**6**

votes

**2**answers

259 views

### Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?
Since weak ...

**6**

votes

**2**answers

652 views

### If $G \times G \cong H \times H$, then is $G \cong H$? [duplicate]

Let $G,H$ groups. Suppose that $G \times G \cong H \times H$. Then is necessarily $G \cong H$?
I know that if $G,H$ are finite groups then it is true (you show that the map $FinGrp \rightarrow ...

**3**

votes

**2**answers

297 views

### 2 and 3 pullbacks

If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times_C C^I \times_C B$, where $I$ is the ...

**5**

votes

**0**answers

143 views

### Nice references for injective model structures and Quillen functors between motivic homotopy categories

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...

**1**

vote

**1**answer

155 views

### “order two sequence” in a paper of Waldhausen

In F. Waldhausen's paper "Algebraic K-theory of generalized free products, Part I", page 142,line 19, there is a term "order two sequence". Can anyone explain its meaning to me?
According to the ...

**5**

votes

**0**answers

146 views

### Weak equivalences of left Bousfield localizations

Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1.
If necessary, the model structures can be assumed to be simplicial, ...

**8**

votes

**1**answer

274 views

### Which statements and arguments of Hovey's “Model categories” fail without functorial factorizations of morphisms?

I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations ...

**3**

votes

**1**answer

249 views

### (Fri)end(l)y way to express Kan liftings

I would like to have an explicit description of the left/right Kan lift of a functor $F$ through $G$, $\text{Lift}_GF$/$\text{Rift}_GF$ in terms of coends/ends (this can be done for Kan extensions, so ...

**2**

votes

**0**answers

108 views

### Is an integral simple fusion ring, categorifiable?

A fusion ring $\mathcal{F}$ (see here p 28) is integral if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers. Its rank is $r$ and its dimension is $\sum ...

**8**

votes

**2**answers

257 views

### What properties do “large topoi” share with actual topoi?

Fix Grothendieck universes $\mathcal{U} \in \mathcal{V}$ and suppose that $C$ is a locally $\mathcal{U}$-small category which is $\mathcal{V}$-small. Denote by $Set$ the category of ...

**5**

votes

**2**answers

168 views

### Relationship between pullbacks and the Ore condition

Let $C$ be a category and assume either that $C$ has all binary pullbacks or that $C$ satisfies right calculus of fractions. In both cases the localization of $C$ at every morphism (i.e. the ...

**13**

votes

**5**answers

689 views

### is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...

**3**

votes

**1**answer

124 views

### Constrained factorization of monads

Suppose $A\dashv B\colon \cal H\to K$ are adjoint functors; if the comonad $AB$ has a right adjoint $T\colon\cal K\to K$ then $T$ is a monad in a natural way.
I would like to tell when "$T$ can be ...

**3**

votes

**3**answers

227 views

### Higher dimensional pasting diagram of cubes

A pasting diagram is the analogue of composition of arrows in higher categories. The nlab page gives an example of a two dimensional diagram and how to compute it as a composite of two morphisms. I am ...

**3**

votes

**1**answer

155 views

### Factorization of morphisms in a diagram category

Let us suppose that $I$ is a small category and $\mathcal{E}$ a combinatorial model category. Then there exists two Quillen equivalent combinatorial model category structures on the diagram category ...

**1**

vote

**1**answer

122 views

### arrows in the injective representations of quivers

Let $Q$ be a quiver (a directed graph) and $E$ be a representation of $Q$ by R-modules, that is, for each vertex $v$ in $Q$ there exists an R-module $E(v)$ and for each arrow $a:v\to w$ there exists a ...

**4**

votes

**1**answer

367 views

### Splitness of commutative diagrams

Consider the following commutative diagram in the category of $R$-modules where $R$ is an associative ring with identity and all modules are unital.
$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ ...

**6**

votes

**1**answer

318 views

### Morita theorem for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting?
I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...

