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Stream: deprecated: id my structure

Topic: categories with the same objects sharing some morphisms

Carlos Zapata-Carratala (Aug 12 2022 at 11:03):

Consider two categories $A$ and $B$ over the same family of objects $\text{Ob}(A)=\text{Ob}(B)$. Consider the subset of (pairs of) morphisms in $A$ that are splittings, i.e. a morphism with a one-sided inverse, which form a subcategory, $S\subset A$. What I have found is the situation where there is a faithful functor $i: S \to B$, so informally one can think that there are two categories over the same objects sharing some morphisms, i.e. $\text{Hom}(A) \cap \text{Hom}(B) = \text{Hom}(S)$, but otherwise they may not interact further. Any ideas about theory on this sort of situation?

Mike Shulman (Aug 12 2022 at 17:15):

An [[M-category]] is two categories with the same set of objects and where the morphisms of one are included in the morphisms of the other. This can be formalized as a category enriched over the cartesian monoidal category $\mathcal{M}$ whose objects are injections of sets and whose morphisms are commutative squares. Something like your situation could probably be formulated similarly in terms of enrichment over a category whose objects are spans of injections.

Valeria de Paiva (Aug 12 2022 at 18:49):

@Mike Shulman thanks for the tip!

dusko (Aug 13 2022 at 00:03):

sorry if i am being thick, carlos, but it seems to me that there are two ways to interpret what you write:
Carlos Zapata-Carratala said:

Consider the subset of (pairs of) morphisms in $A$ that are splittings, i.e. a morphism with a one-sided inverse, which form a subcategory, $S\subset A$.

there are two kinds of morphisms with one-sided inverses: split epis and split monos.

if we take all morphisms with one-sided inverses, as you seem to be suggesting, the spanned subcategory $S$ consists of the morphisms that have a split epi-split mono factorization.

if we assume that you forgot to mention that you are looking only, say, at the morphisms with a right inverse, then the subcategory $S$ is simply the subcategory of split epis in $A$. (this is the case where john power's categories enriched over sets with subsets applies, to which mike shulman seems to refer.)

in the first case, the subcategory $S$ seems like the largest absolutely complete subcategory of $A$. if $S$ is also the largest absolutely complete core of $B$, then the functors out of $A$ and $B$ must coincide on this part but may differ on the rest.

in the second case, if $A$ and $B$ share the same split epis, but the actual splittings are different then this might be some interesting AC subtlety, of two universes with different choice functions and different well-orderings...

i am sure that i am way off in at least one of the two cases, but both are interesting, so thank you :)

Carlos Zapata-Carratala (Aug 13 2022 at 01:02):

Sorry @dusko I was being vague. Let me give some further detail: $A$ and $B$ are categories of algebras (a particular kind of ternary structure called a semiheap), $A$ is the usual category of universal algebra where morphisms are homomorphisms between the underlying sets, $B$ however, has the same algebras as objects but the morphisms are pairs of maps $\varphi: X_1 \to X_2$ $\psi: X_2 \to X_1$ which satisfy the following "diamorphism" condition:
$\varphi( (a \psi(b) a')_1) = (\varphi (a) b \varphi (a'))_2$.
These diamorphisms can be composed in the obvious way and thus form the category $B$. The observation that prompted my question above is that given morphisms in the category $A$: $f:X_1\to X_2$ and $g:X_2\to X_1$ such that $g$ is a right inverse of $f$, what I called a splitting $f\circ g = \text{id}_{X_2}$, then by the fact that they are homomorphisms they satisfy the diamorphism condition above. This implies, in particular, that all the homomorphic isomorphisms of semiheaps (the isos of the category $A$) are also isomorphisms in the category $B$ of diamorphisms.

We have recently identified this notion of diamorphism for semiheaps and are looking to investigate the natural categorical properties that one can wish for, that's why I thought of posting it in this forum. Many thanks for all your comments!

dusko (Aug 13 2022 at 10:31):

@Carlos Zapata-Carratala oh man. noncommutative geometry of some sort?

so if i imagine that this ternary operation is like the malcev algebra from an inverse semigroup, ie your abc is some a-b+c for noncommutative + (sorry, but you already used the "multiplication" for the ternary operation), then all of your diamorphisms would boil down to semigroup retractions like you describe. so if we perform some sort of completion to represent each algebra as obtained from an inverse semigroup (and under some conditions embedded into it?), then the category $B$ would be the category of some sort of inverse semigroups with just retractions. a big lattice, likely modular.

if we generalize dimorphisms so that they correspond to either the epi or the mono part of a retraction, we land in the absolute core case from my first msg... where do these semiheaps come from? maybe completing them to semigroups is just the wrong thing to do? i thought i had seen everything, but your semiheaps and a president of the US being investigated for espionage are completely new phenomena for me and made my day. life is fun :)

Carlos Zapata-Carratala (Aug 13 2022 at 10:57):

dusko said:

so if i imagine that this ternary operation is like the malcev algebra from an inverse semigroup, ie your abc is some a-b+c for noncommutative + (sorry, but you already used the "multiplication" for the ternary operation), then all of your diamorphisms would boil down to semigroup retractions like you describe. so if we perform some sort of completion to represent each algebra as obtained from an inverse semigroup (and under some conditions embedded into it?), then the category $B$ would be the category of some sort of inverse semigroups with just retractions. a big lattice, likely modular.

That's precisely what we are working on, there is a rich theory about the relation between inverse semigroups and heaps (this is somewhat classical and it is found in Chapter 8 of Wagner's Theory of Heaps ). These heaps are essentially affine versions of groups and the correspondence is precisely the Malcev algebra construction you mention.

We are studying semiheaps which come from cubic matrix algebra and ternary hypergraph adjacency, you can see this in our recent preprint:

Heaps of Fish: arrays, generalized associativity and heapoids

These "fish semiheap(oid)s" are not affine versions of group(oid)s in general since "ternary inverses" don't generally exist. We are working on results that link semiheaps and semigroups but it turned out that we needed generalized notions of morphisms (what I called diamorphisms above) to make sense of simple algebraic constructions like retracts, hence my question.

Like you say, this is quite a fun and mysterious algebraic territory. Your comment made my day :)

dusko (Aug 14 2022 at 00:55):

downloaded heaps of fish. very nice! will probably read through. keep us posted :)

Carlos Zapata-Carratala (Sep 21 2022 at 18:36):

dusko said:

downloaded heaps of fish. very nice! will probably read through. keep us posted :)

For those interested in these ternary algebras, we have a new paper on generalizations of units in semiheaps:

https://arxiv.org/abs/2209.07203