**2**

votes

**1**answer

44 views

### What is the cokernel of a map of presentable stable $\infty$-categories?

Let $C$ and $D$ be presentable stable $\infty$-categories, and let $f:C \to D$ be a continuous functor between them. Let $0$ be the trivial stable $\infty$-category. What is the colimit of the ...

**2**

votes

**2**answers

80 views

### Is the $\infty$-category of presentable $\infty$-categories presentable?

Let $\mathit{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and continuous functors in some universe. Is it presentable itself a larger universe?

**3**

votes

**0**answers

48 views

### Rectifying the definition of a closed category

The definition of a closed category I'm using is here.
Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal ...

**0**

votes

**0**answers

198 views

### Set as a (strict) infinite-category?

First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold:
1) trying to ...

**1**

vote

**0**answers

45 views

### Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"
In Part (2) of Theorem 19.8.4 of that book it says
If ...

**2**

votes

**0**answers

49 views

### TTF triples are recollements

The notion of recollement
$$
\mathcal{A}'
\stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}}
...

**1**

vote

**0**answers

69 views

### $n$-recollements and perverse t-structures

A recent preprint on arXiv brought my attention on the notion of $n$-recollement (def. 2) a generalization of the notion of recollement among three abelian or triangulated categories behaving like a ...

**1**

vote

**0**answers

143 views

### Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...

**3**

votes

**3**answers

399 views

### What are a couple of examples of finite sized but interesting categories?

I'm studying category theory and, given that I don't have a background in topology, I'm struggling to think of some finite categories that interesting.
The main one I know of is finite preorders -- I ...

**0**

votes

**0**answers

30 views

### Explicit computation of a limit of a cosimplicial object

Let $\Delta$ be the simplex category. Let $T_{n}$ be the standard topological $n$ simplex, i.e. it is the set of points of $\mathbb{R}^{n}$ such that $0\leq t_{1}\leq \dots \leq t_{n}\leq 1$. Its ...

**13**

votes

**2**answers

536 views

### Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...

**3**

votes

**0**answers

116 views

### When there exists some “cone” of a morphism of (ind-representable) cohomological functors?

I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups.
The question is: given a tranformation $F\to G$ of two functors of this sort, is it ...

**2**

votes

**2**answers

150 views

### A conservative, non faithful functor between triangulated categories

Suppose that we have:
1) triangulated categories $C,D$, each equipped with a $t$-structure.
2) triangulated functor $F: C \to D$ which is $t$-exact.
3) $F$ reflects isomorphisms, i.e. is ...

**2**

votes

**0**answers

90 views

### Do smooth manifolds create colimits for complex manifolds?

Suppose we have a diagram $D$ in the category $\textrm{Diff}_\mathbb{C}$ of complex manifolds, and suppose this diagram has a colimit $L$ after inclusion in the category $\textrm{Diff}$ of smooth ...

**1**

vote

**1**answer

250 views

### Van Kampen colimits

nLab uses the following definition of van Kampen colimits --- a colimit in a category $\mathbb{C}$ is called van Kampen iff it is preserved by the internal indexing functor $\mathbb{C}/(-) \colon ...

**6**

votes

**1**answer

434 views

### Lurie's Endomorphism Space vs. Endomorphisms

In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of ...

**15**

votes

**0**answers

293 views

### Is there a symmetric monoidal 2-category “SuperDuperVect”?

Recall that the category $\mathrm{SuperVect}$, as a category, consists of pairs of vector spaces, thought of as formal direct sums $V \oplus W\,\Pi$, where $\Pi$ is the "odd line". (Called "$\Pi$" ...

**2**

votes

**0**answers

101 views

### Universality of the Simplex Category

This is not a research level question, but I have not been able to find a satisfactory proof of functoriality of a certain map, and have posted this on several sites, with no luck. I wonder if anyone ...

**6**

votes

**1**answer

213 views

### Categorical definition of infinite symmetric product

Let $(C,\otimes,I)$ be a symmetric monoidal category with coequalizers and directed colimits.
Fix some object $X$ and morphism $\tau\colon I\to X.$
Using $\tau$ one can construct a sequence of ...

**3**

votes

**0**answers

82 views

### About Quillen equivalences between Bousfield localizations

Let $\mathcal{M}$ be a locally presentable category equipped with two left proper and left determined combinatorial model structures $\mathcal{M}_1$ and $\mathcal{M}_2$. There exist two sets $S_1$ and ...

**4**

votes

**1**answer

287 views

### $\Omega X$-action on spectral $X$-bundles

I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the ...

**18**

votes

**3**answers

2k views

### Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...

**8**

votes

**0**answers

358 views

### Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question.
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...

**0**

votes

**0**answers

122 views

### Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. And a repost from this MSE question.
The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the stacks project is ...

**1**

vote

**0**answers

69 views

### Monadicity of profinite algebras

We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples.
In case that I was ...

**9**

votes

**2**answers

360 views

### A cosmos where coproduct injections are not monic

The injections (coprojections) of a coproduct in a category are very often monomorphisms. For instance, this happens in any extensive category (essentially by definition) and also in any category ...

**0**

votes

**1**answer

84 views

### Trying to relate the fundamental groupoid to vector bundles

Fix a topological space $X$. Now consider a functor from the fundamental groupoid of $X$ to the category $Vect$. In other words, we assign a vector space to each point of $X$, we allow ourselves to ...

