Finally split two-sided fibration off of Grothendieck fibration. Thanks to Emily Riehl for adding the definitions here.

]]>Created essential fiber in order to refer to it on the categories list.

]]>I added more details in essentially surjective functor. Please check for details (Mike?).

]]>The page fully formal ETCS is convincing regarding the foundation of the **Category of Sets** as a first order theory. Good !

What about the formal defintion of **FUNCTOR** that implies the connection between two categories ?

Usual textbooks are defining a **FUNCTOR** as two **MAPS** with some additonnal properties. The first **MAP** is between two **COLLECTIONS** of objects. The second **MAP** is between two **COLLECTIONS** of arrows. But these usual textbooks never define **MAP**, neither **COLLECTION**.

What are the formal systems capable to express the concept of **FUNCTOR** ?
Do those systems rely on set theory ? On some type theory ?

It is possible (may be not handy) to setup a first order theory that formalizes in complete autonomy both the concepts of **CATEGORY** and **FUNCTOR** ?

Can we view an integral somehow as transfinite composition of “small” arrows, in a rigorous (but possibly nonstandard) way?

For example, if we view a real 1-form as a functor with values in $B\mathbb{R}$, is its integration along a curve a transfinite composition of its values at all the tangent vectors of the curve?

I don’t know if the question is clear enough, in case it isn’t, feel free to ask. (It may also be that none of this makes any sense.)

]]>Let $U,V$ be functors between categories $C$ and $X$.

Why a dinatural transformation $\{*\}\xrightarrow{\cdot \cdot}\hom_X(U-,V-)$ is a function which selects a natural transformation $U\xrightarrow{\cdot} V$ ?

As far as I understand, a category is morally speaking a special kind of graph (say associative unital). Structures are therefore totally encoded into the shape of the graphs of the category (ie of relations), and the nature of objects and morphisms are totally irrelevant. By asking that a functor preserves the identities, the latter can be considered to have an "absolute" (ie non relational) nature (or say, globally defined to the whole category).

Yet, if one wants to have a purely relational approach to algebra, it is legitimate to try to remove as much as possible "absolute" meaning to objects studied, and only to characterize them by how they interact with their surrounding. In this respect, being an identity of some object C is relative to the whole category and is not a "true" essence of the morphism : it is a global property.

Now, a functor F: J -> D being a diagram of shape J in D, shouldn't it be legitimate to only ask that F(j°k) = F(j)°F(k) ? That is to say, to translate only the shape of the diagram into D and as a result to only preserve the relations between the mapped morphisms ? Therefore the functor would be a "local" injection of J into D without caring about where it starts ? (ie, not at the level of the identity which is defined globally).

An identity with respect to some "collection" of morphisms is just an idempotent element that acts as a unit on this collection. One could even want to propose an alternative definition of category (without unit necessarily), as an associative graph but only to impose the presence of some idempotent element for every object. Such idempotent element could be either a unit (if is absorbed by everyone globally) or an absorbing element (it absorbs all endomorphisms, so it absorbs everyone locally).

Are there people studying that ? Would it be a good idea to study that and why ?

PS : note that it is in my opinion bad that algebra forgot to study stuctures with absorbing elements. One can easily reinterprete many structures (semi-rings, so rings and fields, ...) as special case of "annulus" (M,+,.,0,i), where

1) (M,+,0) is a commutative monoid,

2) (M,.,i) is such that . is associative and i is absorbing

3) . is distributive over + ]]>

Let

be a commutative square of categories and functors. Assume that $L_1$ and $L_2$ have right adjoints $R_1$ and $R_2$, respectively. Under which conditions do we have $FR_1\cong R_2G$?

The thing is that we have a concrete situation in which this does seem to be the case, but we would like to have an easy-to-check criterion which implies it.

In our case, all four categories are actually functor categories and the right adjoints correspond to taking Kan extensions.

]]>I have added a new paragraph to direct image about direct image functor with compact support $f_! F$. Eventually I would create a separate entry direct image with compact support, but not yet.

