Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,394
questions
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822
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About the definition of quasi-equivalent dg-categories
In the article Grothendieck Ring of pretriangulated categories by Bondal-Larsen-Lunts, two dg-categories $\mathcal A$ and $\mathcal B$ are called quasi-equivalent if there is a chain of dg-categories ...
- 2,069
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votes
1
answer
521
views
What is the right way to define the nerve of an unbiased monoidal category?
I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...
- 7,107
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votes
2
answers
352
views
How strict can I be in the definition of "2-group"?
Recall that a group is an associative, unital monoid $G$ such that the map $(p_1,m) : G \times G \to G\times G$ is an isomorphism of sets. Here $p_1$ is the first projection and $m$ is the ...
- 53.1k
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169
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Variation on definition of logical functors avoiding power objects
Without power sets in meta-theory not every Grothendieck topos is an elementary topos, Set is still Grothendieck, but it lacks power objects.
Now I am looking for a definition of a logical functor ...
- 699
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140
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Does the 2-category of double categories and vertical transformations have flexible limits?
Consider the 2-category of pseudo-double categories (with the weak composition in the horizontal direction and the strict composition in the vertical direction), strong double functors, and vertical ...
6
votes
1
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195
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In a weak factorization system, the left class is left cancellative iff the right class is what?
Let $\mathcal K$ be a locally presentable category, and let $(\mathcal L, \mathcal R)$ be a cofibrantly-generated weak factorization system thereon. We say that $\mathcal L$ is left cancellative if $...
- 61.6k
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Checking 2-dualizability
Let $(\mathcal C, \otimes, I)$ be a symmetric monoidal 2-category, and let $X \in \mathcal C$ be a dualizable object, with dual $X^\vee$, unit $coev: I \to X \otimes X^\vee$, and counit $ev : X^\vee \...
- 61.6k
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2
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425
views
Domains that may require a good categorical background
I'm a PhD student in category theory, more specifically I study 2-dimensional category theory, that means bicategories, pseudofunctors, careful definitions of various structures you can put on this ...
- 351
6
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1
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779
views
Domain of left adjoint from condensed sets to anima
$\DeclareMathOperator\Hom{Hom}$Let $X$ be a condensed set in the sense of Clausen-Scholze. If there is a universal anima $Y$ (or $\infty$ groupoid, or homotopy type) together with a map of condensed ...
- 386
6
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1
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167
views
Subobject classifier in $\mathsf{Top}^{D^{\text{op}}}$?
Let $D$ be a small category. Does the category of diagrams $\mathsf{Top}^{D^{\text{op}}}$ have a classifier of (strong?) subobjects? I tried following the "sieve construction" for the ...
- 637
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236
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Stability properties of essential geometric morphisms
Notation.
$\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints.
$\mathsf{Topoi}_{\text{ess}}$ is the bicategory of topoi, essential ...
- 8,186
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206
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What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified?
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 1,685
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1
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522
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When does the loop functor $\Omega^\infty:Sp(C) \rightarrow C$ commute with filtered colimits?
Let $C$ be a pointed $\infty$-category which admits finite limits.
Let $Sp(C)$ denote the $\infty$ category of spectrum objects. One way to define, i.e. 1.4.2.24, is by taking the homotopy limit in $...
- 343
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159
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Tensor product of unit and co-unit in a closed compact category
Consider a compact closed category, i.e., a symmetric monoidal category with a unit $\eta$ and co-unit $\epsilon$. It seems natural to demand that the tensor product of two units (for different ...
- 2,901
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465
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Category of spaces/sheaves
Consider the following category $\mathcal C$:
An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$.
A morphism $(X,\mathcal F)\to(Y,\mathcal G)$...
- 18.3k
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215
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monoidality of $ A\otimes (-) $ with $ A $ monoid belonging to the center
Let $(\mathcal{C}, \otimes)$ a monoidal category, and $(A, m, e)$ a monoid (where $m: A\otimes A\to A$, $e: I\to A$ ecc. ), with $(A, u)$ belonging to the centre of $(\mathcal{C}, \otimes)$: $u: A\...
