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Questions tagged [ct.category-theory]

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

5
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1answer
125 views

Sheaves over a sheaf

Everything I write I mean in the in the sense of Lurie's HTT. Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C}...
4
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1answer
57 views

Flattening categories

Motivating example: Given a ring, we can construct its lattice of ideals functorially (with morphisms mapping to the preimage maps on ideals). Next, we may flatten this category of lattices, obtaining ...
3
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0answers
63 views

Motivation for using etale topology in representability of functors problems

I am reading a paper that proves the representability for certain functors whose domain is the category of superschemes. The paper claims that to prove representability of functors (or possibly just ...
14
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1answer
649 views

The state of the art in the rectification of homotopy-coherent structures

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a ...
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75 views

Vertex algebras and factorization algebras

It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "...
4
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1answer
129 views

Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$?

Given a morphism of Lie groups $ \theta:G\rightarrow H$  and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle. See that the morphism of Lie ...
8
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116 views

Adjoints to forcing

Forcing over a partial order $P$ can be viewed in a category theoretic sense as constructing the presheaf topos ${\bf Set}^{P^{op}}$ over the partial order (viewed as a category) then passing through ...
14
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1answer
492 views

Are fully general Frobenioids necessary?

Mochizuki's notion of a Frobenioid introduced in The geometry of Frobenioids I is rather elaborate. However, he also introduces a myriad of further properties that a Frobenioid may satisfy, and his ...
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1answer
110 views

Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?

In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization---the quotient of $F(UC +_{obj W} W^{op})$, the free category on the ...
2
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1answer
171 views

A module associated to an endomorphism of a vector bundle

Let $E$ be a vector bundle over a compact connected Hausdorff space $X$. To an endomorphism $\alpha \in End(E)$, we associate a $C(X)-$ module $\Gamma(E,\alpha)$ consisting of all $\beta\in End(E)$ ...
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3answers
940 views

The Kan construction, profunctors, and Kan extensions

It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "nerve-realization" adjunction $$ \text{Lan}_y F \dashv N_F = \hom(F,1) $$ that exists among the left ...
2
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1answer
181 views

Understanding the definition of $G$-gerbe

In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following. Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\...
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1answer
1k views

Locally presentable abelian categories with enough injective objects

I came to the following question when thinking about the (infinitely generated) tilting-cotilting correspondence, where it appears to be relevant. Does there exist a locally presentable abelian ...
5
votes
2answers
249 views

Semantics-structure adjunction

In the discussion on the nLab article for monadic adjunctions, John Baez suggests and Mike Shulman confirms that the relationship between adjunctions and monads itself constitutes an adjunction called ...
6
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0answers
106 views

Kleisli category with equalizers?

I was reading Hosseini's paper on "Equalizers in Kleisli Categories", yet I couldn't figure it out an example of a Kleisli category with (all) equalizers with the extra requirement of not being ...
9
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1answer
311 views

Lex $\infty$-colimits

In the paper lex colimits, Garner and Lack gave a general characterization of "exactness properties" for categories (and enriched categories). A "lex-weight" is a weight for colimits whose domain ...
7
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2answers
288 views

Explicit generating acyclic cofibrations and right properness of a model category

Let $\mathcal{C}$ be a cofibrantly-generated model category. My impression is that the following two conditions are highly correlated: $\mathcal{C}$ is right proper. There is an explicitly-...
16
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2answers
818 views

A multicategory is a … with one object?

We all know that A monoidal category is a bicategory with one object. How do we fill in the blank in the following sentence? A multicategory is a ... with one object. The answer is fairly ...
6
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1answer
161 views

Derived category of a quotient

Le $A$ be an abelian category and $B$ a Serre subcategory of $A$. Under which conditions do we have $$D^*(A/B) \simeq D^*(A)/D^*(B),$$ where $D^*$ stands for the derived category with $* = +,-,b$ or ...
7
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0answers
93 views

Stability of accessible $\infty$-categories under some operations

I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories. In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...
7
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2answers
232 views

Infinite Krull-Schmidt categories?

In a Krull--Schmidt category, if $$ X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s}, $$ where the $X_{i}$ and $Y_j$ are all indecomposable, then $r = s$, ...
7
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1answer
283 views

$\Theta$-Sets and Higher-QuasiCategories

In his well-known paper "Disks, Duality and $\Theta$-Categories, Joyal defines at the very end a $\Theta$-Category to be a cellular set suitably "fibrant". I was wondering whether someone has worked ...
25
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3answers
583 views

What is the structure preserved by strong equivalence of metrics?

Let $X$ be a set. Then we can define at least three equivalence relations on the set of metrics on $X$. We say that two metrics $d_1$ and $d_2$ are topologically equivalent if the identity maps $i:(...
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4answers
691 views

Good introduction to statistics from a algebraic point of view?

