What do you call it when you have one (small) category being a (full) subcategory of another , and every object in the big category is isomorphic to one in the small category ? This is the case for the category given by objects hom(S,A) ,and morphisms given by the equivalence relation hom(T,A) ,as a subcategory of stack(A) . Is there an equivalence of categories ? jim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Theorem 1 of IV.4 in Mac Lane's _Categories_for_the_Working_Mathematician_ shows in particular that for a functor S:A-->C, the following are equivalent: (i) S is an equivalence of categories, (iii) S is full and faithful, and each object c \in C is isomorphic to Sa for some object a \in A. The proof of the implication (iii)->(i) appears to depend, in general, upon an analogue of the axiom of choice which is applied with respect to functions between classes. We say that S is essentially surjective on objects (e.s.o) if each object c \in C is isomorphic to Sa for some object a \in A. Thus, any full subcategory inclusion which is essentially surjective on objects is an equivalence of categories, and, in particular, such an inclusion is not only an equivalence but is also injective on objects. If a full subcategory inclusion A --> C is e.s.o. and, moreover, each isomorphism class of A is a singleton, then we say that A is a skeleton of C. Cheers, Rory Lucyshyn-Wright P.s. Perhaps someone on the list could provide a history of the different formulations of equivalence of categories and the associated terminology, with appropriate references. On Sun, 20 Sep 2009, jim stasheff wrote:
What do you call it when you have one (small) category being a (full) subcategory of another , and every object in the big category is isomorphic to one in the small category ? This is the case for the category given by objects hom(S,A) ,and morphisms given by the equivalence relation hom(T,A) ,as a subcategory of stack(A) . Is there an equivalence of categories ?
jim
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Jim: I don't understand your context precisely. (I heard a talk the other day where the speaker used "category" to mean "A_{infinity}-category" without any explanation.) However I can tell a story which uses some of the words you have. Without the axiom of choice (such as in a topos), there are two different conditions on a functor f : A --> X for it to be an "equivalence": 1) there is a functor g : X --> A such that f g and g f are isomorphic to identity functors; and, 2) f is full, faithful and essentially surjective on objects (this last means each object of X is isomorphic to a value of f). Clearly 1) implies 2). The converse holds when epis split (Ax Choice) in the ambient world. Stacks are designed not to see the difference between equivalences of types 1) and 2); that is, if you hom out of an equivalence of type 2) into a stack [for an appropriate topology], you get an equivalence of type 1). See old papers of Paré, Bunge, Joyal, . . . Ross On 20/09/2009, at 11:21 PM, jim stasheff wrote:
What do you call it when you have one (small) category being a (full) subcategory of another , and every object in the big category is isomorphic to one in the small category ? This is the case for the category given by objects hom(S,A) ,and morphisms given by the equivalence relation hom(T,A) ,as a subcategory of stack(A) . Is there an equivalence of categories ?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Jim Stasheff asked,
What do you call it when you have one (small) category being a (full) subcategory of another, and every object in the big category is isomorphic to one in the small category ? ...
One adjective that *had* been used for such a subcategory (whether small, or full, or not) was "replete". I'll defer to others on the question of whether that terminology is still in use today, or is ... um ... *deprecated* :-) . Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Sunday 20 September 2009 14:21:13 jim stasheff wrote:
What do you call it when you have one (small) category being a (full) subcategory of another , and every object in the big category is isomorphic to one in the small category ? This is the case for the category given by objects hom(S,A) ,and morphisms given by the equivalence relation hom(T,A) ,as a subcategory of stack(A) .
In Adámek, Herrlich and Strecker's book "The Joy of Cats", the small category is said to be an "isomorphism-dense" subcategory of the big category. I don't know how widespread this terminology is, though.
Is there an equivalence of categories ?
Yes. Whenever A is a full, isomorphism-dense subcategory of B, then the inclusion functor from A to B is an equivalence (Remark 4.10 in that book). -- Robin Adams <robin@cs.rhul.ac.uk> Royal Holloway, University of London [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
My understanding of this ancient terminology ids that a replete subcategory is one that is closed under the forming of isomorphic copies.A subcategory which contains an isomorphic copy of every object in the containing category is called skeletal A subcategory ois both replete and skeletal if and only if it contains all objects of the larger category. ---John On 9/22/09 3:04 AM, "Fred Linton" <flinton@wesleyan.edu> wrote: Jim Stasheff asked,
What do you call it when you have one (small) category being a (full) subcategory of another, and every object in the big category is isomorphic to one in the small category ? ...
