Mike, On 18/11/13 20:40, Michael Shulman wrote:
Thanks again to everyone who replied with examples and comments. I've created an nLab page which hopefully includes everything I learned: http://ncatlab.org/nlab/show/continuous+algebra
Some more examples and comments follow. In about 1996, I considered the special case in which the 2-categorical structure is a poset-enrichment, because I had in mind applications to topological spaces and locales wrt the specialization order. In that case, your Theorem 1 in the nLab has an additional equivalent condition, when T is a Kock-Zoberlein monad: (4) The underlying object of the algebra A is injective wrt to T-embeddings, where a T-embedding is a map f:X->Y such that Tf has a reflective left adjoint. (I deemly recall that Jonathon Funk may have mentioned to me in 2004 in Cambridge during a PSSL in a pub conversation that (4) holds for the general 2-categorical case, but he would have to confirm or reject that, as I can't find references after a quick search.) This was in the paper (i) Properly injective spaces and function spaces, Topology and its Applications, v89, n1-2, pp75-120, 1998, as Theorem 4.2.2. In this paper and a series of other papers following that, a number of examples are given in the categories of topological spaces and of locales, namely (ii) Injective Spaces via the Filter Monad, Topology Proceedings, 1997. (iii) Semantic domains, injective spaces and monads, ENTCS, 1999, with R. Flagg. (iv) Injective locales over perfect embeddings and algebras of the upper powerlocale monad. Applied General Topology, volume 4, number 1, pp. 193-200, 2003. Using the theorem with conditions (1)-(4), one can unify Alan Day's characterization of the algebras of the filter monad as the continuous lattices with Dana Scott's characterization of the injective T0 topological spaces as the continuous lattices under the Scott topology. This is in paper (ii), which also considers the densely injective topological characterized as the continuous Scott domains under the Scott topology (again Scott's result), using the proper filter monad. Then there are many other examples of classes of filters one may consider, to get new and previously known injectivity results using the theorem with the added condition (4). These examples are in the paper (iii) with Flagg: The stably compact spaces are the injective spaces with respect to flat embeddings (as defined in Johnstone's book Stone spaces). Here one uses the monad of prime filters, and Harold Simmons characterizations of their algebras as the stably compact spaces (under another name) in his paper "A couple of triples". The injectives over completely flat embeddings are the sober spaces (use the monad of completely prime filters). The injectives over perfect embeddings in the category of core-compact spaces are the continuous meet-semilattices under the Scott topology (use the monad of Scott open filters). The injective spaces over locally dense embeddings in the category of locally connected T0 spaces are the L-domains under the Scott topology (use connected open filters). The injectives over open embeddings are the spaces with a least element in the specialization order. The injectives over closed embeddings are the spaces with a largest, isolated point in the specialization order. The injectives over semi-open embeddings are algebras of the lower hyperspace. In the paper (iv), I considered locales. The injectives over perfect embeddings (=embeddings such that the right adjoint of the frame map preserves directed joins) are the algebras of the upper powerlocale monad (perfect + Frobenius = proper). They can also be characterized by a sort of "continuity" condition. Define U<V for opens of a locale to mean that every Scott closed collection of opens that covers V has U as a member (Scott closed = lower set closed under directed joins). Then a locale is injective/an algebra iff every open V is the join of the opens U<V and the relation < is multiplicative. In the section "Remarks" I give many more examples like this. For example, we get stably compact locales via the ideal-completion monad. Of course, this is related to the paper by Jiri Adamek, Lurdes Sousa and Jiri Velebil advertised on the 11th this month. And Dirk Hofmann did much work related to that. Best, Martin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]