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Is there an easy way of remembering the direction of arrows between morphisms in Categories?

The direction of arrows so confuses me: products and co-products, (EDIT- Also, pull-backs, pushouts, contra/co-variant functors) and their universal mapping properties. I have to look back into Lang's Algebra and revise it every time.

Please tell if the question is not appropriate.

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up vote 11 down vote accepted

I think that the best way to remember these things is to have a good example of each of these constructions in your head. If you remember the direction of the arrows for them, and if the example is "natural" enough the direction will be obvious, you will have the direction in the general case. Just watch out, sometimes two constructions will yield the same things in specific examples so you want to have examples that distinguish between the various notions. An example of this is that finite products and finite co-products have a habit of being the same in categories in which I search for examples (additive categories like categories of representations).

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Adding to Grétar's answer, here are some of mine: products and coproducts(=disjoint union) in Sets, homology is covariant (as are homotopy functors) but cohomology is contravariant, pullbacks and pushforwards I think of as "fibered products" and "fibered coproducts"...the latter of which "is" tensor product. As for universal properties, they're just "this is the smallest/largest thing that does X". – Charles Siegel Feb 11 '10 at 20:39

Aside from having a good clear example in mind: Cartesian products for products, disjoint unions for coproducts, pullback bundles for pullbacks,... one way I've found useful to remember directions of arrows involving universal objects is that arrows should factor through the universal object.

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For most universal properties the universal object is the smallest one which satisfies a certain property. For instance the product is the "smallest object" which allows you to complete its diagram, so anything which also satisfies it will project onto the product (this requires remembering the maps for the product are projections and not injections). Likewise the coproduct is supposed to be the "smallest object" allowing you to complete its diagram, so there should be an injection from the coproduct to anything else. The intuition for them being small is because the maps in the universal property are unique.

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I had the same difficulties. I remember the directions of the arrows in the case of a product (because there is just one direction that makes sense for general products) and the fact that a product is a limit. The arrows in all other limits go in the same direction, the arrows in all colimits go in the other directions.

All you have to remember now is whether a object is a limit or a colimit. In many cases this is clear from the name: If there is a X and a co-X, then most likely the co-X thing will be the colimit.

On the other hand: It isn't really necessary to remember the arrow-directions in full detail. In most examples of basic category theory the arrow directions are clear from the application and even if they are not: It doesn't matter for the proof since most elementary category-proofs work the same way for Xs and co-Xs.

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Johannes: should "limes" and "colimes" be "limit" and "colimit"? – Tom Leinster Feb 11 '10 at 17:02
You're right. The german word is "(ko)limes" and so... – Johannes Hahn Feb 11 '10 at 17:36

I like to think of such properties in the forms they take for sets. For instance, I remember being rather surprised that the fibered product was actually a rather familiar thing (ordered pairs that lie over the same point in the base) after I read the section on it in Hartshorne, which just talked about the universal property, Yoneda's lemma means that you can restrict yourself to the case of Sets in proving results about them (Grothendieck spends some time on this in EGA 0 and gives plenty of detail).

For co things, though, you have to hom out of them. A map out of a quotient A/B is the same thing as a map out of A that vanishes on B. So, more generally, maps out of co things satisfy nice properties.

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