Month: November 2017

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On the two definitions of adjunctions

(Note that we write compositions from left to right and accordingly apply most maps on the right.)

Def.1
A functor F:\ct A\to\ct B  is called a left adjoint to a functor U:\ct B\to\ct A  if there is a bijection between the homsets:

    \[\ct B(A^F\ig B) \simeq \ct A(A\ig B^U)\]

natural in both A and B. In this case U is called a right adjoint to F, and we write F\adj U.

Def.2
A functor F:\ct A\to\ct B is called a left adjoint to a functor U:\ct B\to\ct A if there are natural transformations \eta:1_{\ct A}\tto FU and \eps:UF\tto 1_{\ct B} satisfying the zig-zag identities:

    \[\matrix{\eta F\cdot F\eps = 1_F &\sep U\eta\cdot \eps U=1_U}\]

So, why are these two definitions equivalent?
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Morita equivalence by categorical bridges

Let \mathcal Iso denote the category of 2 objects (name them 0,1) and 2 nonidentity arrows, f:0\to 1,\ g:1\to 0 which are inverses of each other.

We define a  bridge  as a category over \mathcal Iso, i.e. a category \mathcal H equipped with a functor B:\mathcal H\to\mathcal Iso. The full subcategories \mathcal A:=B^{-1}(0) and \mathcal B:=B^{-1}(1) are called banks of the bridge, and we write \mathcal H:\mathcal A \rightleftharpoons \mathcal B.
The arrows in the B-preimage of the two arrows in \mathcal Iso are referred to as through arrows or heteromorphisms in the bridge.
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