# Tagged Questions

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### On the existence of a square root for a unitary modular tensor category

The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category.
Question: What about the converse, i.e., can we characterize every unitary modular tensor ...

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4 views

### For which category (if any) are Lie algebras the algebras of a monad? [migrated]

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...

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**1**answer

51 views

### Isofibrations and Diagonal Functors

Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor.
Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any ...

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**1**answer

195 views

### Is the assignment of a root system to a complex semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...

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59 views

### Existence of a $\lambda$-generated Model Category Structure

Apologies if this is a stupid question:
Let $C$ be a cofibrantly generated model category ($\mathbf{Edit}$: Combinatorial) and let $[X,C]$ be functor category equipped with the projective model ...

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117 views

### Is the theory of weak $n$-categories a cofibrant replacement of the theory of strict ones?

I have algebraic models of $n$-categories in mind. By "theory of (weak) $n$-categories", I mean "[monad / operad / whatever] whose algebras are (weak) $n$-categories".
To be more precise: fix an ...

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58 views

### A right adjoint to the truncated Witt functor?

For any ring $A$, let $\mathrm{wEt}_A$ be the category of weakly etale $A$-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor
$$
W_r : \mathrm{...

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107 views

### Yoneda extension of a faithful functor is faithful

Let $F: \mathcal C \to \mathcal D$ be a functor with $\mathcal D$ cocomplete, and let $\mathscr P \mathcal C$ be the free cocompletion of $\mathcal C$ (i.e., the category of small presheaves on $\...

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120 views

### When does every $\infty$-localization correspond to a Bousfield localization?

Let $\mathcal{M}$ be a model category presenting an $\infty$-category $\mathcal{C}$. I believe that every left Bousfield localization $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ corresponds to a ...

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103 views

### Stable unions without stable images

A regular category is one with finite limits and pullback-stable images (i.e. (regular epi, mono) factorizations). A coherent category is a regular category that also has pullback-stable finite ...

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53 views

### Is the category of prederivators cartesian closed?

The question is in the title. ${\bf PDer} = Fun({\bf Cat}^\text{op}, {\bf CAT})$ is obviously cartesian since $\bf CAT$ is. The usual argument for presheaf categories does not apply directly since 1-...

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29 views

### Strict/strong functors are co/reflective inside lax functors, the coendy way

Bozapalides' remarks on lax presheaves show that the category $[{\cal A}^\text{op}, {\bf Cat}]$ is reflective and coreflective inside the category of lax functors, lax natural transformations and ...

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523 views

### When is a functor a right derived functor?

Suppose we have Grothendieck abelian categories $\mathcal{A}, \mathcal{B}$. Suppose also we have given an exact functor of triangulated categories
$$
F \colon D(\mathcal{A}) \to D(\mathcal{B})
$$
...

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92 views

### Is there a schema category for hyperstructures?

I am completely fascinated by Niels Baas' notion of hyperstructures, chiefly because I can see how such gadgets could be used in modeling both biological and social systems, or other evolutionary ...

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843 views

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### History: What was the Lemma? (Grothendieck Harvard Lectures; Mumford)

In an article about the life of Grothendieck, available here:
http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf
Allyn Jackson writes about how Mumford was profoundly impressed:
Mumford ...

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58 views

### What's the connection between Galois objects and Galois closed objects?

An object in a free coproduct completion is Galois closed if it has no nontrivial coverings, i.e every covering morphism is split by the identity.
An object of a Galois category is a Galois object if ...

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68 views

### Realization/embedding for (weakly) finite linear categories

I am trying to determine the status of the following claim. I know how to prove this (unless I made a stupid mistake), so the question is mostly
Is it in the literature?
If not, is there something ...

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143 views

### Is there a compact generated triangulated category which does not have a compact generator?

Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums.
A ...

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92 views

### Galois categories and the connected components functor

In stacks 0BMQ, a Galois category is defined to be a functor $F:\mathsf C\longrightarrow \mathsf{FinSet}$ such that $\mathsf C$ is finitely bicomplete, every object ...

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69 views

### Quotient of triangulated category? (quiver)

This maybe a stupid question, but I really want to know the answer:
Background: Given a quiver with potential, one can consider the derived category of the complete Ginzburg algebra of it, then ...

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110 views

### Whiskering approach to strict 2-categories: literature reference needed

I am familiar with the nLab web page that nicely lays out the axioms needed to define strict 2-categories using whiskering as opposed to horizontal composition of 2-cells. However, I am old fashioned ...

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114 views

### Balanced Tensor Product of Module Categories

(Moved from MSE)
Let C be a k-linear (Vectk-enriched) monoidal category and consider the 2-category Mod_C of k-linear (C,C)-bimodule categories in the sense of Ostrik (https://arxiv.org/abs/math/...

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83 views

### What is the definition of pure exact sequences in the category of chain complexes?

Let $\mathcal{C}$ be a closed symmetric monoidal Grothendieck category. Then there are two general notions of purity in $\mathcal{C}$, the $\lambda$-purity and the $...

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187 views

### Open problems where Haskell meets Category theory or Hopf algebras [closed]

I couldn't find any idea to obtain a problem where Haskell programming language meets Category Theory, Algebraic Topology or Hopf algebras for an original and interesting problem.
Also, I wonder ...

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1k views

### Nuances Regarding Naturality

It's frequently said, informally, that a natural isomorphism is one that doesn't depend on arbitrary choices.
But the phrase "arbitrary choices" lends itself to different interpretations. Consider ...

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248 views

### The naive approach to deriving profunctors - What's wrong with it?

Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ ...