**1**

vote

**0**answers

98 views

### Unique extendable functions… Is there a theory?

I also made this post in MSE, but I think it may fit here as well.
Motivation: Take a continuous function $f$ from a topological space $X$ to a hausdorff space $Y$. If $g$ is a continuous function ...

**3**

votes

**0**answers

316 views

### 3 possible tensors in 2-categories?

let $\mathcal{A}$ be a 2-category, consider:
$$ \mathcal{A}(W \otimes_i A, X) \simeq_i \mathcal{C}at(W, \mathcal{A}(A, X))
\;\;\; i = 1, \: 2, \: 3. $$
where $W$ is a category, and $A$, ...

**1**

vote

**1**answer

294 views

### Construction of Highly Structured Quotient Groups in Quasicategories

Suppose we have a map of $E_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there ...

**8**

votes

**2**answers

294 views

### Reference for an unbiased definition of a symmetric monoidal category

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor
$$
\mathcal{C}\times ...

**11**

votes

**1**answer

213 views

### How many Fréchet manifolds are there?

Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small.
...

**5**

votes

**0**answers

157 views

### Is there a reasoned derivation of the coherence conditions for symmetric rig categories?

I know what the coherence conditions are, I can look them up in
M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72.
In theory, ...

**1**

vote

**2**answers

153 views

### Are monoids with zero and partial homomorphisms related?

Context: Let $\Sigma=\{U,C,A,G\}$ and $L\subset\Sigma^*$, i.e. $L$ is a language over the alphabet $\Sigma$. Let $\Sigma'=\{0,1\}$ and define a homomorphism $f:\Sigma^*\to\Sigma'^*$ by extending $U ...

**2**

votes

**1**answer

58 views

### Morphisms $P \to M$ in the derived category of a dg-category, if $P$ is h-projective

Let $\mathbf A$ be a dg-category. Denote by $\mathsf{C}_{\mathrm{dg}}(\mathbf A)$ the dg-category of right $\mathbf A$-modules, and by $\mathsf{C}(\mathbf A) = Z^0(\mathsf{C}_{\mathrm{dg}}(\mathbf ...

**1**

vote

**1**answer

100 views

### partial pullback-completion of a category

Let $\mathcal{C}$ be a (possibly enriched) category with all finite product, and $\mathbf{M}$ a class of morphisms.
Can one construct completion of $\mathcal{C}$ w.r.t. all pullbacks along morphisms ...

**2**

votes

**0**answers

78 views

### A generalization of the Spanier-Whitehead construction

What I call "Spanier-Whitehead stabilization" is a construction which extends a category $\bf C$ to a bigger one $\mathcal{SW}_\Omega({\bf C})$ where a given endofunctor $\Omega$ is invertible. The ...

**0**

votes

**0**answers

80 views

### Representing topoi by topological groupoids

i was reading an article written by Butz and Moerdijk (https://www.math.uu.nl/publications/preprints/984.ps.gz) and i have a problem in understanding their proof of theorem $5.1$ (The one in which ...

**3**

votes

**0**answers

77 views

### Problem with a proof in Wellfounded trees in categories

I'm reading the paper Wellfounded trees in categories by Moerdijk and Palmgren. I'm having trouble understanding the proof of Theorem 7.2 (page 216), i.e. that in $\mathbf{ML}_{< \omega} ...

**4**

votes

**2**answers

283 views

### The classifying space of an infinite totally ordered set is contractible

I asked this question on math.stackexchange, but no one answered.
Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...

**6**

votes

**1**answer

426 views

### Vectorisation of a category

I have no experience with category theory at all, but I recently stumbled upon the following construction. Since it is extremely elementary and seems rather natural, it should be known, but I have not ...

**5**

votes

**1**answer

209 views

### Can you “combine” Ord and Mon to get Cat?

Mon is the category of moniods, which can be seen as categories with one object. Ord is the category of preorders, which can be seen as categories with up to one morphism in each homset.
Is there ...

**6**

votes

**1**answer

276 views

### If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?

Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily ...

**4**

votes

**0**answers

237 views

### Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces

Warmup:
Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric ...

**1**

vote

**1**answer

148 views

### A delicate question about derived functors

Let $A\subseteq B \subseteq C$ be three triangulated categories, such that $A$ is a full triangulated sub-category of $B$, and $B$ is a full triangulated sub-category of $C = K(R)$.
Let $F: C \to D$ ...

**3**

votes

**2**answers

257 views

### How to show the following two definitions of homotopy monomorphism are equivalent?

Let $M$ be a model category. In Toen's The homotopy theory of dg-categories and derived Morita theory Page 11 it is written:
a morphism $x \to y$ in a model category $M$ is called a homotopy ...

**2**

votes

**1**answer

95 views

### A question about the morphisms in the homotopy category of dg-Cat

Let $dg-Cat$ denote the category of (small) dg-categories and $Ho(dg-Cat)$ denote the localization of $dg-Cat$ at quasi-equivalence. Using the model structure on $dg-Cat$ we can describe the morphisms ...

**5**

votes

**1**answer

130 views

### Regular epimorphisms in the category of simple undirected graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...

**13**

votes

**2**answers

366 views

### A model category of abelian categories?

Let $\mathcal{M}$ be the following category:
The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels.
The morphisms are functors that preserve the ...

**9**

votes

**2**answers

162 views

### Reflective Localizations vs. categories of local objects

Given a category $\mathcal{C}$ and a set (let's not bother with size issues here) $\mathcal{W} \subseteq \text{Mor}(\mathcal{C})$ we may form the category $\mathcal{C}[\mathcal{W}^{-1}]$ obtained by ...