Talk:Introduction to Category Theory/Categories

Tlep - if you think I've messed up your line of development, feel free to be ruthless :) I think a project like this needs a Benign Dictator. How do you do the illustrations such as the one for functions in the sets and function - that would help beef up the discussion of the 'categories as a generalization of monoids' point.

If you look at any wikibook on mathematics you will notice that projects like this need active participants. Most wikibooks are at most half done and no longer actively developed. I create illustrations with Inkscape, it's free, try it. Only problem is wikimedia can't render all SVG files correctly. For example arrowheads should be drawn as triangles. Tlep 08:09, 1 November 2007 (UTC)

I'm wondering about putting all of the examples in their own section, not just free categories

I've inserted various more verbiage into the monomorphism section, with the idea of making it a bit gentler for what I imagine the plausible audience to be, but doesn't the current last paragraph assume a bit more than they're likely to know, or am I underestimatig the target audience?

Yea, you are right, too much copy-pasting from wikipedia. Fixed.

I also expanded section on directed graphs, I'll probably want to define limits on all diagrams and not just functors. (exercise: prove that limit of graph is isomorphic to limit of subcategory generated by graph) Tlep 10:39, 5 November 2007 (UTC)

I added a more 'demotic' version of the original conclusion to monomorphisms. Will be on the road for a week so contributions might be sparse

I'm wondering if the full Lawvere&Rosebrugh definitions of injection and surjection shouldn't at least be foreshadowed, although probbly not introduced here, since this lesson is getting to be a bit on the long side, and I think one problem of learning CT is the very large amount of basically trivial terminology and detail that can be piled on at the beginning, without getting used to actually do much of anything. AveryAndrews 20:15, 10 November 2007 (UTC)

I think it's confusing to use word 'element' for arrows. Many categories have sets as objects and the correspondence between set elements and category 'elements' isn't simple (elements of a ring R correspond to arrows $$\mathbb{Q} \to R$$). Arrows $$S \to X$$ really should be call 'S-arrements' of X or something. Not here, but maybe some later lesson. Tlep 16:34, 11 November 2007 (UTC)

As a complete beginner, I couldn't make any sense out of the notion of generalized elements,so they should definitely not be mentioned at this point, but elements-as-arrows-from 1 seemed pretty straightforward. Understanding the equivalence is also good drill in basic set theory. However, I think this lesson is probably getting to be too long, so should perhaps be rethought & reorganized AveryAndrews 22:10, 18 November 2007 (UTC)

The section on the free category on a directed graph is a bit confusing. I found this definition to be very useful which i found on wikipedia: "Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph. Composition of morphisms is concatenation of paths. This is called the free category generated by the graph." I didnt want to edit the page, since I am still learning category theory and dont want to disseminate false information :) 137.158.152.206 23:00, 26 May 2008 (UTC)