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In Shastri's Basic Algebraic Topology Remark 1.8.8, the author wrotes that

The difference between covariance and contravariance is simply in the fact that covariance preserves the direction of the arrow whereas contravariance reverses it. However, in practice, it turns out that contravariance has more mathematical structure in it whereas covariance is more geometrical and easy to understand.

I am really curious as to why the author thinks that way. Unfortunately, he doesn't provide further justification of that. My question is:

  1. Is this a serious claim? (i.e. Is there a sense in which it is true?)
  2. If the answer to 1 is 'yes', what's the underlying reason for this phenomenon? (maybe there is a categorical justification?)

Though I am just a beginner in algebraic topology, answers from all levels are welcomed.

As Connor Malin comments below, every contravariant functor is a covariant functor from $\mathsf{C^{op}}$ to $\mathsf D$, so the two concepts are duals of each other, which further mystifies the claim.

YuiTo Cheng
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    I am skeptical. One can easily argue that contravariant functors are inherently more geometric in daily life because we tend to consider sheaves more than cosheaves in geometric contexts. Also, it really isn't right to say contravariant constructions have more algebraic structure because even if it were true that contravariant functors come with more familiar algebraic structure, covariant functors would then often come with the dual structure. For example, homology with field coeffecients forms a coalgebra, which is not a trivial thing. – Connor Malin Dec 18 '20 at 06:17
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    Maybe he is saying that in cohomology you have the cup product while in homology you don't. – Yuval Dec 18 '20 at 06:36
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    Maybe it would help to have more context for this statement. After all, it's not at all true in general category theory, because of the duality in your final paragraph. Presumably the context is algebraic topology (hence Yuval's guess), but maybe you can give us a more specific context than that? – Toby Bartels Dec 18 '20 at 07:46
  • Coordinates are contravariant, bases covariant. In practical computations you are more likely to work with coordinates and define structures from it. This may mean that your intuition is biased to this direction. I would understand the statement not as inherent properties, but as a difference in perception. – Lutz Lehmann Dec 18 '20 at 07:51
  • My 2 cents to expand on the comment by @LutzLehmann, covariance is probably a more 'natural' concept to pick up for beginning students - 'geometric' not in the sense of (algebraic) geometry but 'geometric' in the diagrammatic sense. On the other hand, contravariance is less 'easy to pick up', but arises quite often in various applications (e.g. cohomology). – mi.f.zh Dec 18 '20 at 08:14
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    From mathematical point of view covariance and contravariance only differ by the direction of an arrow. And they are equivalent. So we are in the domain of pure personal opinion. I'm not even sure what that comparison (regardless of how accurate) is supposed to say? I mean even if one has more mathematical structure (whatever that means) and the other is more geometrical (whatever that means), so what? Meaningless and confusing. – freakish Dec 18 '20 at 08:22

1 Answers1

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Is it a serious claim? I claim this doesn't have a serious answer: It is opinion-based.

Remark 1.1.8 follows the definition of covariant and contravariant functors, thus it is certainly a statement about functors used in practice. There are other contexts where the words "covariant" and "contravariant" are used, for example in the context of (multi-)linear algebra; see Lutz Lehmann's comment and this question. I guess that most people (including physicists and engineers) who have heard these word associate it with tensors. But this context is irrelevant here.

Have a look at the contents of the book. It has 13 chapters, but one cannot say that contravariant functors (essentially cohomology functors) are overrepresented and provide more mathematical structure than covariant functors.

There are some nice things which you can do with cohomology, for example you can introduce the cup product and get the cohomology ring or define cohomology operations. But all that can be dualized, for example you can give homology a coring-structure. Many people have never heard about this concept, thus it may appear somewhat strange.

Contravariant functors are often representable ("Brown's representability theorem") which led to the fruitful concept of a spectrum. Another example is $K$-theory which arises via classifying vector bundles on spaces $X$; this gives a very important (generalized) cohomology theory. The construction is geometrical and easy to understand although it is contravariant.

This shows that many important concepts of algebraic topology are based (in a natural way) on contravariance. However, other things like homotopy groups are covariant.

I think we need on equal terms both co- and contravariance. The whole complex of duality, for example duality in manifolds, relies on both types.

Paul Frost
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