New answers tagged ct.category-theory
4
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Comonoid homomorphisms in the bicategory of profunctors
It's helpful to consider the case of posets and $\mathbf{2}$-enriched profunctors between them, where $\mathbf{2}$ is the cartesian closed poset $(0 \leq 1)$. (This obviates the complications of ...
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Does the oriental inject into the cube?
Yes, though I don’t know a clean description. Several relevant constructions are discussed in §8 of Aitchison 1986/2010, The geometry of iterated cubes, especially §8.1–8.3; the following description ...
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Is there a category theoretic definition of a cryptographic commitment scheme?
Answer from Dusko Pavlovic:
the question is a very nice case for categorical cryptography because the commitment schema seems like an extreme case of bad notations causing bad confusions. the use ...
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When is the cofibrant replacement of a product the product of the cofibrant replacements?
I asked this question a very long time ago, when I was just starting to do research in abstract homotopy theory. This is a classic case of an xy problem, where I had a proof in mind, and by asking ...
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9
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A result on symmetric closed monoidal categories
This is a special case of:
Let $F\dashv G$ be an adjunction. If there exists an isomorphism $id \cong GF$, then the unit $id \to GF$ is an isomorphism.
Indeed, $[-,A]$ is left adjoint to itself as ...
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7
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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 ...
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What are the internal adjunctions in the bicategory $\mathsf{Span}$?
Let $\mathcal E$ be a category with pullbacks. A span $A \xleftarrow a X \xrightarrow b B$ has a right adjoint in the bicategory $\mathbf{Span}(\mathcal E)$ if and only if $a$ is invertible in $\...
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Most striking applications of category theory?
Uniform Manifold Approximation and Projection (UMAP), a method for dimensionality reduction:
The theoretical foundations for UMAP are largely based in manifold theory
and topological data analysis. ...
0
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Most striking applications of category theory?
I think that "Causal-net condensation" should be an example.
see:
https://www.researchgate.net/publication/369369549_Causal-...
2
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A question about rigid objects in monoidal categories
Below is a graphical calculus:
For more details, you can read Sections 2 and 7 of my preprint arxiv:2203.06522 or watch my course of current semester on YouTube, from Session 3 at 15:00 here, and ...
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3
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Duality in a monoidal category as a functor
Yes. This is covered in Davydov, A. A. (1998). "Monoidal categories and functors". Journal of Mathematical Sciences. 88 (4): 458–472, which is linked from the wikipedia page on rigid ...
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3
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Do objects in the derived category behave stackily?
What do you mean vanishes in those degrees? Do you mean the cohomology sheaf vanishes? This would just follow from the analogous vanishing statement for sheaves.
Or for the literal complex do you mean ...
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4
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Interesting Grothendieck topologies or coverages on the category Prob
This answer is motivated by the related question here.
I think it can help to write $\mathbf{Prob}$ as a disjoint union of the categories $\mathbf{Prop}_\lambda$, with $\mathbf{Prop}_\lambda$ the full ...
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3
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Transitivity axiom for a Grothendieck Topology
I will make some assumptions about the definitions you are using, let me know if you are using a different approach.
A map $f: X \to Y$ is measure-preserving if $\mu(f^{-1}(A)) = \mu(A)$ for each ...
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Why is "everything staying correct" for simplicial spaces?
In general, it is not true that in the context you are interested in, simplicialization of statements preserves their truth. For example, already “every epimorphism of sets is a retraction”, but not “...
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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 ...
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Generalization of category algebra
The category algebra is best understood in terms of the category algebroid: the free $R$-linear category generated by $C.$ The algebra just adds a bunch of zero products for non-composable morphisms ...
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5
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In a weak factorization system, the left class is left cancellative iff the right class is what?
$\newcommand\Inj{\mathit{Inj}}$The answer to Question 1 is as anticipated: for a wfs $(\mathcal L, \mathcal R)$, we have that $\mathcal L$ is left cancellable iff $\mathcal R = ({}^\square\{X_i \to 1\}...
1
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Comparing the exit path category and the nerve of a stratified space
This is false if the stratification is bad. (I first posted a more difficult example, but edited for simplification.)
Example. Let $X = \mathbf R$, let $Z$ be the closure of $\big\{\tfrac{1}{n}\ \big|\...
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3
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Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
I answer the question "where can I read the formal definition of the presentation of ∞-categories by generators and relations?"
You can read about this in the Unicity paper by Barwick and ...
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