21
votes
Accepted
History of the pullback corner notation
In an email to me dated 17 February 1992, Peter Freyd said:
I was using a different notation in 1974 in lectures at Montreal. A high school teacher named Butler suggested the right-angle. It was an ...
- 8,481
20
votes
Accepted
Why do elementary topoi have pullbacks?
I'll give a counter-example to the claim that having a subobject classifier and being cartesian closed implies the existence of all finite limits. However, this is based on the definition of sub-...
- 42.4k
15
votes
History of the pullback corner notation
If it is of any use, on page 251 of Taylor's "Practical Foundations of Mathematics", shortly after introducing pullbacks, he writes:
Pullbacks are often indicated with the right angle symbol, which ...
6
votes
Accepted
Homotopy pullback is right adjoint in the derived category
There is no such functor $\mathcal D/Y\to \mathcal D/X$. It's clear what is meant to be on objects, but it is not well-defined on morphisms.
Let $f$ be the inclusion of a point $p$ into a circle $C$. ...
- 55.9k
6
votes
Accepted
Tensor product of intersections in an abelian rigid monoidal category
Here's a useful lemma to check whether a square in an abelian category is a pullback square:
Lemma: A square $A_0\to (A_1,A_2)\to A_3$ with $A_0\to A_1, A_2\to A_3$ being monomorphisms is a pullback ...
- 15.8k
5
votes
Accepted
Intersection of subalgebras in an abelian monoidal category
Yes, such an algebra structure exists.
In fact, there is a unique algebra structure on $A \cap B$ such that $j_A : A \cap B \to A$ and $j_B : A \cap B \to B$ are algebra morphisms.
To construct the ...
- 339
5
votes
Accepted
Intersection of Frobenius subalgebra objects
Let $k$ be any field. Consider the commutative $k$-algebra $k[x,y,z]/(x^2,y^2,xz,yz,xy-z^2)$. This is a five dimensional algebra with basis $1$, $x$, $y$, $z$, $u$ where $u=xy=z^2$. It is clearly a ...
- 16.2k
5
votes
Accepted
Homotopy pullback in the category of DG algebras
I was wondering if somebody could tell me the definition of homotopy pullback.
Suppose $\def\C{{\cal C}}\C$ is a relative category, i.e., a category equipped with a subcategory,
morphisms in which ...
- 37.8k
5
votes
Behavior of divisors under push forward and pull back
Of course, just choose a point on each (divisorial) component of the exceptional locus of $f$, then for each of these points the divisors in $|H|$ passing through form a hyperplane in $|H|$, and any ...
- 39.3k
4
votes
History of the pullback corner notation
A small comment, that does not fit in its proper place. The department of Algebra in the University of Santiago de Compostela has studied categories since the second half of the sixties under the ...
- 9,229
4
votes
Accepted
Pullback of localizations
$\newcommand{\Cat}{\mathrm{Cat}_\infty} \newcommand{\Spaces}{\mathrm{Spaces}} \newcommand{\map}{\mathrm{map}} \newcommand{\Fun}{\mathrm{Fun}}$ Any coCartesian fibration which is a localization is a ...
- 15.8k
4
votes
Accepted
Properties of codimension under pull back
This is false as stated: taking for $f$ the embedding of a closed subscheme $X\subset Y$, it would mean that $\operatorname{codim}(X\cap Z,X)\leq \operatorname{codim}(Z,Y) $. There are well-known ...
- 38k
3
votes
Accepted
Pullback of monomorphisms between selfdual objects in a tensor category
This is not true in the non-semisimple case. Here is an example from finite group representation theory.
Let $G$ be $A_5\cong \operatorname{SL}(2,4)$, and let $k$ be an algebraically closed field of ...
- 16.2k
3
votes
Pull-back a section of a vector bundle
abx is right. This kind of things is built through a partition of the unity. Or if you like better, put a Riemannian structure on $M$. The vector bundle $D^\bot$
orthogonal to $D$ in $TM$ is ...
- 2,479
3
votes
Accepted
Does Spec functor sends pushouts of rings into pullbacks of sets?
Let me work directly with schemes (you can of course restrict to affine schemes if necessary).
For $S$-schemes $X,Y$ there is a natural map
$$|X \times_S Y| \to |X| \times_{|S|} |Y|,$$
which is ...
- 63.1k
2
votes
Does "symmetry" of a pullback connection should be obvious?
Here is a proof which is not very elegant, but avoids coordinates. First of all, you define $$\hat T(X,Y)=\nabla_X^{\phi^*TN}d\phi(Y)-\nabla_Y^{\phi^*TN}d\phi(X)-d\phi([X,Y])$$ and observe that this ...
- 6,825
2
votes
Accepted
Are the injections of a coproduct a cover in the canonical pretopology?
This will not be the case in general. A family is a cover in the canonical topology if all its pullbacks are jointly regular epimorphism.
So this will for example be the case if coproducts are ...
- 42.4k
1
vote
Accepted
Pullback of Lie algebras
A reference for pullbacks for modules over a ring: Proposition 5.11, p.222 in "An Introduction to Homological Algebra" by Rotman. For Lie $k$-algebras (or just $k$-algebras), you could refer ...
- 2,821
1
vote
Pullback of Lie algebras
This follows from the existence of free Lie algebras. More precisely, the forgetful functor $U: \mathbf{Lie}_k \to \mathbf{Vect}_k$ has a left adjoint $F$ which takes $V$ to the free Lie algebra on $V$...
- 2,338
1
vote
Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?
It suffices to understand the special case of a linear map $T:U\to V$ where $U,V$ are Euclidean vector spaces. (Think $U=T_pM$, $V=T_{f(p)}N$, $T=df(p)$.)
Suppose first that $n=\dim V\leq \dim U=m$.
...
- 34.7k
1
vote
Accepted
Injectivity of the cohomology map associated to the pullback of line bundles
The unit of adjunction map $L\rightarrow f_\ast f^\ast L$ is an isomorphism if the fibres of $f$ are connected (I assume this in what follows). Then the Leray spectral sequence $E^{pq}_2=H^p(R^q f_\...
- 2,808
1
vote
Accepted
Pullback of homogeneous twisted differential operators
This is true.
One definition of htdo is analogues to that of equivariant sheaf: a htdo is a tdo $\mathcal D$ equipped with an isomorphism $act^\sharp \mathcal{D} \cong pr_X^\sharp \mathcal{D}$ ...
- 403
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