**3**

votes

**1**answer

147 views

### About (co)limits of accessible categories

I am reading the paper colimits of accessible categories. In the introduction, the authors summarize what is known about limits and colimits of accessible categories. I believed that there was ...

**5**

votes

**3**answers

390 views

### Exponentials in the opposite category of finite separable algebras

Let $K$ be a field and $G=Gal(K_s/K)$ is its absolute Galois group. Then, by Galois theory, the category of finite separable algebras over $K$ (denoted by $Sep(K)$) and the category of finite ...

**10**

votes

**2**answers

719 views

### Fields aren't group objects in Ab, so what are they?

This might be a vague question, but I am troubled by the fact that fields do not admit a nifty categorical definition. An obvious attempt such a definition would be to say that fields are commutative ...

**3**

votes

**3**answers

185 views

### Slice-category-like terminology question

Let $\mathcal C$ be a category, and consider a new category $\mathcal C'$ with
$Obj(\mathcal C') := \{$pairs $(X \in Obj(\mathcal C), T \in End_{\mathcal C}(X)) \}$
$Hom_{\mathcal ...

**3**

votes

**1**answer

157 views

### What is the suitable setting for supercoherence with value in a bicategory?

It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as ...

**3**

votes

**0**answers

160 views

### Ends and Coends - Analogues for higher arity - Horn Filling

Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$.
We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$
Recall the definition ...

**3**

votes

**0**answers

132 views

### Proof that the category of presheaves on a category $C$ is equivalent to the category of presheaves on its Karoubi envelope

Let $C$ be category, let $PSh(C) = [C^{op},$ *Set*$]$ be the category of presheaves on $C$, the Karoubi envelope of $C$, denoted $\overline{C}$, is defined as the full subcategory of $PSh(C)$ which ...

**3**

votes

**1**answer

154 views

### Given an exact category, viewed as a site, do there exist non-additive sheaves?

Suppose given an exact category $\mathcal{C}$. The following question arises while proving the Gabriel-Quillen-Laumon embedding theorem following Laumon [1].
Laumon constructs an abelian category ...

**2**

votes

**1**answer

191 views

### how do automorphisms act on the right in grothendieck's galois theory

So, I'm reading through some notes on the etale fundamental group (mostly Murre, but also some other notes I have), and I find it confusing how in a galois category $\mathcal{C}$ with fundamental ...

**5**

votes

**1**answer

379 views

### Is Logic/Set Theory necessary for studying Topos Theory?

I have just completed a postgraduate course, in which I studied Category Theory, without having a background in Set Theory and Logic - this probably already sounds absurd to many. This did not seem to ...

**3**

votes

**0**answers

64 views

### When is the localic reflection of a topos discrete?

Recall that the inclusion of locales into topoi has a left adjoint, called the localic reflection. It sends a topos $\mathcal{E}$ to the poset of subobjects of the terminal object, which is a locale. ...

**7**

votes

**1**answer

194 views

### Are topoi and etale geometric morphisms locally small?

This question is related to this one: Local smallness and (higher) topoi which has not yet been answered.
The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in ...

**1**

vote

**0**answers

44 views

### Completion of stabilization versus…?

There is a passage in HA.1.4.1 which sound totally obscure to me:
This "formal argument" should look like the following equivalence:
$$
\text{Ind-}\Big( \varinjlim\big\{ ...

**1**

vote

**1**answer

130 views

### Pullback as a local property

Given a commutative square in a nice category, say, manifolds $Mfd$. Suppose all edges are submersions (I guess transverse should be OK), then the square is a pullback if and only if it locally is, ...

**3**

votes

**2**answers

225 views

### Analogues of 'cone' distinguished triangles for pointed model categories?

For an additive $A$ and any morphism $f:X\to Y$ in $C(A)$ one has the following distinguished triangle in the homotopy category $K(A)$: $X\to Y\to Cone(f)\to X[1]$.
What is the closest analogue of ...

