# Tagged Questions

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### Intersection and union of torsion classes

One of the main result in
Cassidy, C., M. Hébert, and G. M. Kelly. "Reflective subcategories, localizations and factorization systems." Journal of the Australian Mathematical Society (Series A) ...

**6**

votes

**3**answers

476 views

### Category of Gödel Codings? [Reference Request]

Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with ...

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votes

**1**answer

197 views

### Establishing Duality in Tannakian Categories

I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid.
However, I find the definition of rigid categories somewhat difficult. I don't ...

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votes

**2**answers

233 views

### Serre's Theorem for Coherent Sheaves

I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...

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113 views

### When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?

Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal ...

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285 views

### Definition of internal field objects

Let $\mathcal{C}$ be a category with finite limits and a strictly initial object $\mathbf{0}$. The final object is denoted by $\mathbf{1}$. I propose the following definition of a field object ...

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110 views

### How do you categorify the cycle index series?

Let $F$ be a combinatorial species. The exponential generating series of $F$ is defined to be
$$ \sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!} $$
It was observed by Baez and Dolan in their paper ...

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127 views

### Sum-epimorphisms and prod-monomorphisms

Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
...

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votes

**1**answer

113 views

### Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?

In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory ...

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**1**answer

165 views

### strictifying tricategories

Every tricategory is equivalent to a Gray-categories. However any Gray-category is not equivalent to a 3-category. As far as I know, this is similar to the fact that braided monoidal categories are ...

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62 views

### Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$.
Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...

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**1**answer

270 views

### Are chain complexes over a field always injective?

Question: Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and ...

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65 views

### Subdivision of a small category

I am reading about subdivision of a category from this paper:
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Delgado.pdf
At the page 4, the author of this paper gives the first example ...

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**1**answer

266 views

### Tensor product of dendroidal sets: counter-examples

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.
...

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353 views

### Primitive Recursive Arithmetic via Universal Algebra

From the wikipedia article on Primitive Recursive Arithmetic (see http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic):
"Primitive recursive arithmetic, or PRA, is a quantifier-free ...

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117 views

### Reference Request: Category of explicit maps between primitive recursive sets?

[Edited]
Let $\mathsf{PR}$ be the category defined as follows:
Choose a specific presentation of Primitive Recursive Arithmetic, that is, with a specific set of terms for primitive recursive ...

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148 views

### How is a MacNeille completion “universal” like a beta-compactification is “universal”?

The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping ...

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445 views

### Why are pushouts the right tool in these setups

$\newcommand{\cat}[1]{\mathcal{#1}}$
$\newcommand{\cod}{\operatorname{cod}}$
$\DeclareMathOperator{\dom}{dom}$
$\DeclareMathOperator{\colim}{colim}$
The question is about two pushout constructions ...

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**1**answer

85 views

### Lax monoids where only the unit triangle is lax

I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper ...

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**1**answer

212 views

### In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective?
Recall that an object ...

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**1**answer

282 views

### Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?

Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume ...

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88 views

### Hilb as a Colimit in the Category of Scott Complete Categories (foundations)

Here is a paper I found by Adamek that generalizes Domain theory into categories of categories called Scott Complete Categories. The category of Scott Complete categories is denoted SCC. For years, ...

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158 views

### Computable Categories in the most direct sense?

While there is a lot of work in category related to notions of realizability and computability, etc... I've failed to find work on categories that are computable in the sense of having object and ...

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154 views

### Kan extensions in concrete 2-categories

Kan extensions make sense in any 2-category. I am interested in Kan extensions in "concrete" 2-categories consisting of actual categories with some sort of structure (e.g., finite products, finite ...

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365 views

### Lifting DG-categories to characteristic zero

The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...

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456 views

### Etale spaces using Kan extensions

Mac Lane - Moerdijk's "Sheaves" gives this cryptic hint in page 91 that the equivalence between etale spaces and sheaves on a space $X$ can be cooked up using formal methods.
More precisely, we are ...

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**1**answer

169 views

### Equivariant Derived Category

If $G$ is an connected unipotent group over $k$,and $X$ a scheme of finite type over $k$, (an algebraic closed field of positive characteristic) then we can define the bounded derived categorie of ...

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163 views

### For a quasicategory $C$, why is $\mathrm{Fun}(\Lambda^2_0,C) \to \mathrm{Fun}(\Delta^{\{2\}},C) \cong C$ a cocartesian fibration?

More generally, I expect that the following is true:
Let $D$ be a diagram quasicategory, let $d \in D$ be a vertex, and use this to define $D' = D \amalg_{\Delta^{\{0\}}} \Delta^1$. Then ...

