of an surjection and an injection, normally.

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matt grime
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But this is a good post. If you give me time to find my references, and refamiliarize myself with the topological viewpoint, after the weekend I'll gladly talk some more on the subject.

Thanks. I'd appreciate it.

mathwonk
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just as a guess, it seems to me that fibrations are usually defiend in topology as maps with a certain lifting property, i.e. in terms of the existence of other maps that factor your map. once you have a concept defined in terms of maps, it makes sense in any category.

this sounds a little like the "projective" property mentioned above.i.e. projectives are objects such that any map to them lifts through any surjection.

fibrations are usually more special, i.e. special surjections such that any map say from an interval lifts through them, and homotopies are ropeserved. but its been 40 years since i took topology, but as i recall serre defined a fibration to be any map with the "homotopy lifting" property.

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I'm starting to suspect that it's best to leave them as undefined primitives that satisfy the model category axioms. The best "definition" I have of weak equivalences is a map that induces an isomorphism in the homotopy category. But the homotopy category depends on how the w.e. are defined. Moreover, the axioms that they must satisfy seem to make them be interdependent. That is, a (co)fibration is the dual of a co(fibration) and a weak equivalence induces a homotopy isomorphism in a category that depends on the weak equivalences. Any attempt to find necessary and sufficient conditions seems circular. I think the axioms of model categories give the similiarly shaped diagrams, but not conversely, and that's all we can say.

So unless you guys think otherwise, I'm going to read them axiomatically.

matt grime
Homework Helper
The idea of a weak equivalence is that of a quasi-isomorphism.

All these things are just families of maps that have certain properties. Given any particular category there is no intrinsic way of defining them. Or if you will, any category has many different classes of morphisms that can be fibrations, cofibrations, and weak equivalences.

The usual thing to do is to pick some class that will be interesting.

You're right, in general you won't be able to pick (simple) categorical axioms that define (co)fibrations and weak equivalences and that are independent of the category.

Here are some examples:

let Ch(A) be the category of chain complexes over an abelian category A, then fibrations are cones, and weak equivalences are quasi-isomorhpisms. The corresponding 'stable homotopy category' is the derived category D(A).

Part of my thesis was proving that there are generically infinitely many stable categories one can create from the module category of a group algebra (in characteristic p, where p divides |G|).

I am from a non-topological background, so I don't really get the topological view point. The difference is that in topology you don't have abelian/exact/additive categories around until you pass to the stable homotopy category (you can't in general 'add' maps of topological spaces unless the domain is a suspension of an object - in the stable case everything is by fiat the supsension of some other space).

You will, in general, need you sets of morphisms to satisfy some extra hypotheses. These rules are so that when you form the localization (add formal inverses for the weak equivalences) then you might create something that requires a strictly larger set theoretic universe from the one you started in.

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That's very helpful, thanks. I'm most comfortable with topology, so I like my algebra to stem from that. I'm in the second semester of a homological algebra course, and it's interesting seeing the ideas in a purely algebraic context. One problem with that, I guess, is that I try to relate all of what I learn to topology. Chain complexes "=" chain complexes of singluar homology. Homology itself I still think of as counting n-dimesional "holes" in spaces. So when I came across fibrations, I wanted to relate them to the Serre fibrations I learned in homotopy class.

Thanks for the insight.

mathwonk
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homology measures failure of exactness. i.e. the failure of any necessary condition to be sufficient.

in the case of topological holes, the conditions are "having a boundary" and "being a boundary".

i.e. to be a boundary, it is necessary but not sufficient to NOT have a boundary.

e.g. the boundary of a disc is a circle, which has no boundary, but not every circle bounds a disc in a given space.

dually, in differential form calculus, dd = 0, so for [ ] to be d of something it is necessary but not sufficient, to have d[ ] = 0.

related concepts measure the ability to extend a map from a subspace to the whole space. e.g. a map of a circle in the subspace, which bounds in the larger space, must go to a circle which bids in the target, if the extension to the whole space is to be possible, but this si not sufficient.

special spaces, called cw complexes or cell compexes are defiend which are built in predictable ways form attaching cells, and for which enough of these necessary conditions can be formuklated to be sufficient.

one gets theorems like whiteheads thm that a map which induces isomorphism on all homotopy groups, perhaps is a homotopy equivalence. the proof is an extensions problem of extendiong homotopy from cells in the space to the whole space.

such techniques are amenable to enormous generalization. they are inductive methods with built in "obstructions" at each step.

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I had realized that homology measures how far from exactness a chain comples is, and of course I've seen CW complexes and cellular homology and de Rham cohomology, but I guess I didn't make the connection between holes, boundaries, etc. Nice explanation.