# 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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### 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 ...
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### 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 ...
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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 ...
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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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### 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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### 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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### 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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### 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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### 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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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 $$W_r : \mathrm{... 1answer 110 views ### Yoneda extension of a faithful functor is faithful Let F: \mathcal C \to \mathcal D be a functor with \mathcal D cocomplete, and let \mathscr P \mathcal C be the free cocompletion of \mathcal C (i.e., the category of small presheaves on \... 1answer 121 views ### When does every \infty-localization correspond to a Bousfield localization? Let \mathcal{M} be a model category presenting an \infty-category \mathcal{C}. I believe that every left Bousfield localization \widetilde{\mathcal{M}} of \mathcal{M} corresponds to a ... 2answers 105 views ### Stable unions without stable images A regular category is one with finite limits and pullback-stable images (i.e. (regular epi, mono) factorizations). A coherent category is a regular category that also has pullback-stable finite ... 0answers 53 views ### Is the category of prederivators cartesian closed? The question is in the title. {\bf PDer} = Fun({\bf Cat}^\text{op}, {\bf CAT}) is obviously cartesian since \bf CAT is. The usual argument for presheaf categories does not apply directly since 1-... 0answers 29 views ### Strict/strong functors are co/reflective inside lax functors, the coendy way Bozapalides' remarks on lax presheaves show that the category [{\cal A}^\text{op}, {\bf Cat}] is reflective and coreflective inside the category of lax functors, lax natural transformations and ... 2answers 535 views ### When is a functor a right derived functor? Suppose we have Grothendieck abelian categories \mathcal{A}, \mathcal{B}. Suppose also we have given an exact functor of triangulated categories$$ F \colon D(\mathcal{A}) \to D(\mathcal{B}) $$... 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 ... 0answers 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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/... 0answers 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 ‎‎... 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 ... 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 ... 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} ... 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{... 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 ... 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 ... 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. 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 ... 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 ... 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 \... 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? 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 ... 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 ... 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 ... 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 $P_\lambda(X)... 2answers 803 views ### Corollaries of the Yoneda Lemma in Analysis? This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: http://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis. I am looking for some ... 1answer 193 views ### Coherence theorem for symmetric lax monoidal functors Let$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)$. Consider the following statements: 1) ... 0answers 94 views ### Cubical model category Question. Call a cubical model category a model category enriched over cubical sets equipped with a tensor product$X \otimes K$and a cotensor product$X^K$where$X$is an object of the model ... 1answer 193 views ### Does the inclusion of presheaves into families of sets have a left adjoint? Consider the inclusion of presheaves on$\mathbb{C}$into families of sets indexed by$\mathbb{C}$-objects (which proceeds by forgetting the action on morphisms). Is there a left adjoint to this ... 1answer 292 views ### Is the natural isomorphism$|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$. ... 2answers 209 views ### Constructively, are all fibrations cloven? A "cloven fibration" is a fibration for which we have an explicit choice of cartesian liftings; this is often phrased as, "We can pick a lifting without using the axiom of choice". Firstly, I'm a bit ... 2answers 374 views ### What is the intuitive meaning of the coskeleton of a simplicial set? Consider the inclusion$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^...
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-...
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 \$\...