Mathematics – Category Theory
Scientific paper
2011-01-06
Mathematics
Category Theory
34 pages; shortened, revised version with new terminology (e.g. "adjunction contexts" in the first version are now called "pai
Scientific paper
The well-known bilateral correspondence between the theory of monads on categories and the theory of adjoint pairs of functors between categories depends heavily on the unitality of the monads. As it turned out, for numerous applications the requirement of unitality for a monad is too restrictive and monads may be replaced by {\em demi-monads (weak monads)}. Now the question arises, which generalised form of an adjunction does correspond to these generalised types of monads. An answer is given in this paper and to this end we consider, for functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em full pairing} given by maps $$\xymatrix{\Mor_\B (L(A),B) \ar@<0.5ex>[r]^\alpha & \Mor_\A (A,R(B))\ar@<0.5ex>[l]^\beta,}$$ natural in $A\in \A$ and $B\in \B$. We call $(L,R)$ a {\em demi-adjoint pair of functors} provided $\alpha = \alpha\circ \beta\circ \alpha$ and $\beta = \beta \circ\alpha\circ\beta$ (regularity) and $\alpha$ satisfies some symmetry condition. More generally, we relate any full pairing of functors with a {\em quasi-monad}, that is, an endofunctor $F:\A\to \A$ with an associative product $FF\to F$ and a natural transformation $I_\A\to F$ with no further conditions ({\em quasi-unit}) and define the category of {\em quasi-$F$-modules}. From this configuration we derive the notion of a {\em regular quasi-monad} which becomes a {\em demi-monad} (in the sense of B\"{o}hm, Lack and Street) provided the quasi-unit satisfies a symmetry requirement.
Wisbauer Robert
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