Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
1,788
questions with no upvoted or accepted answers
6
votes
0
answers
113
views
Properties of the Yoneda extension of $F$ from properties of $F$
Given a presheaf $F: \mathcal{A} \to \text{Set}$, a classical result shows a connection between the limits preserved by $F$, the limits preserved by its Yoneda extension, and a property of the ...
- 13.2k
6
votes
0
answers
98
views
Conservative $\text{Ho}\mathcal{V}$-functor
Let $\mathcal{M}$ be a $\mathcal{V}$-model category.
Which hypotheses on $\mathcal{M}$ guarantee the existence of a conservative $\text{Ho}\mathcal{V}$-functor $$\text{Ho}\mathcal{M} \to \text{Ho}...
- 8,186
6
votes
0
answers
162
views
Cartesian liftings in double categories
The question: I wonder whether the following definition, or something similar, has appeared somewhere (see below for motivations). Any reference or pointer is welcome!
(In what follows, I denote ...
- 1,536
6
votes
0
answers
287
views
Representation-finiteness vs. $\tau$-tilting-finiteness
Setting: Throughout, $\Lambda$ is a finite dimensional associative algebra, $\operatorname{mod} \Lambda$ is the category of all finitely generated left $\Lambda$-modules, and all subcategories are ...
- 483
6
votes
0
answers
449
views
What are the most general conditions under which the inverse image of sheaves of abelian groups has a left adjoint?
If $f: E \to X$ is an étale map (a local homeomorphism), then the inverse image of sheaves of abelian groups $f^{-1}$ has a left adjoint, as shown by Roland in his answer here. This subsumes as a ...
- 5,933
6
votes
0
answers
393
views
Hodge Realisation of Mixed Tate Motives
For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper,
https://www.uni-due.de/~bm0032/publ/TateMotives.pdf
There exists an abelian category of mixed Tate ...
- 2,961
6
votes
0
answers
185
views
Mixed Hodge modules of product spaces
Let $X$ be an algebraic varietiy (as good as you want, say affine and smooth) and let us denote by $MHM(X)$ the category of mixed Hodge modules as descrived by Saito (see for example this or this).
...
- 497
6
votes
0
answers
129
views
What are regular epimorphisms of coalgebras?
It is known that epimorphisms between coalgebras are surjective maps. My question is
What are regular epimorphisms between coalgebras?
I am in particular ...
- 2,302
6
votes
0
answers
315
views
Freed-Hopkins-Lurie-Teleman topological boundary conditions v.s. Lagrangian subspace
This question concerns the comparison of topological boundary conditions of TQFTs on a manifold with some boundary.
For example, we can consider defining the TQFT on a $D^3$ ball with a topological ...
- 10.3k
6
votes
0
answers
145
views
Model structures on varieties of algebras
I say that a category of (say) algebras for a monad[¹] $\text{Alg}(\mathbb T)$ is "uninteresting" if the only model structures on $\text{Alg}(\mathbb T)$ result as transfer of the nine model ...
- 13.2k
6
votes
0
answers
163
views
Under Vopenka, Is every weak orthogonality class in a locally presentable category small?
This is true for orthogonality classes- see Corollary 6.24 in Adamek and Rosicky - but I can't seem to find this result in the literature for weak orthogonality.
Here, by a weak orthogonality class ...
- 61.5k
6
votes
0
answers
168
views
What are the regular monomorphisms and regular epimorphisms of the category of smooth premanifolds?
A premanifold is a locally ringed space locally isomorphic to an open subset of Euclidean space equipped with its sheaf of smooth functions. No assumption of paracompactness or the Hausdorff property.
...
- 10.3k
6
votes
0
answers
238
views
Derived functors via coends, again
Some time ago I asked what the co/end of Ext/Tor functors was.
Now I want to play with the same items arranged in different ways: let's start from the fact that the composition maps
$$
\hom(A,X)\...
- 13.2k
6
votes
0
answers
1k
views
Verdier Quotient a quotient?
This question seems trivial, so hopefully it will be resolved quickly.
As pointed out in this question on quotient categories and localization, the two constructions are sometimes related, but in ...
- 2,336
6
votes
0
answers
496
views
N-periodic derived categories
I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places.
Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...
- 18.3k
6
votes
0
answers
166
views
Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II
Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...
- 137k
6
votes
0
answers
477
views
A definition of the homotopy colimit of a coherent diagram
Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...
- 2,270
6
votes
0
answers
247
views
Test categories applied to Dold-Kan correspondence?
Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...
- 211
6
votes
0
answers
310
views
Many-sorted nominal sets as sheaves
The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...
6
votes
0
answers
223
views
$\mathcal{M}(\mathcal{D}_X)$ and $\mathcal{M}^r(\mathcal{D}_X)$ have natural tensor category structures?
Write $\mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S})$ for the category of left $\mathcal{D}$-modules over $X$ and $\mathcal{M}^r(\mathcal{D}_{X/S})$ for the category of right $\...
user78172
6
votes
0
answers
168
views
How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?
It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...
- 8,707
6
votes
0
answers
324
views
(Co)limits in the category of diffeological spaces vs. category of smooth manifolds
I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
- 5,933
6
votes
0
answers
445
views
category theoretic approach to Sylow theorems and finite group theory?
Is there a category theoretic approach to Sylow theorems?
More generally, is there an exposition of finite group theory in terms of category theory which would include Sylow theorems and little facts ...
- 161
6
votes
0
answers
314
views
What is the technical difference between a deformation and a perturbation?
What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?
- 3,850
6
votes
0
answers
169
views
Preservation of universes in presheaves
In Lifting Grothendieck universes Hofmann and Streicher construct a universe in the category of presheaves over a small category given a Grothendieck universe in $\mathbf{Set}$.