**1**

vote

**1**answer

191 views

### A question on the definition of operad

The nlab page says
A (Set-based) operad is a monoid in the monoidal category
$(Psh(ℙ),∘,I)$, where $ℙ$ is the category of $\sqcup_{n\ge 0} S_n$.
The monoidal structure is given by the so ...

**1**

vote

**1**answer

161 views

### Regarding a difficulty in the Fakir article about associated idempotent triple

I just had post this question in SE: http://math.stackexchange.com/questions/518054/about-details-of-the-fakir-theorem-proof-associated-idempotent-triple but dont get any answer.
I understand that at ...

**3**

votes

**1**answer

368 views

### Projective objects in HTT

In HTT.5.5.8.18 Lurie defines a projective object $P$ in a quasicategory $\bf C$ as an object such that its corepresented functor ${\rm Map}(P,-)$ "commutes with geometric realizations". I can catch ...

**9**

votes

**2**answers

352 views

### Do Homotopy Fully Faithful Functors Push-out?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.
Are homotopy fully faithful functors preserved under (homotopy) pushout?
More ...

**3**

votes

**3**answers

475 views

### Yoneda on a not so small category

I am working with "usual" category theory, maybe over ZFC, and I have a functor $F : Set \to Set$. I'd like to apply Yoneda lemma to $F$, i.e. obtain:
$$ [Set, Set](h_A, F) \cong F A $$
However, ...

**4**

votes

**1**answer

232 views

### Two definitions of modules in monoidal category

The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here ...

**2**

votes

**0**answers

106 views

### Hosting Category Theory in a “universe” that is non-LFP

WHen I did my MSc, I was trained by a very talented topologist. I had a passion for the subject before and since. Now I am interested in category theory, but I seem to be very interested in the ...

**2**

votes

**0**answers

56 views

### Representing a small allegory in a tabular allegory?

Let $A$ be a small allegory (like in Freyd and Scedrov book, or in the Elephant of Johnstone), does it always exists a tabular allegory $B$ and a fully faithfull representation of $A$ in $B$ ?
I am ...

**5**

votes

**0**answers

92 views

### Which dense inclusions of sites are ∞-dense?

An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D.
Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ...

**1**

vote

**0**answers

127 views

### Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...

**11**

votes

**3**answers

554 views

### The interplay between certain aspects of interpretability, model theory and category theory

I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would ...

**2**

votes

**2**answers

179 views

### When does a pushout mediating arrow factor through the coproduct?

Setup: Let $\mathbb C$ be a category. Assume that the span $A \xleftarrow{a} X \xrightarrow{b} B$ has a pushout $A \xrightarrow{\mathsf{pinl}} A \sqcup_X B \xleftarrow{\mathsf{pinr}} B$. Let $f : A ...

**2**

votes

**1**answer

70 views

### Is every premodular category the *full* subcategory of a modular category?

In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore ...

**1**

vote

**3**answers

378 views

### Another adjoint pair: Definable sets and set-builder formulas

I see adjointness between the two concepts of "being a definable set" and "being a set-builder formula":
A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : ...

**3**

votes

**1**answer

330 views

### Does the internal axiom of choice imply Lagrange's theorem?

In a topos ${\mathcal{A}}$, given a group object $G$ and a subgroup $H$, the object $H\backslash G$ of right cosets is the coequalizer of two maps $G\times H\rightrightarrows G$, namely the group ...

**5**

votes

**4**answers

423 views

### Configuration topos?

Let ${\bf Fin}$ denote the category of finite sets. If $X$ is a topological space, then for any natural number $k\in{\mathbb N}$, the slice category ${\bf Fin}/X$ contains the configuration space ...

**5**

votes

**1**answer

228 views

### The pushout product as an operation

Motivation: In his utterly famous paper, Rezk (here, (pag. 7)) defines a structure called "Quillen ring". I'm wearing my algebraist's hat today, so I was wondering if this definition is chosen to ...