As is well-known, a faithful functor need not be injective on objects. What are some good examples to illustrate this point?
$\begingroup$
$\endgroup$
4
-
2$\begingroup$ The forgetful functor from any category of modules over some algebra to the category of vector space is an obvious example, just pick two modules which are isomorphic as vector spaces but not as modules. $\endgroup$– AdrienMar 16, 2020 at 12:10
-
1$\begingroup$ Pick a monoid M as one-object category, call it B. Take the disjoint union of any number of copies of M and call it A. There is an obvious functor A -> B that is fully-faithful and not injective on objects. $\endgroup$– G. RodriguesMar 16, 2020 at 12:30
-
10$\begingroup$ "Any" forgetful functor from a category of algebras to $\mathbf{Set}$. The forgetful functor from topological spaces to $\mathbf{Set}$. In fact, most things that are called forgetful functors are faithful, and they're very rarely injective on objects $\endgroup$– Maxime RamziMar 16, 2020 at 12:52
-
$\begingroup$ Generalizing the comment by @G.Rodrigues , the equivalence from any category to its skeleton (if the category has at least two isomorphic objects). $\endgroup$– Kevin CastoMar 17, 2020 at 22:45
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
5
From Categories, Allegories (known as Cats and Alligators) by Peter Freyd and Andre Scedrov:
(1.31) A functor $F:{\mathbf A}\to{\mathbf B}$ is an EMBEDDING if $(A,B)\to F(A,B)$ is one-to-one for all A, B.
(1.33) A functor $F$ is a FAITHFUL if it is an embedding and reflects isomorphisms.
(1.333) A functor between posets is always an embedding. It is faithful iff it is one-to-one on objects.
For myself, I am not so keen on this usage, but the last sentence is a cogent argument for it.
For categories rather than posets, being one-to-one on objects is a mistaken idea, since one should never consider equality of objects anyway. The property of reflecting isomorphisms captures the categorical property that is really intended by this.
I am not myself especially keen on this Freyd--Scedrov usage, although what they say should be considered more carefully than Wikipedia. For further discussion and examples, see Section 4.4 of my book Practical Foundations of Mathematics (CUP 1999).
answered Mar 16, 2020 at 15:50
-
2$\begingroup$ I think this terminology isn't at use anymore - at least I have never seen anyone use "faithful" in that sense $\endgroup$ Mar 16, 2020 at 19:00
-
$\begingroup$ I'm not saying that everything that Freyd or Scedrov said must be taken without question, but it should certainly be taken seriously, and if you disagree with them you should have a good reason. $\endgroup$ Mar 16, 2020 at 19:39
-
3$\begingroup$ Well there certainly is no "proof" of what I'm claiming because it's terminology, but here's strong evidence : all these pages define faithful as only $\hom(A,B)\to \hom(F(A),F(B))$ being injective (en.wikipedia.org/wiki/Full_and_faithful_functors, ncatlab.org/nlab/show/faithful+functor, mathworld.wolfram.com/FaithfulFunctor.html, encyclopediaofmath.org/index.php/Faithful_functor, en.wiktionary.org/wiki/faithful_functor, stacks.math.columbia.edu/tag/0013, unapologetic.wordpress.com/2007/06/05/…) $\endgroup$ Mar 16, 2020 at 20:42
-
$\begingroup$ and I have seen no definition of faithful online that requires the functor to reflect isomorphisms (for instance, the forgetful functor $\mathbf{Top\to Set}$ is always said to be faithful) $\endgroup$ Mar 16, 2020 at 20:43
-
3$\begingroup$ Moreover, there's a (big!) difference between reflecting isomorphisms on the one hand, and on the other hand being injective on isomorphism classes. It seems to me the correct way to interpret "injective on objects" is to mean "injective on isomorphism classes of objects". For instance, any forgetful functor from a category of algebras to $Set$ reflects isomorphisms, but as noted by Max above, such functors are rarely injective on isomorphism classes of objects. $\endgroup$– Tim Campion ♦Mar 17, 2020 at 0:39