Tagged Questions
Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.
-2
votes
0answers
121 views
A definition of ($\infty$-)cat(egorie)s [on hold]
Yesterday I uploaded an updated version of my paper on "Cats". It will be available on the arXiv as of monday; until then, you can download a copy here.
In that paper, I take Simpson's category ...
2
votes
0answers
62 views
Equational theories determined by “identities without variables”
How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...
3
votes
1answer
103 views
In what generality does Eilenberg-Watts hold?
In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...
1
vote
0answers
59 views
Intuitive idea: What is the motivation for relative homological algebra
Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects?
What are some applications? For ...
9
votes
4answers
393 views
Functors and coverings
A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical ...
1
vote
0answers
57 views
Neeman's homotopy limits in stable $\infty$-categories
Related question (I hoped to turn this into an answer for the user, but in the end it became a question on its own right!): this
In the book
Neeman, Amnon. Triangulated categories. No. 148. ...
4
votes
0answers
133 views
Are left adjoints a left adjoint?
Let $\mathcal C$ be a strict, locally small 2-category.
Consider a subcategory $\mathcal L$ of $\mathcal C$ such that $\mathcal L$ has the same objects as $\mathcal C$, and the arrows of $\mathcal L$ ...
5
votes
1answer
95 views
coends of stable infinity categories
Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote ...
5
votes
1answer
124 views
What do algebraic theories with strictly terminal trivial models look like?
By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...
4
votes
0answers
70 views
How to show twisted complexes over a DG category is again a DG category?
In Bondal and Kapranov's paper enhanced triangulated categories, a twisted complex over a DG category $A$ is a set $\{(E_i)_{i\in \mathbb Z}, q_{ij}: E_i\to E_j\}$, where $E_i$ are objects in $A$, ...
4
votes
0answers
93 views
Which endomorphism algebras are not Morita-trivial?
Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that:
$\mathcal C$ is ...
2
votes
2answers
116 views
Integral over integrals with different measures
I was trying to construct a category of measurable spaces and 'nondeterministic functions' (not the usual category of measurable spaces); specifically a map $(\Omega, \Sigma)\rightarrow (\Omega', ...
4
votes
0answers
75 views
+100
Do coherent toposes descend along open surjection?
Let $f:\mathcal{T} \rightarrow \mathcal{S}$ be a geometric morphism between two toposes. Let $g:\mathcal{L}\rightarrow \mathcal{S}$ be an open surjection (of toposes) and assume that the map ...
1
vote
0answers
72 views
The classifying space of the groupoid $\pmb\Delta^n$
Consider the groupoid generated by the category $\{0\to 1\to\cdots\to n\}$; let's call this category $\pmb\Delta^n$ opposed to the category $\triangle^n$, which is "thinner".
I'm trying to figure out ...
5
votes
1answer
310 views
Noncommutative geometry and category theory
The point in which one starts to talk about noncommutative geometry is the Gelfand Najmark theorem. It establishes an equivalence of the catgeories of commutative (non)unital $C^*$-algebras and the ...
3
votes
1answer
129 views
Cogroup objects are to groups what — are to $k$-modules
Let us place ourselves in a category $\mathcal C$ with finite coproducts $X\amalg Y$, even cocomplete if necessary. It is well known that the morphism set $\mathcal C(X,Y)$ carries an abelian group ...
3
votes
0answers
68 views
Categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$
This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains ...
2
votes
1answer
209 views
Expressive power of first-order category theory
Given the signature $\lbrace \mathsf{dom}, \mathsf{cod}, \mathsf{id},\circ \rbrace$ and the axioms of category theory – which are expressible in the signature's first-order (FO) language – ...
5
votes
0answers
175 views
Sign problem for the shift functor on DG modules
Let $\mathcal{A}$ be a Differential graded $k$ category, where $k$ is a commutative ring. I want to extend the shift functor $[1]$ on chain complexes to dg modules over $\mathcal{A}$. Now a right dg ...
6
votes
1answer
251 views
Can the monoidal structure on manifolds be strictified?
I'm asking this question purely out of curiosity.
Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...
1
vote
4answers
221 views
Graphs with dangling edges
In conventional Graph Theory the role of Nodes and Edges is skewed: nodes are perfectly ok being aloof, but poor edges are always drawn between existing nodes (that is, the two maps from EDGES to ...
2
votes
0answers
193 views
Why does this setting imply that a category is Grothendieck?
I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf.
Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...
2
votes
0answers
93 views
Homotopy limits of weak diagrams
This question is somehow related to my question at http://math.stackexchange.com/questions/683915/derived-pseudo-functor
and to the question here: A homotopy commutative diagram that cannot be ...
1
vote
1answer
317 views
On the foundations for large categories
There are some basic discussions on the motivations of large categories and small categories [here] (On the large cardinals foundations of categories) and [here] (Large cardinal axioms and ...
