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    • CommentRowNumber1.
    • CommentAuthorsure
    • CommentTimeSep 12th 2014
    As far as I understand, a category is morally speaking a special kind of graph (say associative unital). Structures are therefore totally encoded into the shape of the graphs of the category (ie of relations), and the nature of objects and morphisms are totally irrelevant. By asking that a functor preserves the identities, the latter can be considered to have an "absolute" (ie non relational) nature (or say, globally defined to the whole category).
    Yet, if one wants to have a purely relational approach to algebra, it is legitimate to try to remove as much as possible "absolute" meaning to objects studied, and only to characterize them by how they interact with their surrounding. In this respect, being an identity of some object C is relative to the whole category and is not a "true" essence of the morphism : it is a global property.

    Now, a functor F: J -> D being a diagram of shape J in D, shouldn't it be legitimate to only ask that F(j°k) = F(j)°F(k) ? That is to say, to translate only the shape of the diagram into D and as a result to only preserve the relations between the mapped morphisms ? Therefore the functor would be a "local" injection of J into D without caring about where it starts ? (ie, not at the level of the identity which is defined globally).

    An identity with respect to some "collection" of morphisms is just an idempotent element that acts as a unit on this collection. One could even want to propose an alternative definition of category (without unit necessarily), as an associative graph but only to impose the presence of some idempotent element for every object. Such idempotent element could be either a unit (if is absorbed by everyone globally) or an absorbing element (it absorbs all endomorphisms, so it absorbs everyone locally).

    Are there people studying that ? Would it be a good idea to study that and why ?

    PS : note that it is in my opinion bad that algebra forgot to study stuctures with absorbing elements. One can easily reinterprete many structures (semi-rings, so rings and fields, ...) as special case of "annulus" (M,+,.,0,i), where
    1) (M,+,0) is a commutative monoid,
    2) (M,.,i) is such that . is associative and i is absorbing
    3) . is distributive over +
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2014

    check out semicategory

    • CommentRowNumber3.
    • CommentAuthorFosco
    • CommentTimeSep 13th 2014

    Check also out the theory of plots (partial margmas with more than one object); see also the comment on this math.SE thread:

    You can define “isomorphisms” (yes, without identities), and notice that “being an isomorphism” and “admitting an inverse” are different notions in this world, and that they collapse in the categorical world (a category is an associative plot, where the composition is defined and every object has a 1, in the same vein a monoid is an “extremely smooth partial magma”). You can then define isoids, i.e. plots where every arrow is an isomorphism.

    We’re even able to define morphisms of plots (p-unctors), natural transformations (trimmings, if I remember well the name Salvatore and I chose), adjoints, limits, and a chain of free-forgetful adjunctions which connects the category (it is a category) of plots to the category of associative plots, semicategories [in this case we’ve even two different adjunctions for two different fully faithful embeddings], and categories. […] Functional analysis and symplectic geometry provide “natural factories” of examples of such structures: one of our two unitization functors applied to the category of symplectic relations gives precisely the Woodward-Wehrheim category.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeSep 15th 2014
    • (edited Sep 15th 2014)

    The morphisms in a category are not so much like the edges in a graph as like the paths in a graph. Edges do not compose, but paths do. Edges are all alike, but paths are not. In particular, some paths are special in a way that identifies them as identity morphisms: the paths of length zero. In particular, an identity morphism is not a loop (and edge from a vertex to itself, which a path of length 1, not 0.) So in identifying the identity morphisms, we do not pick out one loop from all of the loops at a given vertex, label it the identity, and then demand that this labelling be preserved. Instead, we demand that a path of length 0 be mapped to a path of length 0, which is very different.

    This is a moral argument, not a logical one. (Urs's and Fosco's answers are also legitimate, after all.) If you define a strict category as a graph with extra structure, then (besides defining composition) you do indeed pick out one loop from all of the loops at a given vertex. But that is missing the point. Turning a graph into a category is not putting some arbitrary structure on the graph; there is a reason for considering this species of structure rather than another (such as the structure of a mere semicategory). And the reason is that we are rethinking the edges of the graph that we started with as paths in some other (perhaps nonexistent) graph. And so each vertex needs some loop to be rethought of as (not a loop at all but) the path of length zero. And this must be preserved because a path length of zero is preserved, not because some arbitrary labelling is arbitrarily preserved.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2014

    This moral argument may indeed be just what settles the original question (we’ll see) and I entirely agree with the moral.

    But just for completeness maybe one should mention that the distinction between edges in a graph and paths of edges is not so clear-cut. After all, the free-forgetful adjunction between directed graphs and categories, while it sends a graph to its path category, yes, sends a category to the graph whose edges are the morphisms of the category.

    (Of course Toby knows this exceedingly well, I am just mentioning it for completeness in the general discussion.)

    • CommentRowNumber6.
    • CommentAuthorsure
    • CommentTimeSep 15th 2014
    • (edited Sep 15th 2014)
    Right, I totally understand the argument about the length of the path, but the point is that this length is relative to the collection of morphisms you're considering (and fortunately so, this is really the spirit of category theory to care only about how objects and morphisms are interacting with their surrounding, without caring about their hypothetical "true" essence).
    Indeed, suppose I have some category C. Call Id_X the identity at X. Such arrow is a path of length 0 regarding to the collection of morphism C1. I'm always free to add a trivial arrow at X, say Triv_X such that Triv_X is absorbed by everyone. Now, it is Triv_X that is a path of length 0 for "C1 U {Triv_X}". Surely, the category I'm considering now is not the same as the one I started from (can it be equivalent in some case ?), but this shows that being an identity is relative to some collection of morphism, and that the length of an arrow is not absolute.

    With this in mind, for a functor F : J -> C, how is that irrelevant to ask that F sends the identities of J to the identities of Im(F) instead of the identities of C ?

    edit : Let Max be the category with one object and a sequence (f_n)_n of morphisms such that f_n ° f_m = f_n iff n >= m, and f_m °f_n = f_n iff n >= m.
    Let f_0 be the identity (one can interprete the f_m as the integer m and ° as the max function). It is clear that ° is associative, and let Max' be the same category in which we added a morphism f_{-1} that is now the new identity.

    The functor F from Max' to Max that sends f_m to f_{m+1} being essentially surjective, we have an equivalence of categories. It is trivial to see that a functor that would send f_m to f_{n+m} with n > 1 would also preserve the structure. Why not calling that an equivalence of categories too, seriously ? If not an equivalence, an "injection of category" ?