]]>I changed the definition at logical functor, as it said that such a thing was a cartesian functor that preserved power objects. The page cartesian functor says

A strong monoidal functor between cartesian monoidal categories is called a cartesian functor.

which really is only about finite products, not finite limits as Johnstone uses, which I guess is where the definition of logical functor was lifted from. So logical functor now uses the condition ’preserves finite limits’.

So I added a clarifying remark to cartesian functor that the definition there means finite-product-preserving, and that the Elephant uses a different definition.

However, people may wish to have cartesian functor changed, and logical functor put back how it was. I’m ok with this, but I don’t like the terminology cartesian (and I’m vaguely aware this was debated to some extent on the categories mailing list, so I am happy to go with whatever people feel strongest about).

]]>Is there analogous terminology and machinery and formalism for modules over a Lie algebra

*without* passing to the universal enveloping thus `reducing to the previous case'?

cf. Lie algebra cohomology ]]>

Dear all,

I recently got involved with enriched category theory and I want to apply the machinery in a computer science environment. I am interested in non-complete enrichments and here particular in functor categories.

I am aware that if $\mathcal{V}$ is a complete symmetric monoidal closed category and $\mathcal{A}$ and $\mathcal{B}$ are $\mathcal{V}$-categories, then $[\mathcal{A},\mathcal{B}]$ can be enriched over $\mathcal{V}$. In Kelly it is then shown that under this assumption the enriched Yoneda lemma and enriched Yoneda embedding hold. There is also a short explanation of what happens when $\mathcal{V}$ is not necessarily small. I would now be interested what happens when $\mathcal{V}$ is not necessarily complete.

In Borceux (Handbook II Chapter 6) this is made a bit more precise. Here, it is made clear that we do not actually need completeness for the enriched Yoneda lemma. Still, for the enriched Yoneda embedding, we appartently need completeness. But it is not made precise why we actually need it. I assume that this is related to the functor category “problem”. Nevertheless, there is a difference between giving a recipe to get an enrichement when $\mathcal{V}$ is complete, but this does not mean that we cannot find an enrichment when $\mathcal{V}$ is not complete. Is anyone aware of results in this direction?

To reduce the problem, it would be enough to consider functor categories $[\mathcal{A},\mathcal{V}]$, where $\mathcal{V}$ is our enrichment. Here, the enriched Yoneda lemma indicates that $[\mathcal{A},\mathcal{V}](H^{A}, F)$ has a hom-object. In Borceux we can find a definition for an object of $\mathcal{V}$-natural transformations between two functors $F,G:\mathcal{A} \rightarrow \mathcal{V}$ when $\mathcal{V}$ is not complete. This means in the special case we do not have problem to find the right hom-object, but what about a general $\mathcal{V}$-functor $F$. I have not found nice examples that points out a problem. Maybe for some enrichements we can still get an enrichable $[\mathcal{A},\mathcal{V}]$ functor category.

I would appreciate any comment or reference to the literature that might answer some of these questions.

Kind regards,

franeb

]]>I addede a paragraph at Poincaré duality about the generalizations, and created the entry (so far only descent bibliography) Grothendieck duality; the list of examples expanded at duality. All prompted by seeing the today’s arXiv article of Drinfel’d and Boyarchenko.

]]>New entry [[internal diagram]], generalizing [[internal functor]].

]]>I’m struggling to further develop the page on Schur functors, which Todd and I were building. But so far I’ve only done a tiny bit of polishing. I deleted the discussion Todd and I were having near the top of the page, replacing it by a short warning that the definition of Schur functors given here needs to be checked to see if it matches the standard one. I created a page on linear functor and a page on tensor power, so people could learn what those are. And, I wound up spending a lot of time polishing the page on exterior algebra. I would like to do the same thing for tensor algebra and symmetric algebra, but I got worn out.

In that page, I switched Alt to $\Lambda$ as the default notation for exterior algebra. I hope that’s okay. I think it would be nice to be consistent, and I think $\Lambda$ is most widely used. Some people prefer $\bigwedge$.

]]>