One adjective that *had* been used for such a subcategory (whether small, or full, or not) was "replete". I'll defer to others on the question of whether that terminology is still in use today, or is ... um ... *deprecated* :-) . Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I agree with John about the accepted meaning of "replete", but my understanding of "skeletal" (supported by Mac Lane's "Categories for the Working Mathematician") is that it means a category in which every isomorphism class of objects has exactly one member. If I had to find a word for a subcategory which meets every isomorphism class of objects of the ambient category (but possibly in more than one member) I'd call it "representative", or something like that. A skeleton would then be a subcategory which is full, representative and skeletal. Regarding the question of whether such a subcategory is equivalent to the ambient category, I recall that Peter Freyd once showed that each of the following statements is equivalent to the axiom of choice: (a) Every small category has a skeleton. (b) A small category is equivalent to any of its skeletons. (c) Any two skeletons of a given small category are isomorphic. The first equivalence is trivial, but the other two require a bit of ingenuity. I don't think he ever published this. Peter Johnstone On Tue, 22 Sep 2009, John Kennison wrote:
My understanding of this ancient terminology ids that a replete subcategory= is one that is closed under the forming of isomorphic copies.A subcategory= which contains an isomorphic copy of every object in the containing catego= ry is called skeletal A subcategory ois both replete and skeletal if and only if it contains all = objects of the larger category.
---John
On 9/22/09 3:04 AM, "Fred Linton" <flinton@wesleyan.edu> wrote:
Jim Stasheff asked,
What do you call it when you have one (small) category being a (full) subcategory of another, and every object in the big category is isomorphic to one in the small category ? ...
One adjective that *had* been used for such a subcategory (whether small, or full, or not) was "replete". I'll defer to others on the question of whether that terminology is still in use today, or is ... um ... *deprecated* :-) .
Cheers, -- Fred
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Yes, now that you mention it, I agree that this is how the word "skeletal" is used. John ________________________________________ From: Prof. Peter Johnstone [P.T.Johnstone@dpmms.cam.ac.uk] Sent: Wednesday, September 23, 2009 6:00 AM To: John Kennison Cc: Fred Linton; categories@mta.ca Subject: Re: categories: Re: question I agree with John about the accepted meaning of "replete", but my understanding of "skeletal" (supported by Mac Lane's "Categories for the Working Mathematician") is that it means a category in which every isomorphism class of objects has exactly one member. If I had to find a word for a subcategory which meets every isomorphism class of objects of the ambient category (but possibly in more than one member) I'd call it "representative", or something like that. A skeleton would then be a subcategory which is full, representative and skeletal. Regarding the question of whether such a subcategory is equivalent to the ambient category, I recall that Peter Freyd once showed that each of the following statements is equivalent to the axiom of choice: (a) Every small category has a skeleton. (b) A small category is equivalent to any of its skeletons. (c) Any two skeletons of a given small category are isomorphic. The first equivalence is trivial, but the other two require a bit of ingenuity. I don't think he ever published this. Peter Johnstone On Tue, 22 Sep 2009, John Kennison wrote:
My understanding of this ancient terminology ids that a replete subcategory= is one that is closed under the forming of isomorphic copies.A subcategory= which contains an isomorphic copy of every object in the containing catego= ry is called skeletal A subcategory ois both replete and skeletal if and only if it contains all = objects of the larger category.
---John
On 9/22/09 3:04 AM, "Fred Linton" <flinton@wesleyan.edu> wrote:
Jim Stasheff asked,
What do you call it when you have one (small) category being a (full) subcategory of another, and every object in the big category is isomorphic to one in the small category ? ...
One adjective that *had* been used for such a subcategory (whether small, or full, or not) was "replete". I'll defer to others on the question of whether that terminology is still in use today, or is ... um ... *deprecated* :-) .
Cheers, -- Fred
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (7)
-
Fred Linton -
jim stasheff -
John Kennison -
Prof. Peter Johnstone -
Robin Adams -
Rory Lucyshyn-Wright -
Ross Street