Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.
-1
votes
0answers
13 views
Is the Category of $\mathbb{N}$-modules a pre-quantum notion?
I have discovered the following category: the category of $\mathbb{N}$-modules. This category, I believe, has cousins by replacing $\mathbb{N}$ with other structures. Most notably, if we swap in $\...
0
votes
0answers
9 views
A monad that unions sets
Suppose we have a monad that maps types of some kind to other types, and let types be sets. Let $\alpha, \beta$ be types, $\rightarrow$ denote a function between types, and let $a : \alpha \hspace{0....
5
votes
0answers
56 views
What is a spectrum object in $\infty$-topoi?
For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?
To ...
4
votes
0answers
46 views
Categorifying Hyperoperations
Is there some categorical version of tetration or higher hyperoperations?
This is motivated by the fact that coproducts categorify addition of finite cardinals, and products/exponentials categorify ...
0
votes
0answers
52 views
The multi-set monad and modules
I am trying to analyze the category of algebras for the finite free commutative monoid monad, aka the finite multiset monad. This monad is frequently described as having a multiplication that takes a ...
8
votes
1answer
227 views
Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem
This is a crosspost of this MSE question.
In Johnstone's Topos Theory appears the following lemma.
7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two ...
3
votes
0answers
56 views
What is a category of “Lepagean equivalent” or “variation problem”?
I get to know about it form Mark Gotay's work An exterior differential system approach to the Cartan form, in that paper he defined the canonical Lepagean equivalent. The following is cited from it:
...
3
votes
1answer
138 views
$P = [-°,Set]$ is a contravariant co/lax idempotent monad, whose multiplication is determined by the unit
A unidetermined contramonad is a 2-monad $T : {\cal C}\to \cal C$ such that
$T$ is contravariant, i.e. a contravariant endofunctor;
the multiplication $\mu_A : TTA \to TA$ is determined as $T\eta_A = ...
1
vote
0answers
37 views
Localization at the left edge of a coherent left horn inclusion
Let $\Lambda^n_0 \hookrightarrow \Delta^n$ be a left horn inclusion for $n>1$. Then consider $\mathfrak{C}(\Lambda^n_0)\hookrightarrow \mathfrak{C}(\Delta^n)$, and we have a cofibration $[1]_{\...
4
votes
2answers
265 views
When is $\Omega^1$ an equivalence?
Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...
3
votes
1answer
62 views
Is the category of inclusion prespectra bicomplete?
Working in compactly generated weak Hausdorff spaces, is the category of inclusion prespectra bicomplete?
I should probably specify that by inclusion prespectra, I mean prespectra such that the ...
3
votes
1answer
101 views
Codensity monad is idempotent?
Let $j: A \to B$ be a fully faithful functor.
When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$ and thus will be idempotent, because $A$ is ...
1
vote
0answers
46 views
Codensity monad preserves some colimit?
Let $j: A \to B$ be a functor.
When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$. Since a left adjoint preserves all colimits, it is easy to ...
18
votes
3answers
347 views
What does it mean for a category to admit direct integrals?
Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals.
Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...
7
votes
0answers
88 views
Universal Enveloping algebra of a L$_\infty$ algebra
In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras ...
6
votes
1answer
347 views
Enhancing Grothendieck's universes and Grothendieck's axiom: Feferman's universe
A Grothendieck's universe is such a set $U$ so that
$\forall x \in U, x \subseteq U$,
$\forall x,y \in U, \{x,y\} \in U$,
$\forall x \in U, \mathcal{P}(x) \in U$,
given a family $(X_i)_{i \...
11
votes
2answers
280 views
Is there any significance to Bousfield localization in the non-derived context?
The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...
3
votes
0answers
104 views
Is $Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))$ for an additive category $\mathcal{C}$?
Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category.
...
0
votes
1answer
205 views
How to define 0-sphere in a category with zero object?
The 0-sphere $S^0$ is the coproduct of two points,
$$S^0 \simeq \ast \coprod \ast$$
How to define 0-sphere in a category with zero object?
Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, ...
9
votes
0answers
199 views
What are the monomorphisms of ($\infty$-)toposes?
There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...
12
votes
2answers
387 views
Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?
The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}_1=A\operatorname{-mod}$ and $\mathcal{C}_2=B\operatorname{-mod}$, ...
4
votes
1answer
116 views
Behaviour of direct limit with matrices
I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:
Let $(A_n,f_n)$ be a ...
5
votes
1answer
120 views
Dualizable presheaves with respect to Day convolution
This question was posted on MSE and got very little attention, so I'm also posting it here.
Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...
1
vote
0answers
73 views
Axiomatization of the shuffle decomposition
I am trying to figure out an axiomatization for Reedy categories such that the product of representables admits a shuffle decomposition. Cisinski suggested imposing a condition, which we will state ...
7
votes
0answers
137 views
Does each monotonic endofunctor on the category of sets and relations preserve conversion?
