**6**

votes

**0**answers

133 views

### Coherent sheaves and Mitchell's embedding theorem

Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...

**3**

votes

**0**answers

112 views

### When does prolongation preserve sheaves?

Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will ...

**1**

vote

**0**answers

102 views

### Retracts of 2-categories

Let $\mathbf{C}$ be a strict $2$-category and let $M = \{(x_\alpha,y_\alpha)\}$ be a collection of object-pairs so that each hom-category $\mathbf{C}(x_\alpha,y_\alpha)$ has an initial element ...

**6**

votes

**2**answers

298 views

### When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...

**7**

votes

**1**answer

166 views

### When does simplicial localization commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...

**3**

votes

**0**answers

118 views

### When do localizations of presentable (infinity) categories commute?

Suppose that $\mathscr{P}$ is a (locally) presentable $\left(\infty,1\right)$ category (which we can assume WLOG is infinity presheaves on some small $\left(\infty,1\right)$ category) , and $R$ and ...

**6**

votes

**0**answers

116 views

### When does a sheaf of categories represent a homotopy sheaf?

Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...

**-1**

votes

**2**answers

163 views

### Basic category theory: Universality of adjunction unit is justified by Yoneda Proposition in Mac Lane's text [closed]

Background
I am reviewing some category theory, which I did not learn too well the first time around. One text I am using is Mac Lane's. Near the beginning of the chapter on adjunctions (pg 80),
he ...

**3**

votes

**1**answer

207 views

### How do we handle the symmetry condition in nCob and TQFTs?

A $(n+1)$-topological quantum field theory $\mathcal{T}$ is a rigid symmetric monoidal functor from the category $(n+1)$-Cob of $n$-manifolds and $(n+1)$-cobordisms to FdVect.
My question is about the ...

**2**

votes

**1**answer

178 views

### Universal ribbon category of ribbon graphs

I'm skimming through Turaev's "Quantum invariants of knots and 3-manifolds". One of the main results is Theorem 2.5. In my opinion, this Theorem is conceptually half-baked: 1) The ribbon structure on ...

**3**

votes

**1**answer

89 views

### Strictifying strong monoidal functors

Let $C_1$ and $C_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C_1 \rightarrow C_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal ...

**0**

votes

**2**answers

118 views

### Conventional notation for the probabilistic functor

The probabilistic functor $P$ sends a measurable space $X$ to the space of probability measures on $X$ endowed with $\sigma$-algebra generated by evaluation maps, and measurable maps $f:X\to Y$ to ...

**3**

votes

**0**answers

99 views

### On the preservations of certain colimits (by covers) under simplicial localization (of a category of fibrant objects)

Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of ...

**4**

votes

**2**answers

268 views

### Maximum cardinality of a filtered limit of finite sets

Let $(I,<)$ be a directed, partially ordered set. Consider an inverse system $(S_i)_{i \in I}$ of finite sets, i. e. a functor $S:I^{op}\to \mathbf{FinSet}$. What is the maximum possible ...

**2**

votes

**2**answers

210 views

### Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?

The question is in the title, here is my motivation:
$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if ...

**3**

votes

**1**answer

128 views

### Infinite Dimensional Weil Restriction and Ind-scheme

I'm studying the Weil Restriction $\mbox{Res}_{k/\textbf{F}_p}$ where $\textbf{F}_p$ is the field with $p$ elements and $k$ is a perfect field over $\textbf{F}_p$, not necessarily finite.
In this ...

**9**

votes

**1**answer

228 views

### Verdier localization for stable $\infty$-categories

Verdier localization is one of the more intuitive ways to localize a triangulated category, "killing" a suitable class of objects via a functor which is universal with respect to this property.
I ...

**4**

votes

**1**answer

348 views

### Proof without using Yoneda's lemma?

Let $\mathscr{T}$ be atriangulated category.
The third axiom for triangulated categories, namely,
if in the diagram
$$\begin{array} 0X ...

**2**

votes

**4**answers

661 views

### Are there natural examples of non-symmetric Frobenius algebras?

Symmetric Frobenius algebras arise everywhere, but the non-symmetric variety seem difficult to come by. Are there any natural examples/constructions that produce non-symmetric Frobenius algebras in ...

**0**

votes

**0**answers

95 views

### How to prove a Lefschetzesque principle for relative homology and absolute homology

Here is my conjecture:
If H is some homology theory from some abelian category A to some abelian category B, E is the class of all epimorphism in A, E' is some closed class of epimorphisms in A and ...

**2**

votes

**0**answers

107 views

### Which reflexive coequalizer diagrams are projectively cofibrant?

Consider the walking reflexive pair category W,
which consists of two objects 0 and 1 and three generating
morphisms f: 0→1, g: 0→1, and h: 1→0
satisfying the relation fh=gh=id₁.
Consider the ...

**14**

votes

**5**answers

1k views

### What's special about the Simplex category?

I have been wondering lately what makes simplicial sets 'tick'.
Edited
The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of ...

**2**

votes

**1**answer

277 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

**3**

votes

**1**answer

369 views

### building a product of two categories [closed]

MacLane, as well as probably any other category book, does not hesitate to define a product of two categories as a category consisting of pairs of objects, etc.
Now my question is: what law of nature ...

**3**

votes

**1**answer

191 views

### How does associativity get twisted by elements of $H^3(G)$?

In Braided Monoidal Categories by Joyal and Street, §6 a monoidal category is $V =T(G,M,h)$ built using a recipe:
objects are are elements of $G$ ✓
$V_0(x,y) = M$ if $( x=y)$ or else ...

