22
votes
Accepted
Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?
One can take some of the standard violations of CSB with other kinds of mathematical structures and transfer them to categories.
For example, with linear orders, we have the two linear orders $$\...
- 236k
15
votes
Accepted
Loss of cuspidality by Langlands tranfer
You are quite correct that the Langlands transfer map does not preserve cuspidality in general. E.g. if you take a modular form of CM type, coming from a Groessencharacter $\psi$ of some imaginary ...
- 37k
12
votes
Accepted
3-functoriality of the lax Gray tensor product
The lax Gray tensor product is not a two-variable 2-functor.
If it were, we'd have a functor $Fun(A,B) \times Fun(C,D) \to Fun(A \otimes_l C, B \otimes_l D)$ extending the set map $Ob Fun(A, B) \times ...
- 63.9k
9
votes
Accepted
Subobjects as an object in a topos
The general notion you're looking for is a representable functor. For example:
$${\mathcal E}(X\times{-},Y) \sim {\mathcal E}({-},Y^X)$$
$${\textsf{Sub}}({-}) \sim {\mathcal{E}}({-},\Omega)$$
The ...
- 8,481
9
votes
Classification of the functors on the category of cyclic groups
I'll use additive notation, and I'll assume that you are only considering finite cyclic groups. Let $\mathbf{Cyc}_p$ be the category of cyclic $p$-groups. Given $i,j\geq 0$ we can define $Q(p;i,j)\...
- 56.9k
8
votes
Accepted
Is the injective envelope functorial?
One can view $A$ and $B$ as sitting completely isometrically inside their injective envelopes $I(A)$ and $I(B)$. Then by injectivity a unital *-homomorphism (or more generally a unital completely ...
- 3,984
8
votes
What structure do natural isomorphisms preserve?
There's a paper of Peter Freyd that addresses a similar question, not about natural isomorphism of functors but about equivalence of categories. I conjecture that what Freyd proved about equivalence ...
- 73.1k
7
votes
What structure do natural isomorphisms preserve?
The simplest case is that for a fixed (small) category $C$, there is a (multi-sorted) first-order theory whose models are functors $C\to \rm Set$: it has one sort for every object of $C$, and one ...
- 66.7k
7
votes
Accepted
Birkhoff's representation theorem vs matroid-geometric lattice correspondence
Antimatroids
are a good example. We have the syllogism "Antimatroids are to
matroids as join-distributive lattices are to geometric lattices."
Two other examples are the characterizations of ...
- 50.8k
7
votes
Is univalence equivalent to every type function being a functor over equivalence?
Your axiom does not entail univalence.
It is consistent to add to type theory the isomorphism reflection rule
$$\frac{\Gamma \vdash e : A \simeq B}{\Gamma \vdash A \equiv B}$$
which states that ...
- 48.8k
7
votes
Is the injective envelope functorial?
As mentioned by Chris, injective envelopes are brutal when seen as C$^*$-algebras.
Let $A=\text{UHF}(2^\infty)$ and $B$ the hyperfinite II$_1$ factor. Take $f$ to be the inclusion map. We have $I(B)=B$...
- 2,793
6
votes
Birkhoff's representation theorem vs matroid-geometric lattice correspondence
In their recent papert "The fundamental theorem of finite semidistributive lattices" (https://doi.org/10.1007/s00029-021-00656-z and https://arxiv.org/abs/1907.08050) Reading, Speyer, and ...
- 24.2k
4
votes
Subobjects as an object in a topos
To 'turn an object $A$ of a category $\mathcal{C}$ into a set' you can consider the set of objects mapping into/out of that object from/to some other object; if you consider maps out of a terminal ...
- 10.1k
3
votes
Accepted
Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences
An explicit reference for the additivity is
Mac Lane, Saunders, Homology, Classics in Mathematics. Berlin: Springer-Verlag. x, 422 p. (1995). ZBL0818.18001
in particular Chapter III, Theorem 2.1. The ...
- 35.9k
3
votes
Accepted
Functoriality of the Hopf dual
-as suggested after the discussion in the comments-
i am understanding that the OP is asking whether a linear map $j:G \to H$ is functorial, in the sense that the image of the dual map $j^*:H^* \to G^*...
- 7,746
3
votes
Exactness of functors in a $C^*$-tensor category
I agree, and wanted to follow up on @Noah's comment.
Following the definition of $\mathcal C^*$ tensor categories, there are only two assumptions that allow us to construct new objects:
(vi) $\...
- 556
3
votes
Find a functorial zig-zag of spaces
If the diagrams $X_\bullet$ and $Y_\bullet$ are constructed canonically, but not necessarily naturally, from spaces $X,Y$ where $X \simeq Y$, one technique to construct such a zigzag is pick $f: X \...
- 5,829
3
votes
Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?
This is simply a summary that includes all the details.
$\mathbf{Pos}$ usually denotes the category of partially ordered sets as objects and monotone functions as morphisms. For any partially ordered ...
- 405
2
votes
Are (commutative) squares in some sense universal among edge-symmetric double categories?
The answer to this question for (not necessarily commutative squares) may be given implicitly by the following remark in the last paragraph of section 2 of Ronald Brown and Ghafar H. Mosa, "...
2
votes
Birkhoff's representation theorem vs matroid-geometric lattice correspondence
I think I might've stumbled upon the answer I was looking for. It's the correspondence between interval greedoids and semimodular lattices. An interval greedoid is a pair $(E,\mathcal F)$ where $\...
- 3,513
1
vote
Birkhoff's representation theorem vs matroid-geometric lattice correspondence
I learned that there is at least one precise answer to this exact question (more precise than my previous answer). The bijection between finite posets and finite distributive lattices and the ...
- 3,513
1
vote
Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$
I only gather answers for special cases. There is always a natural (functorial) injection $\bigwedge^\ast(A/2) \to H_\ast(A,\mathbb Z/2)$, and furthermore it's a natural isomorphism if ${}_2A=0$. But ...
- 17.5k
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