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272 views

### What's an example of a subcategory whose closure under colimits takes a lot of steps to form by iteration?

Let $\mathcal{C}$ be a cocomplete category and $\mathcal{S} \subseteq \mathcal{C}$ be a full subcategory. The colimit completion $\mathrm{Colim}^\mathcal{C}(\mathcal{S})$of $\mathcal{S}$ in $\mathcal{...

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114 views

### When is a Module category monoidal?

Let $\mathcal{C}$ be a monoidal category and $M$ a left module category over $\mathcal{C}$. That is, a category equipped with an exact bifunctor $F:\mathcal{C}\otimes M\rightarrow M$ satisfying some ...

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92 views

### Reversing the arrows-dual theorems

When one studies homological algebra one learns some basic stuff-diagram chasing, long exact sequences associated to short exact sequence of complexes and so on. Usually one works out the details with ...

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69 views

### Presheaves of Dendroidal Sets?

Are there any references available for presheaves of dendroidal sets?
Seems like a natural extension of simplicial presheaves.

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222 views

### Equivalent definition of a homotopy of functions

It is well known that given $X,Y$ arbitrarily topological spaces, $I$ the unit interval, and continuous functions $f, g : X \rightarrow Y,$ a homotopy between the functions is a continuous function $H ...

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538 views

### Global elements in categories with no terminal object?

Let $\mathcal{C}$ be a category. Suppose $\mathcal{C}$ contains a terminal object, which I will denote by $\boldsymbol{1}$. Then for any object $B$ in $\mathcal{C}$, a global element of $B$ is a ...

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142 views

### Spans as binary relations: reflexivity, transitivity, and completeness?

Let $\mathcal{C}$ be a category, let $B$ be an object in $\mathcal{C}$, and let $\mathcal{R}$ be a span from $B$ to itself. (That is: $\mathcal{R}$ is a diagram $B \stackrel{r_1}{\longleftarrow} R \...

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121 views

### Completeness of 2-category of Monoidal Categories

Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?

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166 views

### Are accessible $\infty$-categories closed under accessible localizations?

The defining problem of homotopy theory is that often when one localizes a nice category at a reasonable class of morphisms, the result is a very bad category. Does passing to the $\infty$ world fix ...

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2k views

### Linear algebra in terms of abstract nonsense?

The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think.
I was wondering what portions of basic linear algebra (first couple of courses) fall ...

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436 views

### Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products

A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...

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172 views

### On the cardinal arithmetic of accessible categories

If $\lambda, \mu$ are regular cardinals, say that $\lambda \trianglelefteq \mu$ if $\lambda \leq \mu$ and
$$\forall X, \, |X| < \mu \implies \mathrm{cf} (P_\lambda(X)) < \mu$$
Here $P_\lambda(X)...

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800 views

### Corollaries of the Yoneda Lemma in Analysis?

This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: http://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis.
I am looking for some ...

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**1**answer

192 views

### Coherence theorem for symmetric lax monoidal functors

Let $V$ and $W$ be symmetric monoidal categories. Let $F:V\to W$ be a lax symmetric monoidal functor with multiplication $\nabla:FA\otimes FB \to F(A\otimes B)$. Consider the following statements:
1) ...

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94 views

### Cubical model category

Question. Call a cubical model category a model category enriched over cubical sets equipped with a tensor product $X \otimes K$ and a cotensor product $X^K$ where $X$ is an object of the model ...

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187 views

### Does the inclusion of presheaves into families of sets have a left adjoint?

Consider the inclusion of presheaves on $\mathbb{C}$ into families of sets indexed by $\mathbb{C}$-objects (which proceeds by forgetting the action on morphisms). Is there a left adjoint to this ...

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292 views

### Is the natural isomorphism $|FX_\bullet| \cong F|X_\bullet|$ lax symmetric monoidal?

Let $\mathcal{V_1}$ and $\mathcal{V_2}$ be cocomplete symmetric monoidal categories, each endowed with a cosimplicial object $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. ...

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209 views

### Constructively, are all fibrations cloven?

A "cloven fibration" is a fibration for which we have an explicit choice of cartesian liftings; this is often phrased as, "We can pick a lifting without using the axiom of choice".
Firstly, I'm a bit ...

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372 views

### What is the intuitive meaning of the coskeleton of a simplicial set?

Consider the inclusion $i_n: \varDelta_{\leq n} \hookrightarrow \varDelta$.
This induces a truncation functor $tr_n:\mathbf{sSet}\to\mathbf{sSet}_{\leq n}$, which has a left and a right adjoint, $tr_n^...

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165 views

### Left adjoint to Double Nerve?

The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor $N^2:2Cat\rightarrow sSSet$ from 2-...

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58 views

### When is a functorial coverage a sheaf, and what universal property does it have?

In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is
a function assigning to each object $A$ of $\...

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193 views

### Representability of the Weil restriction reference and proof

Proposition 2 of 7.6 of Néron Models [BLR] provides a sufficient condition for the representability of a Weil restriction $R_{S'/S}(X')$. The theorem is attributed to Grothendieck.
Is there an ...

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**1**answer

96 views

### A map between direct limits

Let $C$ be a category which has all small colimits.
I have the following situation:
$\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$ are two directed systems in $C$,
with transition maps $\alpha_{i_1,i_2}...

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82 views

### Saturated classes and cofibrantly generated model structures

There seem to be two definitions of what a saturated class should be:
A class of morphisms closed under retracts, pushouts and transfinite composition.
A class of monomorphisms containing all ...