On Feb 23, 2010, at 4:43 PM, peasthope@shaw.ca wrote:

When S is a set, the notation "a \epsilon S" is familiar. Is this ever extended to CT? All the texts I recall use natural language such as "A is an object of C". What if a more symbolic notation is required?

I've seen $a \in Ob(C)$ numerous times, and also - though primarily from Barr & Wells - $a \in C_0$, with the rationale that a category is a graph (consisting of vertices C_0 and edges C_1), with extra conditions introduced to capture the composition operation, showing up as functions defined on composable sequences C_n of n edges (most often for n=2, or 3 for associativity). -- Mikael Vejdemo Johansson [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

\epsilon is often used informally to mean an object or an arrow is contained in a category C. I.e. a \in C is shorthand for a \in ob(C) or a \in ar(C), when its interpretation is obvious from context. Barr and Wells (Toposes, Triples and Theories, 1984) uses set inclusion notation in an interesting way, thinking of arrows rather as "generalised elements" of objections. So, they'll write something like f \in^A B instead of f : A -> B, which should be read "f is an A-element of B". When the category is sets and A = {*}, this is the usual notion of element. This notion gives the right kind of intuition when working and categories that are a lot like sets, especially toposes. The language also admits a particularly beautiful way to express the Yoneda lemma. "The (normal) elements of FA are the same as the hom(-,A)-elements of F." a On Wed, Feb 24, 2010 at 12:43 AM, <peasthope@shaw.ca> wrote:

When S is a set, the notation "a \epsilon S" is familiar. Is this ever extended to CT? All the texts I recall use natural language such as "A is an object of C". What if a more symbolic notation is required?

Thanks, ... Peter E.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Besides Peter Easthope's question on the appropriateness of considering an object of C as an element of C there is also (in fact more generally as I'll mention near the end) the question of whether the notion of "element" is well established in a presheaf category C = Set^{J^op} on J, made a topos by choice of a final object 1 and a subobject classifier O, and (by Yoneda) with the further choice of a full embedding of J in C. With that arrangement, and with the idea that distinct provenances for sets give rise to distinct notions of "element," it seems to me that at least four natural kinds of set arise in a presheaf topos in ordinary mathematical practice. First kind. As a homset from 1. This notion makes the connection with set theory that toposes were developed for, with the caveat that it should be understood in the light of the third and fourth kinds so as not to overstate its applicability. Second kind. As a morphism to O. This represents a subobject of the domain of the morphism (the subobject itself being in general a proper class and therefore not a fit entity for ordinary mathematics). For example in the topos Set with natural numbers object N it is natural to write {2,3,5,7} for the set of 1-digit primes, understood as (the subobject of N represented by) its characteristic function as a morphism from N to O. Third kind. As a homset to O. This is the power *set* C(X,O) of subobjects of the domain X of the homset, which by the cartesian closed structure of C is in a natural bijection with the power *object* O^X when considered as a set C(1,O^X) of the first kind. Fourth kind. As a homset from the image Y(j) under the embedding Y: J --> C of some object j of J. For example if J has objects V and E making each object of the topos a graph G then we think of G as formed from two sets, and refer to the morphisms to G from Y(V) and Y(E) as respectively the vertices and edges of G. In this example the set of vertices of G also happens to be a set of the first kind, but not the set of edges, pointing up the need I mentioned earlier not to overstate the significance of sets of the first kind while also addressing Peter's original question in a roundabout way in terms of the underlying graph of a category. A morphism of a presheaf category is properly understood as an ob(J)-indexed family of functions mapping sets of the fourth kind to their counterparts in the codomain of the morphism. Among these the monics as families of injections furnish the notion of subobject in a presheaf topos with its intuitive meaning complementary to that of the second kind of set, the remark about proper classes notwithstanding. Of course any homset of a topos is a set, but it seems to me that the above four kinds deserve special recognition as sets commonly encountered in mathematical practice having readily distinguishable provenances. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]