11
votes
2answers
266 views
Varieties where every algebra is free
I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
19
votes
5answers
1k views
(Short) Exact sequences with no commutative diagram between them
This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...
0
votes
1answer
142 views
Homotopy limit of a cosimplicial category
Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?
...
2
votes
1answer
148 views
n-limits and the Descent Category
Probably, this question could be at http://math.stackexchange.com/, but, for some reason, I can't access that site right now. So, I apologize in advance.
At this article ...
2
votes
0answers
124 views
Is the collage of two spatial toposes a spatial topos?
Consider the collage operation along a profunctor, defined between two categories ${\bf C}, \bf D$.
Suppose now that the two categories are toposes, say $Sh(X), Sh(Y)$ for two topological spaces ...
6
votes
1answer
268 views
Misunderstanding in the hypotheses of Schlessinger's criterion
Good day to everyone.
In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group ...
3
votes
0answers
130 views
Does there exist a terminal surjective discrete fibration out of $C$?
Let $DF$ denote the category whose objects are categories and whose morphisms $F\colon R\to S$ are the discrete fibrations. This category has applications to the real-world problem of structuring ...
5
votes
1answer
268 views
Not quite adjoint functors
What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
1
vote
1answer
145 views
Expression for functor strengths?
It's well known that endofunctors on $Set$ have an unique strength.
A strength for a functor $T : Set \to Set$ is a natural transformation $t_{A,B} : A \times T B \to T (A \times B)$ such that certain ...
9
votes
0answers
118 views
Does every commutative variety of algebras have a cogenerator?
By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...
1
vote
1answer
187 views
What is this name of this 2-category without very much structure?
I was wondering if there is a name for this 2-category which is like the 2-category of natural transformations, but does not actually require the 1-morphisms to be functors or the 2-morphisms to be ...
4
votes
1answer
160 views
Reference request: “unoriented composition” in generalized categories
I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category ...
1
vote
0answers
46 views
A categorical analogue of Debreu's independent factors theorem
Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...
3
votes
3answers
257 views
When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?
Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...
3
votes
0answers
165 views
Invariants of groups that are invariant under passage to finite index subgroups
This question is mostly idle curiosity.
Recall the following standard terminology: if $P$ is a property of groups, a group $G$ is said to be virtually $P$ if it has a subgroup of finite index which ...
2
votes
1answer
183 views
Cartesian closed category
Let $\bf{C}$ be a category with finite products.
(1) An object $X$ of $\bf{C}$ is called cartesian if the functor $(-)\times X$ has a right
adjoint.
(2) A morphism ...
9
votes
1answer
338 views
Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category?
Following Anton Deitmar, let $\mathcal B$ be an "$\mathbb F_1$-linear category" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is balanced, pointed, contains finite products, kernels, and ...
1
vote
1answer
133 views
comparison between two monadic definitions for an operad
According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is ...
24
votes
4answers
724 views
What other monoidal structures exist on the category of sets?
I know of the following monoidal structures over $\mathbf{Set}$ (taken from here):
The Cartesian product $\otimes=\times$ (categorical product)
The disjoint union $\otimes=+$ (categorial coproduct)
...
2
votes
1answer
168 views
A Criterion for a morphism to be a counit of an Adjunction
Suppose we have two functors $F:C\leftrightarrow D:G$ and a morphism $\varepsilon:FG\rightarrow\operatorname{Id}_D$. I am looking for a way to check whether $\varepsilon$ is the counit of an ...
2
votes
0answers
198 views
When is the realisation of a simplicial set a manifold?
It is known that a simplicial complex is homeomorphic to a manifold if the link of every vertex is a simplicial sphere, for which there exists a definition. (I know that for high dimension, the ...
10
votes
1answer
305 views
Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)
Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
2
votes
2answers
267 views
Are algebraic structures uniquely identifed by their free objects?
It might be a naive question, as I am not a specialist in this field.
This is a follow-up to this question.
I want to study varieties of objects generalizing ordered monoids, in particular using an ...
4
votes
1answer
86 views
Does trace handle composition in a traced symmetric monoidal category?
Suppose that $(C,\otimes,I)$ is a traced symmetric monoidal category (TSMC) with symmetrizor $\sigma$ and trace $Tr$. Given two morphisms $f\colon A\to B$ and $g\colon B\to C$, I can tensor them to ...
9
votes
1answer
361 views
Pushouts in the category of adjunctions
Let $\mathcal A$ be the category whose objects are all categories, and whose morphisms from a category $\mathcal C$ to a category $\mathcal D$ are all adjunctions $F \dashv G$ with $F : \mathcal C \to ...
4
votes
1answer
141 views
Push-outs of fully faithful (enriched) functors
I'd like to know a reference where the following property, that I believe to be true, is checked: given a diagram of categories and functors $B\leftarrow A\rightarrow C$ (I'm actually interested in ...
