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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2
votes
2answers
176 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 ...
5
votes
2answers
395 views

Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...
4
votes
1answer
198 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 ...
19
votes
1answer
1k views

Analogy between the exterior power and the power set

The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$. The usual definition of the exterior ...
17
votes
30answers
764 views

Formalizations of category theory in proof assistants

What are the existing formalizations of category theory in proof assistants? I'm primarily interested in public-domain code implementing category theory in a proof assistant (Coq, Agda, ...
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
0answers
102 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}= ...
13
votes
0answers
277 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
105 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 ...
14
votes
0answers
281 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 ...
1
vote
1answer
157 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
77 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
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 ...
-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
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
5answers
540 views

Resources for learning domain theory?

I'm a computer programmer who's caught on to denotational semantics. I mostly work with Ruby, JavaScript and C, but I know a little Haskell and ML. I've taken my first steps towards reasoning about ...
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
162 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
295 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
2answers
348 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
0answers
160 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 ...
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 ...
5
votes
1answer
144 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
4
votes
1answer
151 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 ...
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 ...
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
357 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), ...
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 ...
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 ...
12
votes
4answers
600 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 ...
0
votes
2answers
165 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 ...
15
votes
2answers
672 views

Internal categories in simplicial sets

Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories? Note that this ...
-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 ...
3
votes
1answer
195 views

Generalisation of the Grothendieck construction for presheaves as a lax pullback

It is well-known that for any presheaf $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Set}$, the category of elements (obtained by the so-called Grothendieck construction) of $F$ is a comma category ...
8
votes
1answer
299 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 ...
0
votes
1answer
437 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 ...
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 ...
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 ...
2
votes
0answers
102 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 ...
25
votes
2answers
2k views

Intuition for coends

Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end $$ \int^{c\in C} F(c,c) $$ as the co-equalizer of $$ \coprod_{c\to ...
9
votes
1answer
406 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 ...
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 ...