**2**

votes

**1**answer

126 views

### Product and coproduct for bipartite graphs

Consider the category $BiGraph$ of bipartite graphs (vertices are named "places" and "transitions" like in Petri nets) and continous maps. A subgraph is named open iff it is place-bordered, and a map ...

**1**

vote

**1**answer

86 views

### When modular tensor categories are equivalent?

I asked this question at math stack exchange math stack exchange but I haven't got any answer yet there.
I would like to know when we say that two modular tensor categories are equivalent.
Is it ...

**2**

votes

**1**answer

77 views

### Direct limit closure of Serre subcategories

Let $C$ be a Grothendieck category and $T$ a Serre subcategory of $C$. Let $\tilde{T}$ be the full subcategory of $C$ consisting of all direct limits of objects in $T$. Is $\tilde{T}$ a Serre ...

**2**

votes

**0**answers

66 views

### Deligne tensor product of semisimple tensor categories

Let $T_1, T_2$ be two semisimple tensor categories over a field $k$ (i.e. symmetric rigid monoidal abelian $k$-linear). Then is the Deligne product, $T_1\otimes T_2$ also a $k$-tensor category?
...

**2**

votes

**1**answer

208 views

### 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 ...

**8**

votes

**0**answers

125 views

### 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 ...

**7**

votes

**0**answers

111 views

### 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 ...

**23**

votes

**1**answer

394 views

### 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 ...

**1**

vote

**0**answers

157 views

### 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 ...

**7**

votes

**1**answer

185 views

### Monomorphisms in operad algebras

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 ...

**6**

votes

**1**answer

172 views

### 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 ...

**4**

votes

**1**answer

79 views

### 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.
...

**9**

votes

**1**answer

220 views

### A general version of the 5 lemma

Suppose you have an abelian category $\bf A$, and $A\to B\to C$, $A'\to B'\to C'$ two exact sequences, in a diagram
$$
\begin{array}{cccccccc}
0 &\to & A &\to& B &\to& C ...

**3**

votes

**0**answers

79 views

### What do you call the coherence cells in a lax morphism?

The original question a friend asked me is what to call the coherence cells in a lax monoidal functor. After looking around, I was surprised to realize that when it comes to monoidal functors, ...

**4**

votes

**1**answer

149 views

### When does the projective model structure on functors exist?

What this boils down to is: if $\mathcal{K}$ is cofibrantly-generated model category which permits the small object argument and $\mathcal{D}$ is a small category, then when does ...

**5**

votes

**1**answer

116 views

### Uniqueness of $\infty$-adjoints

Adjoints in a 2-category are essentially unique, in the following strong sense. If $\mathbf{2}$ denotes the "walking arrow" category $(\cdot \to \cdot)$, then there is a 2-category $\mathrm{Adj}_1$ ...

**5**

votes

**0**answers

141 views

### Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors
$$
...

**5**

votes

**0**answers

178 views

### 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 ...

**6**

votes

**0**answers

95 views

### 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 ...

**9**

votes

**0**answers

100 views

### 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 ...

**3**

votes

**2**answers

132 views

### When are subcategories of continuous functors reflective?

Let $J$ be a collection of small categories (to be thought of as diagrams in a category). Let $C$ be a small category with all $J$-limits (i.e. for every $J_0 \in J$ and every functor ...

**16**

votes

**2**answers

778 views

### Does the functor Sch to Top have a right adjoint?

Let $S$ be a scheme, let $T$ be an $S$-scheme, and let $M$ be a set. Let $M_{S}$ be the disjoint union of $M$ copies of $S$, considered as an $S$-scheme. (Notation from [SGA 3, Exp. I, 1.8].) Then ...

**5**

votes

**1**answer

168 views

### Heyting algebras originating from directed graphs

The category RefGph of reflexive directed graphs is the functor
category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is
the simplex category truncated at level 1.
Hence the poset Sub(X) of ...

**2**

votes

**1**answer

309 views

### A morphism-revealing category? [closed]

The constructs can be considered as subcategories of Set but when considered as subcategories of the category SubSet, of pairs of sets with pairs $(X,S)$, $S\subseteq X$, as objects and functions ...