- 4,125
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335
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What is the formality behind passing from Number Fields to Number Rings
In algebraic number theory, one constructs for each number field $K$ a subring $\mathcal{O}_K$, the integral closure of $\mathbb{Z}$ inside $K$, which carries many of the properties which make $\...
user30211
6
votes
1
answer
309
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}...
- 595
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272
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Comonad for normalized pseudofunctors for strict higher categories
Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{...
- 19.4k
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347
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Complexes in stable categories
Generalizing from 1-category theory, there's a simple definition of a "naive complex" in a stable $\infty$-category. Considering bounded positive graded chain complexes, they are a sequence of maps
$$...
user13113
6
votes
1
answer
388
views
Do topological commutative monoids model all 0-connective spectra (after group completion)?
Of course, before group completion, topological commutative monoids do not model all connected $E_\infty$ spaces -- among the grouplike ones, they model only products of Eilenberg-Mac Lane spaces. But ...
- 61.6k
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530
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Is there a theory of decomposition into indecomposables? What's the relation to idempotents?
Call a nonzero object of a pointed category simple if it has no proper quotients, and indecomposable if it's not the product of two objects (dual to connected).
Idempotents seem to pop up in many ...
- 10.3k
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276
views
What structure of a monoidal simplicial model category is preserved by taking the opposite category?
Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, etc.)...
- 10.1k
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votes
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456
views
Universal covering and double cover functors
Initially posted on MSE
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...
- 319
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1
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265
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Relations between functors in a recollement
Consider a recollement situation like the following
by the very definition of the various functors it follows that $i^* j_*=0$, and $j^! i_* = 0 = j^* i_!$. Also, $j^! i_! = 0 = j^* i_*$ by ...
- 13.2k
6
votes
1
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831
views
Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?
A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...
- 6,169
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votes
1
answer
617
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Generalization of analytic functors
It's been a long time since I posted the following question on stackexchange.
Now I think it's better to adress it to you, in the hope I will reach the right audience: Martin's comment, albeit useful,...
- 13.2k
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votes
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332
views
When is this braiding not a symmetry?
Given a topological space $X$ instead of forming the fundamental groupoid $\pi(X)$ which is the category whose objects are the points and morphisms the homotopy classes of paths one can also form the ...
- 1,644
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votes
1
answer
484
views
abelian group objects category
Suppose that $\mathcal{C}$ is a cartesian closed category. When is the category of abelian group objects $\mathcal{Ab}(\mathcal{C})$ a symmetric monoidal closed with respect to something substituting ...
- 205
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votes
1
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492
views
Presheaves on a complete Segal space
Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which ...
- 65.1k
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2
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455
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Does the dual of an object with trivial symmetry also have trivial symmetry?
Let $C$ be a symmetric monoidal category. I am interested in objects $X \in C$ such that the symmetry
$S_{X,X} : X \otimes X \cong X \otimes X$
is equal to the identity. There are many examples of ...
- 61.5k
6
votes
2
answers
748
views
Subobject-poset (co-)homology
Given a group, there is another way to define its "(co-)homology" using a classifying space. Specifically, one takes the partially ordered set of its proper non-trivial subgroups (if they exist), and ...
- 468
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1k
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First Quantization is a mystery... but de-quantizing perhaps not
There is an well-known infamous DICTUM:
-Second Quantization is a functor, First Quantization is a mystery-.
Indeed, second quantization is the "Fock functor", which builds the Fock space in a ...
- 7,613
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1
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487
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Fill in the blanks: "1Cob is the free ____ category on a ____"
This is probably straightforward, but I'm having trouble writing down a precise statement. "Everyone knows" that the cobordism category $\text{2Cob}$ (all manifolds compact and oriented) is the free ...
- 115k
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570
views
Origin of "versal"?
Any number of constructions guarantee the existence of maps $f$ without
guaranteeing their uniqueness. Some time ago, I was introduced to the terminology "versal" for such a construction.
