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Dec
25
accepted What is the name of this construction of an induced order?
Dec
25
revised What is the name of this construction of an induced order?
edited title
Dec
25
asked What is the name of this construction of an induced order?
Dec
21
awarded  Popular Question
Oct
5
accepted On the identity map requirement in the definition of category
Oct
5
revised On the identity map requirement in the definition of category
deleted 9 characters in body
Oct
4
comment On the identity map requirement in the definition of category
@EricWofsey: I repeat: no one is proposing "to define isomorphisms without identities." The proposal is to discard the requirement that all objects have identities, and doing so has no effect whatsoever on the definition of an isomorphism. It remains exactly as it was before. Of course, if there is an isomorphism between two objects, then such objects must have identities, ipso facto, but this can't be the reason for requiring that every object have an identity, since there's no requirement that every object participate in at least one isomorphism.
Oct
4
revised On the identity map requirement in the definition of category
added 987 characters in body
Oct
4
comment On the identity map requirement in the definition of category
@EricWofsey: as I already pointed out in my original post, dropping from the definition of a category the requirement that every object have an identity arrow does not in any way preclude the definition of "isomorphism", "inverse morphism", etc. Of course, these definitions do depend on the concept of an "identity arrow". What seems to me gratuitous is not the concept of an identity arrow per se, but rather the requirement that every object have one.
Oct
4
comment On the identity map requirement in the definition of category
@MaliceVidrine: Maybe so, but the Yoneda lemma dates from 1954, whereas Eilenberg and Mac Lane invented category theory in the early 1940's, therefore it seems to me unlikely that they included the identity arrow requirement in order to preserve the Yoneda lemma. My understanding is that Eilenberg and Mac Lane were primarily after capturing the notion of natural transformation, which, AFAICT, does not need the identity arrow requirement at all.
Oct
4
asked On the identity map requirement in the definition of category
Oct
3
awarded  Popular Question
Sep
20
revised Six object classes | products | co-products in search of a category
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Sep
19
comment Six object classes | products | co-products in search of a category
OK, I see. The bit I missed was the part about taking "the full subcategory of the representatives." Thanks for the clarification.
Sep
19
revised Six object classes | products | co-products in search of a category
added 72 characters in body
Sep
19
comment Six object classes | products | co-products in search of a category
My confusion stems from my failure to see how one can define the desired category without specifying the functor from the original category to it. IOW, the two problems you mention seem coextensive to me. The only way out I can come up with is that maybe we know (somehow) that the skeleton category exists, even if a construction of it (i.e. the functor from the original category) cannot be easily specified. If this interpretation is correct, I'm still a bit mystified; I will have to venture into the pages you linked to to dispel my lingering confusion. Thanks for your patience!
Sep
19
accepted Six object classes | products | co-products in search of a category
Sep
19
revised Six object classes | products | co-products in search of a category
added 440 characters in body
Sep
19
comment Six object classes | products | co-products in search of a category
@KevinCarlson: Thanks. Just now I was having thoughts in that general direction.
Sep
19
revised Six object classes | products | co-products in search of a category
added 158 characters in body