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What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are there interesting facts about this category?

I am particularly interested in the special case of the category of posets and Galois connections.

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You will have to make an arbitrary choice for the direction of morphisms: is the left adjoint "forward" or "backward"? To prevent that, you can add involutions. The resulting category $\mathbf{InvAdj}$ of involutive categories and adjunctions, that I'll define below, has a lot of interesting structure. It is a dagger category, and in fact the `mother of all dagger categories', as it universally embeds any dagger category. In particular, the full subcategory of (ortho)posets and Galois connections has dagger kernels, dagger biproducts, an an opclassifier. See these two papers. Now for the definition (from 3.1.8 of my thesis):

A functor $* \colon \mathbf{C}^\mathrm{op} \to \mathbf{C}$ is called involutive when $* \circ * = \mathrm{Id}$. Define a category $\mathbf{InvAdj}$ as follows. Objects are pairs $(\mathbf{C},*)$ of a category with an involution. A morphism $(\mathbf{C},*) \to (\mathbf{D},*)$ is functor $F\colon \mathbf{C}^\mathrm{op} \to \mathbf{D}$ that has a left adjoint, where two such functors are identified when they are naturally isomorphic. The identity morphism on $(\mathbf{C},*)$ is the functor $*\colon \mathbf{C}^\mathrm{op} \to \mathbf{C}$; its left adjoint is $*^\mathrm{op} \colon \mathbf{C} \to \mathbf{C}^\mathrm{op}$. The composition of $F\colon \mathbf{C}^\mathrm{op} \to \mathbf{D}$ and $G\colon \mathbf{D}^\mathrm{op} \to \mathbf{E}$ is defined by $G \circ *^\mathrm{op} \circ F \colon \mathbf{C}^\mathrm{op} \to \mathbf{E}$; its left adjoint is $F' \circ * \circ G'$, where $F' \dashv F$ and $G' \dashv G$.

(It is not arbitrary to require a left adjoint instead of a right one. A contravariant functor from $\mathbf{C}$ to $\mathbf{D}$ can be written both as a (covariant) functor $F\colon \mathbf{C}^\mathrm{op} \to \mathbf{D}$ or as a (covariant) functor $F^\mathrm{op}\colon \mathbf{C} \to \mathbf{D}^\mathrm{op}$. The latter version has a right adjoint precisely when the former version has a left adjoint.)

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    $\begingroup$ You wrote that "the full subcategory of (ortho)posets and Galois connections has [...] dagger biproducts". But biproducts only make sense in a category with zero object. And, in this case, such zero object would be a poset from/to which there is a Galois connection to/from any other poset. But such poset does not exist. Therefore either the result you claim is wrong or, most probably, there is a misundertanding. Please clarify. $\endgroup$ – Bob Aug 11 '14 at 20:25
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    $\begingroup$ The zero object is the single-element orthocomplemented lattice $\{0=1\}$, see p8 of the first paper mentioned. Notice morphisms are (pairs of) antitone functions. Incidentally, an opclassifier is an object 2 with a natural isomorphism between kernels (subobjects) of an object $X$ and morphisms $2 \to X$ (see Corollary 3.11). $\endgroup$ – Chris Heunen Aug 18 '14 at 11:11
  • $\begingroup$ Is the category of $\mathsf{InvAdj}$ canonically a 2-category? Seems like it should be, right? $\endgroup$ – ಠ_ಠ Feb 27 '17 at 23:49

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