Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.
4
votes
0answers
87 views
$\Omega$ and $B$ as adjoints between symmetric monoidal $(\infty,n)$- and $(\infty,n-1)$-categories
Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= ...
2
votes
0answers
95 views
Morita equivalence via Kan extension
Are there necessary/sufficient conditions for a functor $f\colon \mathcal C\to \widehat{\mathcal A}$ to induce an equivalence $\text{Lan}_yf=F\colon \widehat{\mathcal C}\leftrightarrows ...
4
votes
0answers
141 views
Universal property of module categories over monads
Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...
1
vote
1answer
102 views
Follow up: Is continuity preserved when taking comma object?
Here, I asked wether taking lax pullback preserves continuity, but got no precise answer.
However, I have found this recent article by Riehl and Verity which proves something very similar, but I ...
13
votes
0answers
265 views
The topologies for which a presheaf is a sheaf?
Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on ...
1
vote
1answer
156 views
When are completely positive maps monic/epic?
In the category of C*-algebras and $*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see Mono- and epi-morphisms for C*-algebras). Is ...
-1
votes
0answers
76 views
Quantum Bayesian update and Bayesian update of a model category
We know that we can have internal categories in a model category. We also know that there is a notion of Quantum Bayesian update in a monoidal category. Does the Quantum Bayesian update (equation ...
-1
votes
0answers
154 views
Algebraic topology vs. category theory [migrated]
I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...
3
votes
0answers
81 views
Bicategorical limits with parameters
(This question was asked in http://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.)
Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a ...
3
votes
1answer
173 views
Projectives and Injectives in Functor Categories
Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose ...
3
votes
2answers
114 views
Why is a braided left autonomous category also right autonomous?
In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that:
Lemma 4.17 ([23, Prop. 7.2]). A braided ...
2
votes
0answers
134 views
Is continuity of a functor stable under pullback?
Let $p:C\rightarrow D$, $i:F\rightarrow D$ be functors of 2-categories, and we form the lax pullback of $p$ along $i$ $$
\bar{p}:C\times_D^{lax} F\rightarrow F$$
Q1: Is it true that if $p$ ...
15
votes
0answers
160 views
Finite limits in the category of smooth manifolds?
The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...
15
votes
1answer
293 views
Idempotents split in category of smooth manifolds?
In his paper "Qualitative Distinctions Between Some Toposes of Generalized Graphs" (reproduced here), page 267, Lawvere says that the idempotent-splitting completion of the category of open sets of ...
0
votes
0answers
57 views
Dependent Product Functor [migrated]
I am trying to finish the proof that in category C with Cartesian closed slices, the dependent product functor is the right adjoint of the pullback, so that C is locally Cartesian closed. The proof is ...
7
votes
0answers
264 views
Categorification of the integers
I would like to know a natural categorification of the rig of integers $\mathbb{Z}$. This should be a $2$-rig. Among the various notions of 2-rigs, we obviously have to exclude those where $+$ is a ...
5
votes
1answer
143 views
Making additive envelopes of monoidal categories monoidal
I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$.
(This is defined as the category with ...
3
votes
1answer
203 views
A question on the Grothendieck construction
The usual Grothendieck construction is for pseudofunctors $I^{op}\to Cat$, where $I$ is a 1-category and $Cat$ is the 2-category of 1-categories. The Grothendieck construction produces a category with ...
14
votes
0answers
280 views
a naive question about the second dual of a vector space
Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but ...
2
votes
1answer
80 views
Simple technical adjunction question
Suppose $F,G$ are adjoint, and $\epsilon:F\circ G\rightarrow Id$ is the counit. Is it always true that$$
Id_{FG}\epsilon=\epsilon Id_{FG}
$$
as maps from $FGFG$ to $FG$?
It's true if you precompose ...
1
vote
0answers
47 views
When does a cogenerator determine a variety?
Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they ...
0
votes
2answers
347 views
An isomorphism of categories
(This question was originally asked http://math.stackexchange.com/questions/725421/an-isomorphism-of-categories, with no affirmative answer there.)
Let $C$ be an (finite) extensive category with ...
4
votes
1answer
150 views
defining a bicategory of real-valued matrices
Let $\mathbf{Rel}$ be the bicategory of sets, relations, and inclusions between relations. The following fact is well-known:
Any ordinary function $f : X \to Y$ between sets induces a pair of ...
6
votes
1answer
158 views
Multi-categorical left Kan extensions?
Let ${\bf Set}$ be the category of sets with cartesian product denoted $\times$, and let Sets be the corresponding multi-category of sets, where
$$Hom_{\bf Sets}(A_1,\ldots,A_n;B)=Hom_{\bf ...
