# Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

3,942 questions
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### Statistical models of functions

I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions ...
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### Can natural section/retraction be checked pointwise?

Analogously to this old question, I was asking myself if it is possible to describe left/right invertible natural transformations by their components. Obviously this property is inherited by the ...
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### is the functor of finite order elements in grp representable

My question is whether the functor F:Grp->Set that sends a Group to its Set of elements with finite order is representable. I have the sense that it shouldn't be but I've so far failed to prove it in ...
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### Translating Grothendieck axiom UB into ZFC

In SGA 4, Grothendieck introduced a set-theoretic device called a Grothendieck universe. He and his collaborators worked in Bourbaki set theory, which is practically similar to $\mathsf{ZFC}$ but ...
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### Descent data and trivialization of bundles via coherent isomorphisms of fibers

In this MO question I tried to understand how a trivialization of a bundle (continuous map) $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ is related to a coherent family of isomorphisms ...
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### Limits, colimits and universes

For many purposes in category theory, we consider limit and colimits of diagrams $F\colon\mathsf{J\to C}$ where $\mathsf{J}$ is small category, that is, a category where the classes of objects and ...
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### Derivator of localizations of spectra

Since my question is very specific let me introduce the context. I am trying to apply derivator theory to stable homotopy theory. Theorem 3 of the following preprint by Franke https://pdfs....
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### Free symmetric monoidal category of compactly generated category is compactly generated

Let $k$ be a field and let $\mathcal{C}=\mathbf{StLin}_k$ be the $\infty$-category of stable infinity categories enriched over the $\infty$-category $\mathbf{Vect}_k$, regarded as a symmetric monoidal ...
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### Is there a version of the “infinitary” disjunctive normal form theorem for topoi and slice categories?

According to nLab Proposition 2.3. A complete Boolean algebra is completely distributive iff it is atomic (a CABA), i.e., is a power set as a Boolean algebra. This is basically an infinitary ...
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### Is the double orthogonal of a localizing Serre subcategory still a localizing Serre subcategory?

Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under ...
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### “Scott completion” of dcpo

If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U$ for ...
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### The universal property of composition of morphisms

$\def\K{\mathcal K}$ Preamble. Given a locally small category $\mathcal K$, its "composition law" is a class of maps $$c_{abc} : \K(a,b)\times\K(b,c)\to \K(a,c)$$ with the universal property of an ...
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### Algebras for the codensity monad

Given a functor $j : A \to B$, when this extension exists we call $T_j = Ran_jj$ the codensity monad of $j$. I was wondering if there's a general rule to guess the shape of $Alg(T_j)$ given $j$.
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### Locally presentable categories

Under category Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...
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### Bijection between hom sets of equivalent categories?

I've been trying to prove the following claim, but am now unsure about its truth. Is it true, and if so, where can I find a proof? Claim: For any categories C, D, E such that C and D are equivalent, ...
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### Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra: The space of ...
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### How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
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### Equivalent conditions to be a lax idempotent contravariant monads

$\require{AMScd}$As already said, a contravariant monad is "like a monad, but a contravariant functor": the multiplication and unit are dinatural transformations an algebra is a map $a : TA\to A$ ...