Consider a functor $F : \mathbf{Rel} \to \mathbf{Rel}$ that is monotonic (for all relations $R$ and $S$ with $R \subseteq S$ we have $FR \subseteq FS$). Does such a functor always preserve conversion ...
5
votes
1answer
113 views
Products, coproducts and equalizers in category of lattices
Background: Let $\mathbf{Lat}$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $\mathbf{Pos}$, that is, objects in $\mathbf{Pos}$ that have all finite ...
10
votes
1answer
238 views
Are there continua in $\infty$-topoi?
If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence....
4
votes
0answers
85 views
Koszul duality between QLS algebras and cdg algebras
A Quadratic-Linear-Constant (QLC) algebra $U$ is an algebra which can be written as $T(V)/P$ where
$T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ is the tensor algebra
and $P \subseteq k \oplus ...
4
votes
0answers
202 views
What is the category of algebras for the finitely supported measures monad?
In this post, I was introduced to the monad of finitely supported measures.
$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad.
I have three ...
10
votes
1answer
482 views
What is the correct definition of localisation of a category?
Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange.
There appears to be a discrepancy in the literature regarding the ...
6
votes
0answers
109 views
Virtual generators
Let $\mathcal{K}$ be a category.
Prop 1. If $\mathcal{K}$ has a (strong) generator then there is a faithful (and conservative) functor $U: \mathcal{K} \to \text{Set}$ preserving connected limits.
...
14
votes
2answers
1k views
Fundamental Theorem of Category Theory appropriate for undergraduates?
I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics ...
3
votes
1answer
208 views
Why must the essential image break the principle of equivalence?
I'm having trouble understanding why the "essential image" is defined the way it is.
The nlab article gives the following definition:
(A concrete realization of) the essential image of a functor $...
3
votes
0answers
89 views
Cencov's “categories of figures”
In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-...
2
votes
0answers
39 views
sub relative cell complex
This is a question concerning the proposition 11.4.8 of P.Hirschhorn's book Model categories and their localizations.
The proposition 11.4.8 is an analogous, in my opinion, to the well known ...
19
votes
2answers
619 views
What is the geometric significance of fibered category theory in topos theory?
Often in topos theory, one starts with a geometric morphism $f: \mathcal Y \to \mathcal X$, but quickly passes to the Grothendieck fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, which ...
-1
votes
0answers
56 views
The List-Domain (Co)Monad
I am back again, trying to compute a (Co)Monad, or bimonad as they are called. In particular, I want to define a functor along with a collection of natural transformations that form both a monad and ...
9
votes
0answers
197 views
Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures
I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.
Suppose we are given a (strict) pullback square
...
1
vote
0answers
61 views
What is the value of a polynomial form for a data structure, aka a Container
Data structures like Lists and Trees are often referred to as Containers. They can be given as monads and containers are polynomial functors. The List monad is well known and can be given as a ...
3
votes
1answer
221 views
Measures on sites
Let $\mathcal{C}$ be a category equipped with a Grothendieck topology $\tau$, i.e., a site.
Under which conditions on $\mathcal{C}$ can one construct a Borel $\sigma$-algebra, $\sigma_\tau$, for $\...
8
votes
0answers
204 views
Is there a citeable source for generators and relations of simplicial sets?
Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...
34
votes
2answers
742 views
What parts of the theory of quasicategories have been simplified since the publication of HTT?
It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression ...
2
votes
0answers
137 views
Derived functors and derivatives
When looking at derived functors of a non-exact functor (e.g., ext, tor, sheaf cohomology groups) I am struck by their similarity to derivatives of a non-constant function in that they are both ...
7
votes
1answer
257 views
What are the “smallest” topoi?
Yesterday I was talking to somebody from the Haskell community.
Late in the night we found ourselves discussing possible topoi.
Lets order topoi (up to equivalence, ...) by number of objects/...
6
votes
1answer
167 views
About a canonical model structure on topologically enriched categories
Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying ...
7
votes
2answers
168 views
When a localization of a category is (non-)reflective?
Let $S$ be a class of maps of a category $\mathcal{C}$. The localization of $\mathcal{C}$ at $S$ is reflective when the localization functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ has a fully faithful ...
6
votes
2answers
677 views
What's the point of “created limits”?
Seeing as there's an nLab page about creation of limits, I take it that at least some people think this is an important notion. There's also a discussion here about what the "correct" definition of ...
15
votes
1answer
294 views
When is the category of models of a limit theory a topos?
If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.
Is there a characterization of ...
10
votes
0answers
101 views
Colimits of algebras for $\infty$-Monad
I would like to know in anyone has developed method for constructing colimits in the category of algebra for a monad in the $(\infty,1)$-categorical framework, using transfinite constructions.
I have ...
7
votes
4answers
369 views
Localization of $\infty$-categories
In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...