5
votes
0answers
54 views
Given a 2-category, is the hammock localization wrt the equivalences equivalent to taking the hom-wise nerve of the maximal subgroupoids?
Consider a category $C$ enriched in categories. Here we have a natural class $W$ of equivalences, namely those morphisms having inverses up to 2-isomorphisms. One can take the hammock localization ...
2
votes
0answers
62 views
A question about braiding represented as pseudofunctors
I just asked a very similar question here (About symmetry, braids, and pseudo-functors.), still now I dont get a answere. I hope to express myself more clearly and correctly this time.
Let ...
4
votes
2answers
245 views
Transporting algebraic structure along adjoint equivalences
I have two questions, one general and the other particular to the case I am interested in.
The 'homotopically correct' notion of equivalence of categories is an adjoint equivalence (from one point of ...
0
votes
2answers
164 views
Rank vanishing in tensor categories
Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...
6
votes
1answer
117 views
Does the “free category on a reflexive graph” monad preserve weak pullbacks, and “why”?
Consider the category of reflexive graphs, and the monad $M$ on it taking the free category: $M(G)$ has all vertices of $G$ as objects, and as edges $x \to x'$ all identity-free paths $x \to x'$ in ...
2
votes
0answers
356 views
The link between the subfactors and the motives as enriched Galois theories?
On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...
-1
votes
0answers
18 views
Objects are finite sets, arrows are matrices. How is this a category? [migrated]
I just started to read this book on category theory.
How is this example below a category?
I have difficulty imagining what this construct really is.
Could someone please illuminate me ?
I ...
2
votes
1answer
74 views
About reflective full subcategories and small-orthogonality classes
Let $\mathcal{A}\subset \mathcal{B}$ be two categories with $\mathcal{A}$ full and reflective in $\mathcal{B}$. Let $R:\mathcal{B}\to\mathcal{A}$ be the reflection. That $R$ is the left adjoint to the ...
1
vote
1answer
175 views
Pushout of categories along embeddings gives homotopy pushout?
Consider the pushout of a diagram $A\leftarrow B\rightarrow C$ of categories and assume that at least one of the arrows is an embedding, i.e. injective on objects and arrows. When applying the nerve ...
2
votes
0answers
100 views
Preimages of accessible full subcategories
My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories ...
2
votes
2answers
79 views
Locally finite varieties which are not finitely generated
Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...
2
votes
0answers
26 views
From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?
Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...
4
votes
2answers
249 views
What's an initial object in a poset-enriched category?
I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the ...
8
votes
1answer
298 views
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...
3
votes
0answers
94 views
Recognition principle for 2-categories (2-groupoids)
Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...
8
votes
2answers
334 views
Is the category of schemes wellpowered? regularly wellpowered?
Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...
2
votes
0answers
53 views
Is the category of spherical fusion categories regular? (i.e. is image factorisation possible?)
Consider the category where objects are strict spherical fusion categories and morphisms are strict spherical functors (preserving cups and caps). I am wondering whether there is some kind of image ...
2
votes
1answer
79 views
Induced adjunctions
Suppose $F: C \rightarrow D$ is the left adjoint to a functor $G$. Then is it true that the functor $F^{\star}:[C : Sets]$ defined by prescomposing a functor $P: C \rightarrow Sets$ is still left ...
5
votes
4answers
622 views
The most unexpected and/or the least natural category theory theorem?
Theory of categories is all natural and abstract nonsense. Or is it? What would be the most unexpected and/or the least natural theorem of the theory of category? (It does not really have to be THE).
...
2
votes
1answer
143 views
Showing a functorial isomorphism
I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory.
The exercise in question is from chapter IV.
So, let ...
6
votes
3answers
341 views
Do any Stone-like dualities have some self-dualities hidden inside them?
This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
4
votes
1answer
172 views
Exercises around Diffeological Spaces or a Diffeologic Atlas Theory
Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...
1
vote
1answer
306 views
A question about the proof of Quillen's Theorem A
(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.)
Theorem (Quillen) ...
15
votes
5answers
2k views
Origin of exact sequences
I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
0
votes
1answer
115 views
Sheaffication using a $\lambda$-transfinite colimit
I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway.
I was reading this article ...
7
votes
1answer
224 views
Adjoining adjoints in a 2-category
For a given 2-category $C$, does there exist a faithful and locally faithful 2-functor $C \to C^*$, such that the image of every 1-morphism of $C$ has a right adjoint in $C^*$?
Below are some of my ...