Suppose now I have ...
- 408
6
votes
0
answers
252
views
Are regular epi of locale stably epic?
It is well know that the category of locales is not a regular category, that is the pullback of a regular epimorphism is not always a regular epimorphism: for example, the classical counterexample ...
- 40.2k
6
votes
0
answers
339
views
Limits and colimits of A_{\infty} categories
I have a question related to the discussion (Coequalizer in category of dg-algebras). How do you prove that the category of (small) dg-categories and the category of (small) A_{\infty} categories are ...
6
votes
0
answers
282
views
What are the Čech-local equivalences of (simplicial pre)sheaves?
Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield ...
- 14.9k
6
votes
0
answers
363
views
Construction of 3D Topological Quantum Field Theory from a 2D Modular Functor
I have been reading Bakalov and Kirillov's Lectures on Tensor Categories and Modular Functors in which the authors state that a direct construction of a $C$-extended 3D TQFT from a $C$-extended 2D ...
- 61
6
votes
0
answers
230
views
Nice references for injective model structures and Quillen functors between motivic homotopy categories
It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...
- 16.5k
6
votes
0
answers
310
views
Extension to a right exact functor
Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...
- 61.5k
6
votes
0
answers
368
views
How to compute the abelianization of the representation theory of a Hopf algebra?
I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...
- 53.1k
6
votes
0
answers
313
views
How do I find abelian subcategories of periodic triangulated categories?
If $T$ is a triangulated category, then the formalism of $t$-structures gives a way to find abelian subcategories inside. You're supposed to find two strictly full subcategories, $T^{\le 0}, T^{\ge 0}...
- 8,643
6
votes
0
answers
338
views
Compatibility between strength and costrength of a monoidal monad
Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A ...
- 61.5k
6
votes
0
answers
570
views
Factoring objects in a category
Normally, one builds products in a category. Here I am asking about the inverse operation. Let me be precise.
Given a (monoidal) category $\mathcal{C}$ and an object $X$ of $\mathcal{C}$, does ...
- 11.7k
6
votes
0
answers
501
views
Is there a version of the 2d cobordism hypothesis for surfaces with non-empty incoming and outgoing boundary?
Question: Is there a condition on an object $x$ of an $(\infty,2)$-category $\mathcal C$ which is equivalent to $x = Z(pt_+)$ for a unique TFT $Z$ from the $(\infty,2)$-category of framed bordisms ...
- 6,654
6
votes
0
answers
320
views
Terminology for notion dual to "support"
If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements $i\...
- 20.7k
6
votes
0
answers
356
views
References for semicategories
I am struggling with the notion of structure (for reasons related to Freiman's theory and normed rings), which is the main motivation for my question:
Could you suggest some (good) surveys or ...
- 9,636
6
votes
0
answers
338
views
Free CCC or topos on a cartesian category
$\def\sC{\mathcal{C}}\def\sD{\mathcal{D}}\def\Set{\mathbf{\mathrm{Set}}}\DeclareMathOperator{\Sub}{Sub}\def\op{\circ}$I have a Cartesian category $\sC$. I would like to embed $\sC$ into a cartesian ...
user13113
6
votes
0
answers
319
views
Categories with two objects, or mixed bimodules
A category with just one object is a monoid. A category with two objects (which are distinguished) can be described by the following data (imagine the picture $\stackrel{M}{\curvearrowright} \bullet {...
- 61.5k
6
votes
0
answers
338
views
When are localizations of Lawvere theories flat?
Suppose we have a Lawvere theory $L$, i.e. a category with finite products and objects $[n]$ given by the natural numbers such that $[n] \cong [ 1 ] ^n $, and localize it to a Lawvere theory $S^{-1}L$...
- 12k
6
votes
0
answers
511
views
functor before cat?
As i read the literature, derived functors were there several years before derive categories - right?
- 3,850
6
votes
0
answers
238
views
Category of modules over a coPoisson-bialgebra
Fix a ground commutative ring $k$. A coPoisson-bialgebra is a bialgebra $H$ equipped with a linear mapping $\pi:H \rightarrow H \otimes_k H$ s.t.
$\pi$ is a coLie bracket
$\pi$ is a coderivation
$\pi(...
- 1,368
6
votes
0
answers
381
views
Characterizing the image of a faithful left adjoint.
If there is an adjunction between a category $C$ and $D$ such that the left adjoint $$L:C \to D$$ is faithful (but not full), can one describe the image of $L$ in terms of the co-unit of the ...
- 15.3k
6
votes
0
answers
343
views
Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.
In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:
Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every ...
- 3,161
6
votes
0
answers
299
views
What are these sets in Freyd's model?
Recall Freyd's model of $ZF +\neg AC$ (as recounted in MacLane and Moerdijk's book Sheaves in Geometry and Logic): it arises as the Fourman interpretation of the topos of sheaves on a particular ...
- 33.9k
6
votes
0
answers
440
views
Geometric meaning of gamma sets
First some notation:
Let $\mathscr{F}_* $ be the category of finite pointed sets and pointed maps between them. Then $\Gamma^{op}$ is the full subcategory of $\mathscr{F}_* $ with objects the sets $k_+...
- 988
6
votes
0
answers
234
views
When does a morphism of sketches induce an adjunction between their Set-models?
Sketches come in many flavors, according to "what the user may specify." There are finite product sketches where the user may specify a choice of "discrete cones" to be limits. More generally, there ...
- 8,559
6
votes
0
answers
250
views
"Question-answer" bisimulation
I often come across relations that would be defined as a bisimulation, except that the label match can be "inexact", that is, in the bisimulation game, a move labelled with "a" can be replied to with "...
- 415
5
votes
0
answers
74
views
Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?
In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...
- 681