**2**

votes

**1**answer

62 views

### Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...

**1**

vote

**3**answers

202 views

### Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$
On ...

**3**

votes

**1**answer

160 views

### internal language for the 2-category of small categories

What is the internal language of the category Cat of small categories?
I found an article by Glynn Winskel and his student Mario Jose Cáccamo about such calculus! However it is limited to a fragment ...

**1**

vote

**2**answers

115 views

### Comonads from monoids

The following construction is probably known. I think it should work in any closed symmetric monoidal category, but I will play it safe and formulate the question in the concrete, cartesian closed ...

**3**

votes

**1**answer

123 views

### Exponential objects in a category of abstract automata

I'm working with a more or less standard definition of the category Aut(C) of automata over a category C (where C has finite products) which has tuples $$
A=\langle I_{A},O_{A},S_{A},\sigma_{A}, ...

**1**

vote

**0**answers

61 views

### When can I compute the simplicial mapping space from a presheaf to a simplicial presheaf naively?

Suppose that $\mathscr{C}$ is is a small category, $Y$ is a presheaf (of sets) on $\mathscr{C},$ and $X_\bullet$ is a simplicial presheaf. There is a spectrum of simplicial model structures on ...

**3**

votes

**1**answer

122 views

### How to construct a free 2-group on a groupoid?

Let G
be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor ...

**0**

votes

**0**answers

142 views

### A question about the unbounded derived category of the polynomial ring in infinitely many variables

In this moment I am trying to understand the derived category of the polynomial ring in infinitely many variables over a field $k$, $R=k[x_{1},x_{2},\dots]$ and I am wonder if it is true that ...

**3**

votes

**0**answers

121 views

### Pursuing an abelian categorical proof of the Zassenhaus Lemma

Fix an abelian category and suppose $M',M,N',N$ are sub-objects of a given object such that $M'\subset M$ and $N'\subset N$. Then there exists a canonical isomorphism
$\frac{M'+(M\bigcap ...

**2**

votes

**0**answers

190 views

### Tensor product of pullbacks of abelian categories

Does the tensor product of abelian categories commute with pullbacks? In more detail:
Let $k$ be a field. We consider $k$-linear small abelian categories ...

**5**

votes

**0**answers

99 views

### Does simplicial localization with a 3-arrow calculus 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 ...

**6**

votes

**0**answers

127 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

108 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

97 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 ...

**5**

votes

**2**answers

288 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 ...

**6**

votes

**1**answer

156 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

113 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

110 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

155 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

201 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

171 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

77 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

112 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

96 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

220 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

208 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

121 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

216 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

341 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

638 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

94 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

103 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

273 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

364 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

189 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

124 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

628 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

103 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

82 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

271 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

80 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

75 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

215 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

70 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

57 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

146 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, ...

**83**

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

351 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

124 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 ...