# What is the best way in Category Theory to define all subcategories of a category, but with a specific structure?

For example consider a category $$A$$ which is defined only by the following arrows $$a\to b$$, $$a\to c$$, $$c\to d$$, $$c\to e$$, and with no composition rules.

How would we define a category of subcategories of $$A$$, that contain arrows $$a\to b$$, $$a\to c$$.

E.g. For $$A$$ this category would include the following categories:

• $$a\to b$$, $$a\to c$$

• $$a\to b$$, $$a\to c$$, $$c\to d$$

• $$a\to b$$, $$a\to c$$, $$c\to e$$

• $$a\to b$$, $$a\to c$$, $$c\to d$$, $$c\to e$$

This is a simple example but there could be more complex arrow structures and so the method would need to be expressive enough to do this for arbitrary categories.

• When you say "maintain". I think you mean "contain", in which case you have given a definition of the objects of your category and you just have to decide what the arrows are (inclusion is the obvious choice). In your example, you have missed the cases where one or both of the objects $d$ and $e$ are included but with no arrow from $c$ to them. – Rob Arthan Mar 29 at 14:30
• @RobArthan Ah I don't want to allow objects such as those in the category. So I guess I need to add an extra constraint for this like, all objects of the subcategories must have greater than 1 arrow? – newlogic Mar 29 at 14:38
• Sure: you could say that every object must the domain or codomain of at least two arrows. – Rob Arthan Mar 29 at 14:40
• @RobArthan Thankyou. When defining the structure of A would it be possible to group a->b, a->c in a single category (X) and then say A contains an object which is the category X, and this object has an arrow to d and e, and all this will give the same structure as the original A definition. – newlogic Mar 29 at 14:53
• You can specify which arrow you mean when you write $g\circ f$ if it already has a name; but you aren't "overriding default behavior", you are giving a special name to $g\circ f$. It's alike addition: you don't "override default behavior" of addition when you say that $1+2=3$. You are just giving a name to "$1+2$". And I already saw that link, you don't need to shove it in my direction yet again just to have me say, yet again, "they are not doing what you think they are doing". – Arturo Magidin Mar 29 at 18:06

## 1 Answer

Given any category $$\mathbf{C}$$, with objects $$\mathrm{Ob}(\mathbf{C})$$ and for each ordered pair of objects $$(X,Y)$$ a collection of arrows $$\mathbf{C}(X,Y)$$, with compositions when defined and identity functions, you define a subcategory $$\mathbf{D}$$ as a category with $$\mathrm{Ob}(\mathbf{D})\subseteq \mathrm{Ob}(\mathbf{C})$$, and for each $$X,Y\in\mathrm{Ob}(\mathbf{D})$$, $$\mathbf{D}(X,Y)\subseteq \mathbf{C}(X,Y)$$, with the proviso that the identity functions must be included in $$\mathbf{D}(X,X)$$ for each $$X\in\mathrm{Ob}(\mathbf{D})$$, and the compositions (when defined) of arrows in $$\mathbf{D}$$ must lie in $$\mathbf{D}$$.

If $$\mathbf{D}$$ is a subcategory of $$\mathbf{C}$$, then there is a natural "inclusion functor $$\mathbf{I}\colon\mathbf{D}\hookrightarrow\mathbf{C}$$, given by mapping each $$X\in\mathrm{Ob}(\mathbf{D})$$ to itself in $$\mathrm{Ob}(\mathbf{C})$$, and likewise mapping $$\mathbf{D}(X,Y)$$ to $$\mathbf{C}(X,Y)$$ via the inclusion map of sets/classes.

Given any collection of categories $$\{\mathbf{C}_i\}$$, you can construct a category $$\mathbf{D}$$ whose objects are precisely the categories $$\mathbf{C}_i$$, and where $$\mathbf{D}(\mathbf{C}_i,\mathbf{C}_j)$$ consists precisely of the functors $$C_i\to C_j$$. This is a full subcategory of $$\mathbf{Cat}$$, the category of categories. But you can define a non-full subcategory by restricting the functors you allow to any specific subcollection of functors that you will "allow".

So you can simply start with your category $$\mathbf{C}$$; then specify the subcategory of $$\mathbf{Cat}$$ whose objects are the subcategories of $$\mathbf{C}$$ (categories that have an "inclusion functor" to $$\mathbf{C}$$ as defined above), and that include certain morphisms (and you may also specify that they should exclude "isolated objects", or exclude certain morphisms/objects, if you want); and whose arrows (for these family of categories viewed as a subcategory of $$\mathbf{Cat}$$) are the inclusion functors (or any particular family of functors that you wish to allow).