**6**

votes

**0**answers

133 views

### Picard-Brauer exact sequence for infinity categories

This question may be very naive, or the answer may be well-known. In any case, a good amount of googling did not bring up anything useful (maybe I'm using the wrong words?).
If $f:A\to B$ is a ...

**8**

votes

**3**answers

662 views

### Grothendieck's Homotopy Hypothesis - Applications and Generalizations

Grothendieck's homotopy hypothesis, is, as the $n$lab states:
Theorem: There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$.
What are the ...

**3**

votes

**1**answer

109 views

### Khovanov $sl_2$ homology of a connected sum of some torus knots

Let $T_{p,q}$ be the (p,q) torus knot. Could anybody possibly compute either unreduced or reduced Khovanov $\mathfrak{sl}(2)$ homology of the connected sum $T_{2,3} \sharp T_{3,4}$ of the (2,3) and ...

**5**

votes

**0**answers

89 views

### Non-degenerate limits of topoi

Let $\mathcal{E}$ be an elementary topos with natural numbers object $\mathbb{N}$, and let $\mathbb{C}$ be a cofiltered internal category in $\mathcal{E}$. Suppose we have a diagram of shape ...

**13**

votes

**1**answer

300 views

### Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory

Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...

**0**

votes

**0**answers

82 views

### Merging / combining categories

Given a category X (that will be used as an underlying category) and collection of categories C(i) for i in I with a faithful functor from each C(i) to X, a category C is called the a fibered product ...

**1**

vote

**0**answers

79 views

### A factorization system on ${\rm Ch}(R)$

This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Ch}(R)$ be the category of chain complexes of $R$-modules (eventually bounded).
...

**2**

votes

**1**answer

219 views

### Examples of functors $\mathbf{Set} \to \mathbf{Set}$ which are not analytic

Let $\mathbb{B}$ denote the groupoid of finite sets and bijections.
A functor $F : \mathbf{Set} \to \mathbf{Set}$ is analytic if it is the left Kan extension of some functor $G : \mathbb{B} \to ...

**3**

votes

**0**answers

73 views

### Regarding the definition of S-flows over a category (given a monoid S)

(This question was originally directed to Simone Virili, referring to the answer http://mathoverflow.net/a/103840/2926, but could also be addressed to the greater community.)
I was wondering if you ...

**1**

vote

**0**answers

61 views

### Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions
$\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...

**1**

vote

**0**answers

148 views

### Coarse moduli spaces and rational points [closed]

Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...

**84**

votes

**9**answers

4k views

### What non-categorical applications are there of homotopical algebra?

(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)
More ...

**3**

votes

**0**answers

356 views

### An exact sequence which does not split

Let $X$ and $Y$ be indecomposable modules over a finite dimensional algebra and let $f \colon X \to Y$ be a non-zero morphism which is neither a monomorphism nor an epimorphism.
Suppose that it is ...

**3**

votes

**1**answer

128 views

### Pullbacks of $C^*$-algebras

I am reading the paper of Pedersen: "Pullback and Pushout Constructions in C^*-Algebra Theory". I try to work out the arguments from Proposition $3.1$ of his paper (you can find this article in the ...

**7**

votes

**1**answer

182 views

### Localizing 2-categories about a single morphism

This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and ...

**10**

votes

**0**answers

215 views

### Goodwillie calculus and morphisms of functors

Let $F,G: \mathcal{T}\to \mathcal{S}$ be two functors from topological spaces to spectra (or topological spaces) and let $s: F\to G$ be a morphism between them.
Suppose $F$ and $G$ are analytic and ...

**1**

vote

**0**answers

130 views

### Axioms for a symmetric monoidal bicategory

I start reading the axioms for a symmetric monoidal bicategory. The axioms include so many diagrams to be satisfied. I am wondering if people really use these axioms directly to check a given data is ...

**5**

votes

**1**answer

201 views

### formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian ...

**1**

vote

**0**answers

68 views

### Can a category's partial monoid of arrows be completed to a total monoid?

Categories can be presented in the language of arrows only, without reference to objects (as discussed, say, here: Categories presented with Arrows only, no objects: partial monoids), and this is a ...

**8**

votes

**1**answer

233 views

### What is this operad-like structure called?

I'd like to know what's the name (if any) of the following categorical structure, and also references where it has been considered.
Given a category $C$, let $O=\{O(n)\}_{n\geq 0}$ be a sequence of ...

**5**

votes

**2**answers

342 views

### Homotopy factorization of morphisms of chain complexes

This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...

**6**

votes

**1**answer

329 views

### What information is lost in $X \to \mathrm{Sh}(X)$?

Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure?
Thanks, Adrian.

**10**

votes

**1**answer

349 views

### Categorical interpretation of disjoint cycle notation for tracing permutations

For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$
A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...

**0**

votes

**1**answer

142 views

### Gauss Sums over “semisimple spherical tensor category”?

I read on the arXiv the following:
Let $\mathcal{\mathbf{C}}$ be a semisimple spherical tensor category with simple unit and let
$\mathbf{\Gamma}$ be the set of isomorphism classes of simple ...

**2**

votes

**0**answers

100 views

### Is the following a sufficient condition for being a primal algebra?

I have a question regarding universal algebra and, in particular, primal algebras:
Suppose that A is a finite simple algebra with no proper subalgebra, no automorphism except the identity map, with a ...