**4**

votes

**1**answer

242 views

### 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 ...

**3**

votes

**1**answer

136 views

### 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 ...

**3**

votes

**2**answers

184 views

### 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 $\varphi:F^\bullet \to G^\bullet$ ...

**6**

votes

**1**answer

273 views

### 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
...

**2**

votes

**1**answer

134 views

### 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 ...

**2**

votes

**0**answers

192 views

### 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 ...

**7**

votes

**1**answer

92 views

### 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 ...

**6**

votes

**0**answers

116 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...

**2**

votes

**0**answers

106 views

### DG natural transformation Serre functors

This question might be really easy (or stupid), but I have only vague (heard-about) knowledge of DG categories, so I don't know where to look for an answer.
Let $X$ be a smooth projective variety ...

**3**

votes

**2**answers

195 views

### How to simplify the proof of right-properness?

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback ...

**-1**

votes

**1**answer

218 views

### Order-Perserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...

**0**

votes

**1**answer

61 views

### Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Consider the direct limit of an indexed family $\{a_n\}_{n\in \omega}$:
$\require{AMScd}$
\begin{CD}
a_0 @>>> \ldots @>>> a_n @>>> a_{n+1} @>>> ...

**12**

votes

**1**answer

187 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors ...

**4**

votes

**0**answers

94 views

### 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 ...

**2**

votes

**2**answers

172 views

### The source-side-opposite of the arrow category

Given a category $C$, is there a name for the following category:
$\mathrm{Obj}(D) = \left\{ (x, y, f) \middle| x, y \in \mathrm{Obj}(C), f \in C(x, y) \right\}$
$D((x, y, f), (x', y', f')) = ...

**2**

votes

**0**answers

45 views

### Holonomy 2-functor transformation by transition functions

The holonomy 2-functor on a $\mathcal{G}$-principal 2-bundle associates a bigon:
$$\mathsf{hol}_i(\Sigma):\mathsf{hol}_i(\gamma)\Rightarrow \mathsf{hol}_i(\gamma')$$
in $\mathcal{G}$ to each bigon:
...

**0**

votes

**0**answers

63 views

### Is there a general way to define invariants in a category, using generalized elements?

In Awodey's category theory, page 37, he uses a specific generalized elements $2 \to X$ and $2 \to A$ for posets $A=\{a \leq b \leq c, \}$ and $X=\{x \leq y, x \leq z, \}$to make an invariant in the ...

**4**

votes

**1**answer

101 views

### Is this additive equivalence a triangulated equivalence?

Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, and suppose that there exists an additive equivalence $F: \mathcal{C} \to \mathcal{D}$. Suppose further that $\mathcal{C} = ...

**5**

votes

**1**answer

252 views

### Does this notion related to species/operads/FI-modules have a name?

Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? ...

**6**

votes

**1**answer

195 views

### What is the right adjoint of the tensor product in a closed monoidal functor category?

The nLab says the following about closed monoidal functor categories:
Let $C$ be a complete closed monoidal category and $I$ any small category. Then the functor category $[I, C]$ is closed ...

**2**

votes

**0**answers

77 views

### Factorization system in a derivator

Has anybody attempted to define the notion of a factorization system "in" a derivator?
Something on the lines of this: let $\mathbb{D}$ be a (strong) derivator. We define the orthogonality relation ...

**2**

votes

**1**answer

126 views

### How do you rigidify a Bousfield localization?

I'm learning about Bousfield localizations. For a triangulated category satisfying some axioms, a Bousfield localizations can be described as an idempotent functor $L:D \to D$.
I thought there is a ...

**1**

vote

**0**answers

170 views

### Reference request: Models of isomorphic languages result into isomorphic categories

This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory.
Fix an uncountable universe ...

**5**

votes

**2**answers

167 views

### Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
...

**8**

votes

**1**answer

311 views

### Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...

**3**

votes

**1**answer

149 views

### Action of a strict 2-group on a category gives autoequivalences?

A strict action of a strict 2-group (seen as 2-category with only one object $\star$) $\mathcal{G}$ on a 2-space (principally a category) $\mathcal{M}$ is a strict 2-functor $\Phi:\mathcal{G}\to ...