I would like to have some references for Grothendieck's theory of "Topological algebra": a synthesis of homotopical and homological algebra, with special emphasis on topoi.
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$\begingroup$ The bottom of this page: pages.bangor.ac.uk/~mas010/pstacks.htm suggests that "Pursuing stacks" is a good first step. $\endgroup$– S. Carnahan ♦Commented Aug 8, 2015 at 6:14
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There are 'obvious' places to follow this up. The mentions in PS (Pursuing Stacks) are worth looking into (but skim that as you can easily get bogged down in it and weighted down by its length) and also see the 'letters to Larry Breen' from 1975 (for which look at the Grothendieck circle pages . These latter give some of the wish-list for a theory of $n$-groupoids, including the corresponding higher Galois theory (and here you could skip to 2015 and Marc Hoyois's preprint: http://arxiv.org/pdf/1506.07155v3.pdf). For the development since then the work of Lurie, and Toen and Vezzosi provide sources for some of the ideas, but I am not sure what aspect you are interested in. ('Higher algebra' by Lurie would be a good place to browse.)
Look through Ronnie Brown's page that relates to PS ( pages.bangor.ac.uk/~mas010/pstacks.htm) and follow up some of the links and look at http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/agrb_web.pdf.
There is in PS, I seem to remember, some explicit mention of 'topological algebra' in the sense you want. (Edit: In fact he mentions it early on in the 'letter to Quillen', but not after that if my search was working.)
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1$\begingroup$ For a readable version of PS: thescrivener.github.io/PursuingStacks $\endgroup$– David Roberts ♦Commented Aug 10, 2015 at 6:46