Dear Mamuka, I am also puzzled with your examples of duality. I would like to speculate on some aspects of it. The category of finite dimensional vector spaces is the archtype of a self-dual category. In a sense, vector spaces are showing up in geometry as tangent spaces of smooth manifolds. But the tangent space of a manifold at a point is the "infinitesimal" structure of that maniifold at that point. It seems that linear structures are showing up naturally as infinitesimal structures. Stable categories are showing up as infinitesimal structure of higher toposes: https://ncatlab.org/nlab/show/tangent+%28infinity%2C1%29-category Of course, linear structures may also be obtained by other means. -André ________________________________________ From: Mamuka Jibladze [jib@rmi.ge] Sent: Saturday, September 16, 2017 12:35 PM To: Alexander Kurz Cc: categories@mta.ca Subject: categories: Re: the dual category Alexander's example reminded me of something I always wanted to ask somebody and never did, since it always felt too vague to me. But now I thought - just ask. In at least five very different contexts that I know, one seeks for a nice placement of some very non-self-dual category against the background of another one, "less non-self-dual". In order of my increasing ignorance, these are: Presenting spaces/locales/frames as certain (co/)monoids in the category of sup-lattices, which is as nicely self-dual as it ever gets. Extending the duality between discrete and compact abelian groups to the self-dual category of locally compact abelian groups. There are several closely related similar dualities, like e. g. the duality for (locally?) linearly compact vector spaces by, I believe, Lefschetz. In fact I think working with Banach or Hilbert spaces is largely motivated by the desire to force infinite-dimensional vector spaces to behave more like finite-dimensional ones, which form some of the nicest self-dual categories. Passing from (unstable) to stable homotopy theory is in a sense forcing some amount of self-duality. The main feature of stable categories is that they are additive (i. e. finite coproducts are isomorphic to the corresponding products) but also much more - e. g. most of homotopy cartesian or cocartesian squares in such categories turn out to be homotopy bicartesian; this in particular implies the crucial feature that the adjunction between suspension and loop space functors becomes an equivalence (in a homotopy bicartesian square like A -> 0 | | V V 0 -> B A is (stably equivalent to) the loop space of B iff B is (stably equivalent to) the suspension of A; more generally, in a similar square A -> 0 | | V V X -> B A is the fibre of X -> B iff B is the cofibre of A -> X, etc.) The context mentioned by Alexander, which triggered this post in the first place - the phenomenon called limit-colimit coincidence: it seems that imposing on some categories certain constructivity constraints coming from computer science tends to imply certain amount of self-dual features. Like, initial algebras for endofunctors become forced to become isomorphic with final coalgebras for the same endofunctors. Or, similarly, left adjoints to some functors to become isomorphic to right adjoints to the same functors. In physics, it seems that the main motivation of various quantization procedures is to achieve certain amount of self-duality. For example, evolution of a physical system becomes time-reversible. It seems like in many cases such "self-dualization" can be formulated in terms of forcing certain objects in certain monoidal categories to become invertible but I don't know enough to tell more about it. In any case I am aware of several works by category theorists which provide appropriate formalism for such and similar constructions; the most general formalism that I know is probably the Chu construction. But, if I am not overlooking something obvious, I have only seen explanations of *how* to "increase self-dual features", not *why* do these self-dualization phenomena tend to occur in so many disparate contexts. Does anybody know any underlying *reasons*? Can this phenomenon be explained by the mere fact that "linearizing" the problem makes life easier at the expense of losing certain amount of information, or there actually exist some deeply rooted principles that force self-dual behavior in certain mathematical or physical circumstances? Sorry for this very vague and long post, but I am really eager to learn about opinions of the category-theoretic community about this question that I hardly ever managed to even formulate. Mamuka [For admin and other information see: http://www.mta.ca/~cat-dist/ ]