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Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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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
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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
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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
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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
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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
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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
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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
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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 = ...
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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
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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, $\mathbf{I}$, ...
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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
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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
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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
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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}, \...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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-...
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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
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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 ...
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...