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Jocelyn Ireson-Paine (Feb 05 2025 at 15:59):By which I mean a program running on a remote computer, usually communicating via web protocols, which you can ask for category-theoretic calculations, and which will return the results as data structures for the requesting computer to use, rather than for a human to read. I put a script onto my online category-theory demonstrator which makes it do this. It can be seen working at www.j-paine.org/lds/sheaves1/test_catserver.html . (Don't let your browser put an https:// in the URL, or it will fail.)
On this page, the final form requests a pullback and renders it locally; other forms request sets and functions to check that these features work. This is a proof of concept, but surely would be useful as a resource for examples for stuff like teaching and diagram generation.
Morgan Rogers (he/him) (Feb 05 2025 at 17:25):Cool idea. Do you use an existing symbolic algebra engine on the back end?
Jocelyn Ireson-Paine (Feb 05 2025 at 19:05):Morgan Rogers (he/him) said:
Cool idea. Do you use an existing symbolic algebra engine on the back end?
No. I coded the algorithms in SWI-Prolog, based on Rydeheard and Burstall's Computational Category Theory. I originally used them in an experimental compiler for Joseph Goguen's sheaf semantics of concurrent interacting objects, and then repurposed them for the online demonstrator at http://www.j-paine.org/cgi-bin/webcats/webcats.php .
That's quite a big program, as you can see from the list of modules at the end. These include: a compiler that translates functional notation into predicates; sets and maps; objects and arrows; functors; code for randomly generating example sets and maps (vital to any good Truth Mine); error-checking for sets and maps as inputs; fundamental categorical constructions; another compiler for translating English text with embedded examples in into predicates that serve the text over the web; formatting for numbers, pairs, maplets, Unicode maths symbols, HTML elements and so on; colours, for colouring different components of compositions to be more intelligible; file naming for scratch files to hold GraphViz representations of diagrams, and also the images thereby generated; code for calling GraphViz itself; assorted stuff for translating URLs to filenames within the document root and back; the explanatory pages themselves, as predicates; some experimental stuff for demonstrating analogy and semiotics; more experimental stuff, unfinished, for Yoneda; a demo page I once posted to nLab; and a built-in SWI-Prolog library for converting to and from JSON. Outside Prolog, there are also some PHP scripts, the guestbook which I've still not had time to fix, a Ubuntu batch file for clearing up old images, and a .htaccess file for rewriting URLs to overcome the hosting company's server's odd ideas about MIME types.
The server I started this thread with uses most of the above except for the GraphViz stuff, server-side images, and explanatory pages, as those are rendered in the browser using HTML, CSS and JavaScript. I've got a slowly-growing collection of JavaScript files which draw images client-side in SVG for that. That's where the pullback diagram comes from.
And here's the full list of server-side SWI-Prolog modules. "Grips2" is the compiler from functional notation to predicates; wc_random generates the random sets and maps; set_arrows builds on a general object-and-arrow module to implement arrows in FiniteSet; set_specs and function_specs check sets and functions sent by the user; output and pet_runtime deal with formatting; most of the rest is categorical and obvious from its name. If I were calling a symbolic algebra engine, I'd need to implement a lot of this anyway, but I'd be interested to hear from anyone who has tried it.
:- use_module( functional(grips2) ).
:- use_module( general(errors) ).:- use_module( general(sets) ).
:- use_module( general(maps) ).:- use_module( webcats(set_arrows) ).
:- use_module( general(random) ).
:- use_module( webcats(wc_random) ).:- use_module( webcats(set_specs) ).
:- use_module( webcats(function_specs) ).:- use_module( webcats(set_prodcone) ).
:- use_module( webcats(set_coprodcone) ).
:- use_module( webcats(set_eqcone) ).
:- use_module( webcats(set_coeqcone) ).
:- use_module( webcats(set_iobj) ).
:- use_module( webcats(set_tobj) ).:- use_module( webcats(constant_map) ).
:- use_module( general(output) ).
:- use_module( pet(pet_runtime) ).
:- use_module( webcats(colours) ).
:- use_module( general(gen_timestamp) ).
:- use_module( webcats(page_bits) ).:- use_module( webcats(set_iobj_is_initial_and_universal_page) ).
:- use_module( webcats(set_tobj_is_terminal_and_universal_page) ).
:- use_module( webcats(set_prod_is_universal_page) ).
:- use_module( webcats(set_prod_is_nonunique_page) ).
:- use_module( webcats(set_prod_is_terminal_page) ).
:- use_module( webcats(set_prod_nlab_page) ).
:- use_module( webcats(set_coprod_is_universal_page) ).
:- use_module( webcats(set_eq_page) ).
:- use_module( webcats(set_coeq_page) ).
:- use_module( webcats(set_ternary_product_page) ).
:- use_module( webcats(set_ternary_coproduct_page) ).
:- use_module( webcats(set_pullback_page) ).
:- use_module( webcats(set_pushout_page) ).
:- use_module( webcats(set_limit_page) ).
:- use_module( webcats(set_colimit_page) ).
:- use_module( webcats(set_exp_is_universal_page) ).:- use_module(library(http/json)).
Morgan Rogers (he/him) (Feb 06 2025 at 17:13):Wow! That is a big program!
Morgan Rogers (he/him) (Feb 06 2025 at 17:15):My best suggestion is to encourage some students to play with it to see if it helps them find and understand examples.