There are already lots of questions on this subject like Is there an introduction to probability theory from a structuralist/categorical perspective? Is there a combinatorial/topological treatment ...
4
votes
1answer
201 views

THC situations and the projection formula

I'm trying to find the real nature of the so-called "projection arrow" for $f^* \dashv f_* $: $$ f_*(A\otimes f^*B)\leftarrow f_*A\otimes B $$ which can be found in any monoidal category $(\mathbf V,\...
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1answer
498 views

Is there any work related to Hypertype Theory? [closed]

Update: Since the question was put on hold, work on this will continue on this notebook. So long, and thanks for all the fish!   First, let me state that I am not a formally-trained ...
2
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0answers
79 views

The Kleisli Category of the Monad of Measures of Finite Support and its composition formula

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. Let's call this ...
3
votes
1answer
277 views

Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that, If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable? I could not find the statement (...
2
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0answers
179 views

Are there enough curves (to connect 'points' of f.g. algebras)?

(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the ...
8
votes
3answers
1k views

Is the category of small categories locally presentable?

I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be ...
3
votes
0answers
125 views

final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...
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1answer
212 views

Does the Eilenberg Moore Construction Preserve fibrations?

Say we have a Grothendieck fibration $p : E \to B$ and a monad $T$ on $B$ and a lift $T'$ of $T$ to $E$, i.e. a monad on $E$ such that $pT' = Tp$ and $p$ preserves $\eta, \mu$. Then because the ...
5
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1answer
180 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 ...
7
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2answers
609 views

From tensor algebras in monoidal categories to (commutative?) monoids

Let $D$ be a monoidal category (without the structure of a symmetric monoidal category), with unit object $Id$, and let $L$ be an invertible object in $D$, so that $L$ is dualizable and the pairing ...
11
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1answer
461 views

Is every representable map a submersion?

Recall that a morphism $f:C \to D$ in a category $\mathscr{C}$ is representable if for all maps $g:E \to D$ in $\mathscr{C},$ the pullback $C \times_{D} E$ exists. Let now $\mathscr{C}$ be the ...
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0answers
121 views

Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable

I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu. Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open ...
6
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1answer
362 views

$\Pi$, $\Sigma$, and identity types without $\eta$ in comprehension categories

In comprehension categories, dependent sums are defined as a choice of left adjoints for all reindexing functors along display maps, satisfying a Beck-Chevalley condition. Dependent products are ...
0
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1answer
142 views

A question about combinatorial model categories

I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.) He uses this result to prove Jeff ...
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0answers
79 views

Relation between the inverse of a natural isomorphism and counit of an adjunction

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories. Suppose that $F\colon\mathcal{C}\to\mathcal{D}$ is a functor from $\mathcal{C}$ to $\mathcal{D}$; $U\colon\mathcal{D}\to\mathcal{C}$ is a ...
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0answers
89 views

Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?

Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
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66 views

Notions of connected components in a finite family fibration

Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...
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2answers
324 views

Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...
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0answers
74 views

What minimal structure is required to define Clifford modules in a way as abstract as possible?

Start with a quadratic form $q$ on a vector space $V$. A module $M$ over the corresponding Clifford algebra is determined by a map $\cdot:V\otimes M\to M$ satisfying $v\cdot(v\cdot m)=-q(v)m$. Now ...
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257 views

The “concreteness preorder” on categories

A category is called concrete if it is equipped with faithful functor to $\mathrm{Set}$. Let us say a category $C$ can be made concrete if there exists a faithful functor $F \colon C \to \mathrm{Set}$...
9
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1answer
514 views

Higher Topos Theory Theorem 2.2.5.3

The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem. We have a trivial Kan ...
2
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1answer
178 views

Category with one isomorphism class

What one can say about a category in which, any two objects are isomorphic. I know that this is a strange question. Thanks
6
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1answer
210 views

Does the Dwyer-Kan model structure make dgCat a model $2$-category?

Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category. Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
6
votes
1answer
270 views

Is the embedding ${\sf cat}\subset {\sf Cat}$ dense?

A quick question. Let $j : {\sf cat} \to {\sf Cat}$ be the inclusion of small categories in locally small categories; is $j$ a dense functor? If it is not (as I expect, since it's hard for a ...
4
votes
2answers
284 views

Limits, colimits and universes

For many purposes in category theory, we consider limit and colimits of diagrams $F\colon\mathsf{J\to C}$ where $\mathsf{J}$ is small category, that is, a category where the classes of objects and ...
6
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1answer
225 views

Does there exist a full and faithful embedding of $\mathsf{Poset}$ in $\mathsf{Set}$?

Does there exist a full and faithful embedding of the category of posets into the category of sets? I suspect no, but I don't know how to prove or disprove this.