# What exactly is duality?

In general, I am familiar with this notion of duality (i.e. in category theory, a statement is dualized simply by "reversing all arrows" and leaving objects unchanged). There are a couple of questions I would like to ask:

• I do not understand why the notion of fibration and cofibration are dual to one another. What I don't get in this explanation (which is also the argument given in the lecture where the problem arose) is why $Y \times I$ is substituted with Hom$(I,Y)$ when passing to the dual statement. Aren't the objects to be left alone, and only the arrows reversed? If not, what is the exact, general, correspondence between objects and their "duals"?

• In connection with the last sentence of the question above: one commonly refers to the dual of a $\mathbb{K}-$vector space $V$ as the vector space of linear maps $V \to \mathbb{K}$. Is this a special case of some "dualizing" correspondence between objects (which can be described for a sufficiently general class of categories, for example as a functor $\mathcal{C} \to \mathcal{C}^{op}$, or $\mathcal{C} \to \mathcal{C}$) or is it just an intuitive way of referring to that particular construction, which shouldn't be taken as a broad statement? Edit as seen in this page, the construction of dual vector space is somewhat general in a categorical sense. However I still fail to see a general, formal connection between this notion of "dual object" and the notion of dual property (and in particular in the case of (co)fibrations), except for the fact that in both cases "arrows are reversed".

Some answers and comments have already been helpful, but I stil fail to see the big picture! Thanks for any further help.

• Duality is whatever a mathematician would look at and think "Yep, that's a duality". Not a very satisfying description, but probably accurate in practice. If you want some odd food for thought, think about the dual vector space and how it relates to inner products. Now, think about $\mathcal{C}^\circ$, $\mathbf{Set}^{\mathcal{C}}$ and $\hom$ viewed as a functor. – Hurkyl Nov 9 '13 at 22:49
• @Hurkyl I feel like I get the intuitive idea of duality. What I don't get is whether this idea can (and is conventionally) formalized or not. I was ok with the intuitive definition until the example of the fibration-cofibration duality surprized me, and now I'm not really sure what to think! – Emilio Ferrucci Nov 10 '13 at 2:56

## 1 Answer

For your first question duality in homotopy theory, it is precisely an example of categorical duality. A homotopy between $f,g:A\to B$ is a continuous function $H:A\wedge I\to B$ where $A\wedge I$ can be any cylinder object for $A$. It is common to take $A\times [0,1]$ as a cylinder object, but that is not required. What is important is that the cylinder object has some properties (look up 'cylinder object' to see what they are, I suggest the nlab). Now the dual notion is that of a path object $A^I$. Its definition is precisely the categorical dual of a cylinder object. Again, it is common to take $A^{[0,1]}$ but it is not required. Now the dual notion of a homotopy becomes a continuous function $H:A\to B^I$. It turns out that for ordinary homotopy these two notions of homotopy are equivalent.

As for the dual of a vector space, the object $\mathbb K$ in the category $K-Vect$ of finite dimensional vector spaces is a particular case of what is called a dualizing object. The duality here is a little different than just categorical duality. Categorical duality is just a (very powerful) tautology. Dualizing objects are different. Again, I suggest the nlab for further reading.

• Thanks for your answer. I looked up this page on nlab, which explains how dual vector spaces can be seen as a special case of a general notion of "dual object". As for the first part of your answer, I still don't understand if there is a universal, "algorithmic" way of dualizing a certain category-theorical notion or not. If I had been asked to dualize the notion of fibration, I would have just left the product "Y \times I" in. What is the general rule which explains why this is incorrect? – Emilio Ferrucci Nov 10 '13 at 2:43
• And also, is there no connection between the categorical notion of duality, and the dualization of an object? For example, take a statement about vector spaces (for example that of being a kernel of a certain linear map): when I dualize this notion (cokernel) all I have to do is reverse arrows, I don't exchange all vector spaces in the diagram with their duals (this would probably lead to something else). So what is the general rule whereby direct products are to be considered as cylinder objects? I still don't see the big picture! Thanks for any further help :) – Emilio Ferrucci Nov 10 '13 at 2:49
• I suggest you read any introductory text on model categories. – Ittay Weiss Nov 10 '13 at 2:56
• A homotopy between $f,g:A\to B$ is a continuous function $H$ from a cylinder object of the domain to the codomain such that blah blah. So, dualize that and you'll say that the dual is a continuous function $H$ from the domain to a path object of the codomain such that coblah coblah. – Ittay Weiss Nov 10 '13 at 5:29