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Stream: deprecated: id my structure

Topic: A generalised category of zig-zags

Amar Hadzihasanovic (Mar 22 2023 at 11:40):

I am looking for a name or reference for the following construction, which generalises the [[localization]] of a category.

Suppose we have a category $\mathcal{C}$ and two wide subcategories $\mathcal{B}$ and $\mathcal{F}$.
Then I can define a category, call it $\mathcal{B}^*\mathcal{F}$, whose

• objects are the objects of $\mathcal{C}$,
• morphisms from $x$ to $y$ are zig-zags $([b_0], f_0, \ldots, b_n, [f_n]): x \leftarrow x_1 \rightarrow x_2 \leftarrow \ldots \rightarrow y$, starting and ending with either a backward-facing or forward-facing morphism, where the backward-facing morphisms $b_i$ are in $\mathcal{B}$ and the forward-facing morphisms $f_i$ are in $\mathcal{F}$,

composing by concatenation + composition in $\mathcal{B}$ and $\mathcal{F}$ when the final morphism of one zig-zag and the initial morphism of the other have the same direction, and modulo the following identifications:

• backward-facing identities are equal to forward-facing identities,
• a zigzag $(b_0, f_0): x \leftarrow y_0 \rightarrow z$ is equal to $(f_1, b_1): x \rightarrow y_1 \leftarrow z$ whenever $b_0;f_1 = f_0;b_1$.

When $\mathcal{F} = \mathcal{C}$, then this should be precisely the localisation of $\mathcal{C}$ at the morphisms in $\mathcal{B}$, and more in general any morphism which is in the intersection of $\mathcal{F}$ and $\mathcal{B}$ will be invertible in $\mathcal{B}^*\mathcal{F}$.
However one can, for example, consider examples where $\mathcal{F}$ and $\mathcal{B}$ only intersect at isomorphisms, and in that case the result has a different flavour compared to a localisation.

Amar Hadzihasanovic (Mar 22 2023 at 12:19):

I guess that

• if pullbacks of $\mathcal{F}$-morphisms along $\mathcal{B}$-morphisms exist and are in $\mathcal{F}$, and symmetrically, then $\mathcal{B}^*\mathcal{F}$ is a quotient of the bicategory of spans in $\mathcal{C}$ where the left leg is in $\mathcal{B}$ and the right leg in $\mathcal{F}$,
• dually if pushouts of $\mathcal{F}$-morphisms along $\mathcal{B}$-morphisms exist and are in $\mathcal{F}$, and symmetrically, then $\mathcal{B}^*\mathcal{F}$ is a quotient of the bicategory of cospans in $\mathcal{C}$ where the right leg is in $\mathcal{B}$ and the left leg in $\mathcal{F}$...

JS PL (he/him) (Mar 22 2023 at 12:20):

My first thought is that this seems related to the free dagger category over a category -- see Def 3.1.18 in Chris Heunen's Thesis.

JS PL (he/him) (Mar 22 2023 at 12:22):

Which you might already have known. But I think that free dagger category zigzag category is when $\mathcal{B}=\mathcal{F}=\mathcal{C}$ in your construction

Amar Hadzihasanovic (Mar 22 2023 at 12:24):

Yes and in addition you have the “cospan = span” equation when they form a commutative square, which you don't have in the free dagger category.

Amar Hadzihasanovic (Mar 22 2023 at 12:26):

(Indeed if you added that to the free dagger category, you would obtain the free groupoid over the category)

Amar Hadzihasanovic (Mar 22 2023 at 12:45):

Oh, actually I think the following is the case, more precisely to my previous guess.

Suppose that $\mathcal{C}$ has pullbacks, and

• pullbacks of $\mathcal{F}$-morphisms along $\mathcal{B}$-morphisms are in $\mathcal{F}$,
• pullbacks of $\mathcal{B}$-morphisms along $\mathcal{F}$-morphisms are in $\mathcal{B}$;
  then there is a bicategory of spans in $\mathcal{C}$ whose left leg is in $\mathcal{B}$ and whose right leg is in $\mathcal{F}$.

I believe that $\mathcal{B}^*\mathcal{F}$ is exactly the 1-truncation of this bicategory, i.e. the category with the same objects, and as morphisms classes of 1-morphisms under the equivalence relation generated by “$f \sim g$ if there exists a 2-morphism between $f$ and $g$”;
This picture is an attempt at a single-diagram proof idea:

Amar Hadzihasanovic (Mar 22 2023 at 12:46):

85ef6293-7599-4d32-8459-5bcbc2866e38.jpg

Amar Hadzihasanovic (Mar 22 2023 at 12:49):

The cospan zig-zag $(f_1, b_1)$ is equal to the composite of spans $(\mathrm{id}_x, f_1)$ and $(b_1, \mathrm{id}_y)$, which is exactly the pullback span produced by $(f_1, b_1)$, and there is a morphism from the span $(b_0, f_0)$ to this pullback precisely when $b_0;f_1 = f_0; b_1$.

Carlos Zapata-Carratala (Mar 23 2023 at 10:17):

Amar Hadzihasanovic said:

I am looking for a name or reference for the following construction, which generalises the [[localization]] of a category.

Suppose we have a category $\mathcal{C}$ and two wide subcategories $\mathcal{B}$ and $\mathcal{F}$.
Then I can define a category, call it $\mathcal{B}^*\mathcal{F}$, whose

• objects are the objects of $\mathcal{C}$,
• morphisms from $x$ to $y$ are zig-zags $([b_0], f_0, \ldots, b_n, [f_n]): x \leftarrow x_1 \rightarrow x_2 \leftarrow \ldots \rightarrow y$, starting and ending with either a backward-facing or forward-facing morphism, where the backward-facing morphisms $b_i$ are in $\mathcal{B}$ and the forward-facing morphisms $f_i$ are in $\mathcal{F}$,

composing by concatenation + composition in $\mathcal{B}$ and $\mathcal{F}$ when the final morphism of one zig-zag and the initial morphism of the other have the same direction, and modulo the following identifications:

• backward-facing identities are equal to forward-facing identities,
• a zigzag $(b_0, f_0): x \leftarrow y_0 \rightarrow z$ is equal to $(f_1, b_1): x \rightarrow y_1 \leftarrow z$ whenever $b_0;f_1 = f_0;b_1$.

When $\mathcal{F} = \mathcal{C}$, then this should be precisely the localisation of $\mathcal{C}$ at the morphisms in $\mathcal{B}$, and more in general any morphism which is in the intersection of $\mathcal{F}$ and $\mathcal{B}$ will be invertible in $\mathcal{B}^*\mathcal{F}$.
However one can, for example, consider examples where $\mathcal{F}$ and $\mathcal{B}$ only intersect at isomorphisms, and in that case the result has a different flavour compared to a localisation.

When $\mathcal{F} = \mathcal{B}=\mathcal{C}$, it may be worth mentioning that zig-zags can be given a non-associative higher-order composition which makes $\mathcal{F}$ into a semiheapoid. This perspective may clarify some of the algebraic behaviour of composition and units. You can see some notes on semiheapoids in my recent paper: https://arxiv.org/abs/2205.05456