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

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5
votes
2answers
358 views

What is the co-form of Grothendieck construction?

For simplicity, let us consider only a functor out of a small category $\mathcal{C}$ to $Set$, $$ f:\mathcal{C}\to Set, $$ The Grothendieck construction produces a category (category of elements) ...
0
votes
0answers
112 views

Continuity of Kan extension along the Yoneda embedding

Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a ...
6
votes
2answers
246 views

Weakening simplicial identities

The generators $d_i, s_i$ for morphisms of the simplicial category satisfy simplicial identities: $d_jd_i = d_id_{j−1}$ for $i < j$ $s_jd_i = d_is_{j−1}$ for $i < j$ $s_jd_i = id$ for $i = ...
4
votes
2answers
152 views

symmetric monoidal double categories?

Let me preface this by saying that I don't know much category theory. I am running into a situation where I have a double category and additionally there is a multiplication. Moreover, choosing ...
7
votes
0answers
160 views

What's the Hochschild homology of the category of constructible sheaves?

Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?
6
votes
2answers
281 views

Unexpected interaction between limits and colimits

Let $D$ be a limit-complete category. My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? I'm especially interested in the case that the ...
11
votes
0answers
211 views

Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...
10
votes
1answer
606 views

I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category?

I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need ...
6
votes
1answer
222 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
2answers
278 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
1answer
261 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 ...
2
votes
1answer
220 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 ...
18
votes
0answers
562 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
224 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 ...
0
votes
0answers
85 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 ...
3
votes
0answers
88 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
270 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
126 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
154 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
191 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
376 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 ...
9
votes
0answers
337 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
165 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
290 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 ...
29
votes
0answers
934 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
99 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
65 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
383 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
161 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
184 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
78 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
73 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
262 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
3answers
277 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
198 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
589 views

The link between the subfactors and the motives as enriched Galois theories? [closed]

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), ...
2
votes
1answer
88 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
202 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
115 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 ...
5
votes
3answers
175 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
1answer
92 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
312 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
310 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
105 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 ...
9
votes
2answers
361 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
64 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
89 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 ...
4
votes
3answers
703 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
175 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
433 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 ...