I wonder: ...
- 12.5k
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143
views
Is there a natural way to give a bisimplicial structure on a 2-category?
I mean by the nerve functor.
Given a 2-category $\mathcal{C}$, if we forget the 2-category structure (just view $\mathcal{C}$ as a category), the nerve functor will give us a simplicial set $N\...
- 111
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votes
2
answers
524
views
Embedding abelian categories to have enough projectives
Is it true that the pro-objects of an abelian category form a category with enough projectives?
In general, given an abelian category A, is there a canonical way to embed it a bigger abelian ...
- 25.3k
6
votes
1
answer
220
views
Dissolution of a topos
The dissolution of the locale associated with a frame $F$ is the locale associated with the frame $N(F)$ of nuclei of $F$ (see, e.g., Johnstone, “Stone Spaces” (1982), §2.5). Note that there is a ...
- 30.2k
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votes
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236
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When is the Grothendieck / category of elements construction a fibration on geometric realizations?
Suppose we have a simplicial complex / poset / small category without loops $X$ equipped with a functor $F$ into the category of posets / small categories without loops. Suppose further that for each ...
- 1,986
6
votes
1
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793
views
Solid tensor product of pro-discrete space with Laurent series
Consider the category $\operatorname{Solid}_{\mathbf{Z}}$ of solid abelian groups in the sense of Clausen-Scholze. This category is a full subcategory of condensed abelian groups, $\operatorname{Cond}...
user108998
6
votes
1
answer
706
views
Examples of (co)ends
I am reading texts about (co)ends, and everywhere I see a lack of examples. I am not an expert in this area, and without examples it is difficult for me to use my intuition to grasp the idea. MacLane ...
- 7,274
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398
views
Is there a Grothendieck correspondence for sheaves/stacks?
Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories
$$
\mathsf{DFib}(\mathcal{C})
\cong
\mathsf{PSh}(\mathcal{C}),
$$
whereas the Grothendieck ...
- 10.7k
6
votes
1
answer
151
views
Regularly well-powered iff regularly co-well-powered?
Let $\mathcal C$ be a finitely complete, finitely cocomplete category. Then the following are equivalent:
$\mathcal C$ is regularly well-powered (i.e. every $C \in \mathcal C$ has a small set of ...
- 61.6k
6
votes
2
answers
232
views
When is a locally presentable category (locally) cartesian-closed?
Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is ...
- 8,755
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votes
1
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320
views
When is an object determined by the number of maps from the other objects?
Let $C$ be a category with finite hom-sets.
Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition).
For which categories $C$ does it follow ...
- 347
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votes
1
answer
435
views
Comonoids in the category of monoids
Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids?
...
- 5,402
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votes
1
answer
224
views
Is the category of atomless Boolean algebras with complete embeddings closed under coproducts?
Consider the category whose objects are atomless Boolean algebras (not necessarily complete) and whose arrows are complete embeddings.
Does a coproduct exist in this category for any two atomless ...
- 1,143
6
votes
1
answer
348
views
The category of complexes over a dg-algebra is Grothendieck (it has a generator)
Let $A$ be a dg-algebra over some commutative ring $k$. We have an abelian category $\mathrm{C}(A)$ of (right) $A$-dg-modules. I've read in a few sources that $\mathrm{C}(A)$ is a Grothendieck abelian ...
- 2,069
6
votes
1
answer
203
views
Non-trivial factor splitting from vacuum in TQFTs. F-move not unity?
Consider a unitary modular TQFT, defined by the F and R moves. More specifically, a braided tensor category relevant for anyon models in 2D topologically ordered phases of matter. I am interested in ...
- 295
6
votes
2
answers
694
views
When is a fold monomorphic/epimorphic
Given a functor $F : \mathcal C \to \mathcal C$ with initial algebra $\alpha : FA \to A$, and another algebra $\xi : FX \to X$, we obtain a unique morphism $\mathsf{fold}~\xi : A \to X$ such that $\...
- 469