Tagged Questions
Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.
5
votes
0answers
70 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 ...
1
vote
0answers
111 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:
...
3
votes
1answer
89 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 ...
2
votes
1answer
142 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 ...
1
vote
0answers
55 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 ...
1
vote
1answer
255 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 ...
0
votes
0answers
62 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 ...
9
votes
1answer
246 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 $A$.
...
5
votes
2answers
200 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 ...
0
votes
0answers
112 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 ...
0
votes
2answers
124 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 ...
5
votes
1answer
284 views
+50
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 ...
1
vote
1answer
79 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 ...
6
votes
1answer
192 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 ...
3
votes
1answer
265 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 ...
3
votes
0answers
85 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, ...
2
votes
2answers
150 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 ...
7
votes
2answers
146 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 ...
10
votes
0answers
329 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 ...
13
votes
2answers
448 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 ...
2
votes
1answer
163 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 ...
5
votes
2answers
152 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 ...
12
votes
0answers
207 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 ...
-2
votes
0answers
205 views
Constructing model category from given category [migrated]
Given a model category $\mathcal{M}$, Goerss and Hopkins constructed a subcategory (see Structured Ring Spectra, p. 160) $\mathbf{E}$ of $\mathcal{M}$ such that:
If $X\in\mathbf{E}$ and $Y$ is ...
1
vote
0answers
122 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 ...
13
votes
1answer
638 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 ...
6
votes
2answers
93 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 ...
19
votes
4answers
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?
...
7
votes
1answer
265 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 $\mathcal{C}$ ...
2
votes
0answers
93 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 ...
2
votes
2answers
149 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, $F$ ...
3
votes
0answers
93 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 $(\mathbf{B},V)$, ...
7
votes
2answers
188 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 ...
3
votes
1answer
371 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 ...
13
votes
7answers
759 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, ...
7
votes
1answer
347 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 ...
6
votes
3answers
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 ...
4
votes
2answers
290 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 ...
3
votes
2answers
497 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 ...
1
vote
0answers
103 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 ...
2
votes
0answers
105 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 ...
4
votes
2answers
160 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?
1
vote
0answers
56 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 ...
2
votes
1answer
150 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
...
11
votes
3answers
887 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 ...
1
vote
2answers
125 views
The family of morphisms f such that Qf=identity for some functor Q
Localization of a category deals with the family of morphisms rendered invertible by a functor, i.e. the (saturated) family of weak equivalences, which I think resembles the kernel of a group or a ...
4
votes
2answers
417 views
2-category theory
I know that we can do a lot of 2-category theory, seeing 2-categories as Cat-enriched categories. Yet, I know that there are some limitations of this approach.
I also know that there are many articles ...
6
votes
0answers
143 views
Correspondences as generalized morphism between $C^*$-algebras
While reading about Morita equivalence in the category of $C^*$-algebras I met also the following notion: a correspondence between two $C^*$-algebras $A,B$ is a pair $(X,\varphi)$ where $X$ is a ...
4
votes
0answers
197 views
Reference Request for TQFTs
I had originally asked this question on Math StackExchange but have not obtained any answers, so I decided to post this here. (I have flagged the MSE post to be moved to MathOverflow, for the ...
1
vote
0answers
96 views
The category of discontinuous Banach spaces
A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
