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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Properties of Grothendieck ring for field of characterictic $p$

In this article there is a proof that for field $k$ of characteristic zero Grothendieck ring $K(\mathbf{Var}_k)$ is not an integral domain. In many articles I found statement that similar theorem for ...
0
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1answer
100 views

What is a universal tree?

I came across some slides talking about the Hrushovski construction. One of the examples was the construction of a "universal tree". I was curious because the collection of finite trees does not ...
2
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0answers
31 views

A List-Like Frobenius Monad

Has anyone ever seen a Monad that is very much like the List Monad but is also a co-monad, and hence a Frobenius monad. I was reading a paper that, I think, suggested zipper as an example. I think ...
2
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1answer
72 views

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 ...
1
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1answer
65 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 ...
6
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1answer
262 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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0answers
63 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 ...
6
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120 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 ...
4
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60 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{...
1
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1answer
110 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 $\...
3
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1answer
121 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 ...
5
votes
2answers
105 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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0answers
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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0answers
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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2answers
535 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 $$ F \colon D(\mathcal{A}) \to D(\mathcal{B}) $$ ...
3
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0answers
93 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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884 views

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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0answers
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 ...
4
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0answers
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 ...
3
votes
2answers
144 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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0answers
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 ...
0
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0answers
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 ...
2
votes
1answer
111 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 ...
5
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1answer
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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1answer
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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4answers
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 ...
6
votes
1answer
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 $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ ...
13
votes
3answers
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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1answer
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 ...
2
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0answers
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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0answers
70 views

Presheaves of Dendroidal Sets?

Are there any references available for presheaves of dendroidal sets? Seems like a natural extension of simplicial presheaves.
3
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1answer
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 ...
15
votes
2answers
539 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 ...
4
votes
2answers
143 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 \...
5
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1answer
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?
5
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1answer
167 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 ...
24
votes
2answers
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 ...
10
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3answers
440 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 ...
7
votes
1answer
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)...
15
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2answers
803 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 ...
3
votes
1answer
193 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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0answers
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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 ...
3
votes
1answer
193 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 ...
8
votes
1answer
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 $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. ...
5
votes
2answers
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 ...
8
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2answers
374 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^...
3
votes
1answer
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-...
3
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0answers
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 $\...