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bio website cs.bham.ac.uk/~udr
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seen Jul 18 at 20:41

Professor of Computer Science, with research interests in Programming Languages and Theory of Programming.

My profiles on Theoretical Computer Science, Computer Science, Mathematics, Stackoverflow, and Superuser.

Also on Programmers, where I am shooting for a negative rating (if such a thing is possible)!


Feb
18
comment Definition of (projective ?) limit in a presheaf category.
"Projective limit" is their terminology for what is normally called a "limit". Their "inductive limit" is what is normally called a "colimit".
Dec
7
comment Good books and lecture notes about category theory.
Eilenberg and Mac Lane's original paper: General theory of natural equivalences says that they defined "category" to define "functor", and "functor" to define "natural transformation". But I get the impression that the category theorists of today don't take that remark all that seriously.
Nov
30
comment When should we take direct limit and when should we take inverse limit?
I think a correction is warranted: The direct limit is not a subset of the union. It is a quotient of the disjoint union.
May
8
comment How to get a group from a semigroup
It seems to be that, in the Added note, all occurrences of "right adjoint" should be changed to "left adjoint".
Apr
6
comment Motivation for Tensor Product
Ok, that is nice! Do you have a reference that discusses this?
Apr
5
comment Motivation for Tensor Product
I don't think so. The collection $Hom_R(X,Y)$ of $R$-modules for a non-commutative ring $R$ does not have the structure of an $R$-module. See the paragraph 7.5 of my Notes on Semigroups for example.
Apr
4
comment Motivation for Tensor Product
What happens if $R$ is not commutative?
Apr
1
comment Homsets of group actions related to fixed points
Fantastic! Thanks very much. I was probably too exhausted to understand it last night. But, I had chat with an 'expert' this morning, who made it all clear to me :-)
Feb
7
comment Categorial definition of subsets
I think what you might need is a 10-minute exercise: Prove that, in category Set, the subobjects of $A$ (as clarified above) are one-to-one with the subsets of $A$.
Feb
7
comment Categorial definition of subsets
You cannot make a statement like $\{1,2\} \subseteq \{0,1,2\}$ categorically, let alone distinguish it from another. What you might say is that $\langle \{1,2\}, u\rangle$ is a subject of $\{0,1,2\}$ where $u$ is the injection of $\{1,2\}$ in $\{0,1,2\}$. But then neither $\langle \{0,1\},u \rangle$ nor $\langle\{1,5\},u\rangle$ is a subobject because $u$ doesn't type-check in those cases.
Feb
2
comment Unit of adjunction: Epimorphism?
I can hardly think of any cases where the unit of an adjunction is an epimorphism. Perhaps you meant to ask whether the counit of an adjunction is an epimorphism?
Sep
3
comment Universal property of tensor product
@MTurgeon, This question is not a duplicate of math.stackexchange.com/questions/184627/…. That question asked about the "other direction" (proving the first property from the second). On the other hand, the concern is here is how to produce the second (universal $A$-bilinear map) from the first.
Aug
25
comment Equality of two notions of tensor products over a commutative ring
It is not clear what you are saying. There are two classes of maps being talked about, and each of them has a "universal element" in your terminology. They have no reason to be isomorphic just by virtue of being "universal" (in their own class).
Aug
24
comment Mathematical notation for computer science
There is no single piece of notation that is used in Computer Science, and it is also unlikely that you will find it all in one place. Depending on the subject area you are looking at, you would need to find perhaps a graduate text book on the subject to get all the background you need.
Apr
17
comment Why bifunctors?
@ZhenLin. Yes, I see that now. However, note that nobody thought of calling natural transformations "binatural transformations". Occam's razor should win eventually.
Apr
17
comment Why bifunctors?
Ok, I guess that is a reasonable explanation. I was under the mistaken assumption that the morphism-bimorphism distinction is only made when there is a product and a separate tensor product, because there would then be a real need to distinguish between the two concepts. You are saying that the "bimorphism" terminology can be used whenever there is a closed symmetric monoidal structure.
Apr
17
comment Why bifunctors?
@dtldarek. Yes, that analogy would make sense. In the same way that the tensor product may be seen as an "artificial" construction to make bilinear transformations look linear, one might regard dual category as an artificial construction to make mixed variant functors look functorial. However, the "bifunctor" term is used even when there is no mixed variance.
Apr
17
comment Why bifunctors?
But my understanding is that bilinear maps are not linear maps or vice versa. So, there you do need different terms.