# Tagged Questions

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

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### Special objects in a category - terminology

For an object $A$ in a category $\mathfrak{C}$, consider the following property. ($*$) For every object $B$ in $\mathfrak{C}$, the set of morphisms $\text{Hom}(B,A)$ is either empty or consists ...
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### Model category of cofibrant topological spaces

By browsing the list of open problems in Mark Hovey's book "Model categories", I came across the following one (Problem 8.3): find a model structure on topological spaces, with the same weak ...
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### Modular Tensor Categories: Reasoning behind the axioms

(Sorry for the length of the question, I'm trying to communicate what is bothering me as thoroughly as possible) In the construction of modular tensor categories (MTC) from ground zero, we put ...
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### Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”)

Lemma D4.5.3 of Johnstone’s Sketches of an Elephant states: Lemma. For an object $A$ of a topos $\newcommand{\E}{\mathcal{E}}\E$, the following are equivalent: $A$ is internally ...
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### Semidirect product of semidirect products

For algebraic objects, say for groups $N,K$ and algebra $A$, if we have the semidirect product of a semidirect product, $(A \rtimes_{\gamma} N ) \rtimes_{\theta} K$, are there conditions that would ...
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Setup: Let $\mathcal{O}$ be an operad in the category of sets, and let $\mathcal{O}\text{-Alg}$ denote the category of algebras on it (i.e., operad functors $\mathcal{O}\to\mathbf{Set}$. This category ...
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### Bousfield Localization and Quillen Equivalence

The notion of a (left, say) Bousfield localization of a model category doesn't seem to be invariant under Quillen equivalence. There are a lot of things that could go wrong. But I don't know any ...
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### Filtered Colimit of associative $k$-algebras that are domains

Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains. Then the colimit of the obvious diagram is an integral domain. ...
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### How can a pro-object of the category of finite etale schemes fail to be a profinite-etale scheme?

Let $S$ be a connected scheme. Let $FEt_S$ be the Galois category of schemes $X$ finite etale over $S$. Let $I$ be a directed set, and $\{C_i\}_{i\in I}$ a projective system of objects in $FEt_S$. I'm ...
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### t-structures on the tensor product of stable $\infty$-categories, II

I fork from this thread, a bunch of questions stemmed from a private conversation about that thread. Speculating a bit on the definition of the tensor operation between t-structures generated some ...
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### t-structures on the tensor product of stable $\infty$-categories

It is a matter of checking definitions that the tensor product of presentable $\infty$-categories restrict to a tensor product between stable presentable $\infty$-categories; I was wondering if there ...
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### Which models of set theory are locally presentable?

For the purposes of this question, let me fix a "true" universe of sets, which I will call the "true sets". Recall that a category is locally presentable if it is cocomplete and accessible. Both ...
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### Seeking more information regarding the “rigoidal category” of $\mathbb{N}$-graded sets

Definitions. By an $\mathbb{N}$-graded set, I mean a set $X$ together with a function $|\Box|_X:\mathbb{N} \leftarrow X,$ called the grading. These will simply be called graded sets hereafter. If $Y$...
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### Definition of the differential of the Cone of a morphism of complexes [closed]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$. The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...
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### Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of $S_p$,...
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### Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...
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### About the limitation by size

This could be a big post, so I'll try to summarize my thoughts and divide them into several questions. When working in category theory, I used to choose the following definition. A category $C$ is ...
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### When do powers and ends in functor categories act pointwise?

$\newcommand{\C}{\mathcal C}\newcommand{\I}{\mathcal I}\newcommand{\D}{\mathcal D}\newcommand{\J}{\mathcal J}$Let $(\C, \otimes, I, \multimap)$ be a complete closed monoidal category and $\I$ a small ...
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### Ref request: modelling regular theories as an injectivity condition

The background to the following observation is standard material in categorical logic, and I thought this was was too — I don’t remember learning it, but I don’t think it is original — but I can’t now ...
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### Factorization of a map through a square

Assume to have an abelian category $\mathcal{A}$, and consider its derived category $\mathcal{D^b(A)}$). Let $F:\mathcal{D^b(A)}\rightarrow\mathcal{D^b(A)}$ be a functor between triangulated ...
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### colimits in Cat via coproducts and coequalizers

I am attempting to do a calculation of a colimit in $Cat$, the category of small categories. To this end, people have suggested that I do this by calculating coproducts and using coequalizers. I ...
Assume that $\Gamma$ is a group with neutral element $e$. We associate to $\Gamma$ the following groupoid $G$: \$G=\Gamma \times \Gamma,\;\;\;G^{(0)}=\Gamma \times \{0\},\;\;s(a,b)=(a,e),\;\;\; r(a,...