**2**

votes

**1**answer

299 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

129 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

118 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

416 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

190 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

195 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

181 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

80 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

222 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

158 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 $\mathcal{K}^\mathcal{...

**5**

votes

**1**answer

124 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

**1**answer

176 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

181 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

99 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

107 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

138 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 $F:J_0\...

**16**

votes

**2**answers

792 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 $S$-...

**6**

votes

**1**answer

170 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

315 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 $f:X\...

**4**

votes

**1**answer

244 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

140 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 $Y$...

**3**

votes

**2**answers

188 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

278 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
$S_p$,...

**2**

votes

**1**answer

138 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

198 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

93 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

119 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

111 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

200 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

220 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

62 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

189 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

176 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')) = \left\{...

**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

74 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

102 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} = \text{add}(...

**5**

votes

**1**answer

255 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

202 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

79 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

129 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

173 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

178 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
$$
RHom(C,...

**8**

votes

**1**answer

319 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

150 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 \...

**2**

votes

**0**answers

98 views

### 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 ...

**0**

votes

**1**answer

204 views

### 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 ...

**2**

votes

**1**answer

289 views

### Groupoid isomorphism vs. group isomorphism

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,...

**1**

vote

**0**answers

68 views

### Non-trivial but simple concrete examples for some categories related to Tensor/Fusion categories

I'm writing a note on Tensor and Fusion categories, the readers of which are physicists rather than mathematicians. So instead of giving abstract definitions I have to give examples to inspire each ...