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Created extension system.
I added a couple of references from programming languages (a direct link to Moggi’s paper already referenced in monads in computer science, and another paper by Filinski which studied the relationship between strong monads-as-extension systems and continuation-passing style).
have added to the Idea-section this quote from Marmolehy-Wood
there is an important overarching reason to consider monads in this way. Extension systems allow us to completely dispense with the iterates […] of the underlying arrow. No iteration is necessary. A moment’s reflection on the various terms of terms and terms of terms of terms that occur in practical applications suggest that this alone justies the alternate approach. […] we note that extension systems in higher dimensional category theory provide an even more important simplication of monads. For even in dimension 2, some of the tamest examples are built on pseudofunctors that are difficult to iterate.
Apart from this aspect of simplification, it would be nice to have some discussion as to what intrinsic motivation one could give for the definition of extension systems. What I mean is that the traditional definition of monad is fairly “obvious” or “natural” from various points of views. For instance when regarded as monoids in endofunctor categories, they are well motivated as soon as one accepts that monoids and categories are well motivated.
Now for extension systems one would hope that there is a way to see them as an “evidently relevant” axiom scheme independently of these traditional motivations, probably more from an internal perspective. Of course to some extent this is exactly what happened with mondas (in computer science). But maybe one could give a more purely logical such motivation on this page, too.
Add hatnote pointing to closed category for Street’s notion of “extension system”.
I have adjusted and expanded the wording in the Idea-section here.
Then I have added missing publication data, authors links, document links and page-pointers to the references:
Ernest G. Manes, Sec 3, Ex. 12 (p. 32) of: Algebraic Theories, Springer (1976) [doi:10.1007/978-1-4612-9860-1]
F. Marmolejo, Richard J. Wood, Monads as extension systems – no iteration is necessary TAC 24 4 (2010) 84-113 [24-04]
In
I assume we are looking at the “devices” of chapter I? (Have only scanned it so far, not tried to unravel it yet.)
I assume we are looking at the “devices” of chapter I? (Have only scanned it so far, not tried to unravel it yet.)
Yes, that’s correct. In particular, the “full devices” are equivalent to monads (see Theorem 1.4.1).
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