Reputation
44,075
Next tag badge:
93/100 score
27/20 answers
Badges
2 53 128
Newest
 Enlightened
Impact
~368k people reached

May
18
comment Abstract interpretation of isomorphism between tensor product with dual and hom
No, you can't do it in an abelian category. You need a symmetric monoidal closed category – and in fact that's all you need to define the morphism. Showing that it is (sometimes) an isomorphism is not abstract nonsense, however.
May
18
awarded  Nice Answer
May
18
comment Abstract interpretation of isomorphism between tensor product with dual and hom
Yes, it does. Isn't that what I said? You might like to think about how to do it in, say, the category of modules over a commutative ring.
May
18
comment Abstract interpretation of isomorphism between tensor product with dual and hom
As stated, $\mathscr{F}^\vee \otimes \mathscr{G} \to \mathscr{H}om (\mathscr{F}, \mathscr{G})$ is canonical. You can describe it abstractly if you like. What does require work is showing that it is sometimes an isomorphism.
May
17
revised Compactness and directed systems of subspaces
added 7 characters in body
May
17
asked Compactness and directed systems of subspaces
May
16
accepted When is the closure of an equivalence relation automatically an equivalence relation?
May
16
comment Derived functors - homotopical vs homological approach
You can avoid using the model structure – it is enough to have acyclic resolutions. The proof is easy – as I said, you just calculate explicitly using acyclic resolutions instead of using their respective universal properties.
May
15
comment Proving that a category is cartesian closed
It would perhaps be better to divide the proof into two steps: show that your category is isomorphic to the category of presheaves on the category $\mathbb{B} \mathbb{N}$, which has a unique object and is generated by a non-trivial endomorphism, and then show that every category of presheaves on a small category is cartesian closed.
May
15
comment Derived functors - homotopical vs homological approach
The connection between universal $\delta$-functors and Quillen–Verdier derived functors is a bit subtle. I only know how to prove that they coincide when we can explicitly calculate both in terms of acyclic resolutions.
May
15
comment Right adjoint of covariant hom functor
Well, does this functor preserve colimits or not?
May
15
asked When is the closure of an equivalence relation automatically an equivalence relation?
May
14
comment In a closed monoidal category, is $[-,-]$ always a bifunctor?
It is well known and considered obvious. If you really want a citeable justification, look at Theorem 3 in [CWM, Ch. IV, §7].
May
14
comment How general $ [X,[Y,Z]] \cong [X \times Y, Z] $ is?
It is not "well known". In fact, it is false unless one interprets $\times$ and "algebraic category" in a very particular way.
May
14
comment If a composition of two maps is smooth, as well as one of the maps, then so is the other.
At minimum $g$ must be surjective: for example, we could take $M = \emptyset$, then $g$ and $f \circ g$ are vacuously smooth but $f$ can be arbitrary.
May
14
comment What is a branch point?
That is because the circle is too big and does in fact wind around $0$.
May
14
answered What is a branch point?
May
14
comment Question on the definition of a locally presentable category
Given a fully faithful functor $\mathcal{C} \to \mathbf{Psh} (\mathcal{K})$ with a left adjoint, if $\mathcal{C}$ is accessible, then the functor is also accessible. This is an easy exercise.
May
14
comment Question on the definition of a locally presentable category
You do not need to change $K$. The inclusion is automatically accessible in that case.
May
14
comment Question on the definition of a locally presentable category
They are generators but they are not necessarily presentable.