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239 views

### The topos for forcing in computability theory

My understanding is that forcing (such as Cohen forcing) can be described via a topos. For example this nlab article on forcing describes forcing as a "the topos of sheaves on a suitable site."
My ...

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133 views

### Intuition for hereditary torsion theories

I'm looking for intuition and references for the definition of a hereditary torsion theory and two facts found here. First, the definition and facts:
Definition. A torsion theory $(\mathcal ...

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**1**answer

645 views

### Are all smooth functions composites of 0-, 1-, and 2-ary functions?

I will formalize my question in terms of algebraic theories.
Background:
Recall that an algebraic theory (in the sense of Lawvere) is a category $\mathcal{C}$ which is closed under taking finite ...

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**2**answers

127 views

### An example of two cofibrant dg categories whose tensor product is not cofibrant

I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and ...

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**4**answers

1k views

### Categorical proof subgroups of free groups are free?

This is a crossport of this question from MSE.
Is there a categorical proof that subgroups of free groups are free?
How about the result that subgroups of free abelian groups are free abelian?
...

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**1**answer

290 views

### Can any object in a presentable category be written as a colimit of generators?

Let $\mathcal{C}$ be a presentable category, and let $S$ be a set of objects such that $S$ generates $\mathcal{C}$ under colimits, i.e., such that the smallest cocomplete subcategory of $\mathcal{C}$ ...

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94 views

### Isomorphic subcategories of directed graphs and presets

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and ...

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155 views

### Given functors $F$ and $G$, does $\mathrm{Res}_F \cong \mathrm{Res}_G$ imply $F \cong G$?

Assume $F, G : \mathbf C \to \mathbf D$ be functors. Denote by $\widehat{\mathbf C} = \mathrm{Fun}(\mathbf{C}^{\mathrm{op}}, \mathbf{Set})$ the category of presheaves of sets on $\mathbf C$. Then, $F$ ...

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102 views

### Is a concretely reflective full concrete subcategory necessarily finally dense?

On p.371 of "The Joy of Cats", by J.Adamek H.Herrlich and G.E.Strecker:
Proposition 21.32
If a topological category $(\mathbf{A},U)$ is a finally dense full concrete subcategory of $(\mathbf{B},V)$, ...

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230 views

### How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?

Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original ...

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**1**answer

400 views

### Do hom-sets really live in the category Set?

This isn't really a research-level question (sorry!), but I asked on
math.se (link), and though the question was upvoted a few times, I didn't
get any answers. So since there may well be more ...

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votes

**8**answers

883 views

### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...

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**1**answer

361 views

### Is the functor of points of a scheme cofinally small?

Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full ...

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1k views

### A categorical method to, say, determine the cardinality of a group

I am trying to figure out how much one can figure out about an object using category theory. Ideally, any property that is well defined up to isomorphism should be computable using only category ...

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**2**answers

294 views

### Has this construction, which builds a symmetric multicategory from a commutative monoid, been described or studied anywhere, and if so, where?

Whenever $R$ is a commutative ring, write $R[x^{(n)}]$ for the set of all $p \in R[x]$ such that $p$ is a monic polynomial of degree $n$. Then $R[x^{(n)}]$ is not closed under sums, nor does it ...

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502 views

### A naive question about SGA4

Can someone explain to me the meaning of remark 1.1.2 at the begining of SGA4.1?
It says that if $C$ is a category that belongs to some universe $U$ (which I understand as "$\mathrm{Ob}(C)$ and ...

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116 views

### Construction of Yoneda extension (repost)

This is a reposted question to pull a bit more attention. I unfortunately could not find a detailed construction of the Yoneda extension in literaure, namely, its action on morphisms.
In "Category ...

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votes

**1**answer

129 views

### Terminology question for maps between posets

Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function.
I would like to know whether there is a name and perhaps a different characterizations of such ...

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168 views

### Colimit density and monads

Let $C$ be a cocomplete category, and suppose that it has an object that is colimit dense. Is $C$ automatically monadic over $Set$? And if not, is there an explicit counterexample?

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58 views

### Getting a measure from a premeasure through an adjoint

Let's take the category of measure spaces with objects $(X,\mathcal{F},\mu)$ and avoid the morphisms for now (I'm not sure what they should be), where $X$ is a set, $\mathcal{F}$ is a ...

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**1**answer

155 views

### Reference for “multi-monoidal categories”

I have attempted to find a definition of a monoidal category which incorporates $n$-fold tensor products instead of just binary tensor products.
Definition. A "multi-monoidal category" consists of
...

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**3**answers

912 views

### Non-abelian Grothendieck group

By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations ...