Categorical Interpretation of Strongest/Weakest Topology One way to define the product topology is as the weakest topology for which all projection maps are continuous. Strongest/weakest topologies satisfying a given property are ubiquitous in topology and functional analysis: for example, the subspace topology can also defined as the weakest topology making all inclusion maps continuous.
If we define a partial order under inclusion on the collection of all possible topologies for a given set, then the strongest/finest topology satisfying a given property is the least upper bound of the subset of topologies satisfying that property, and the weakest/coarsest topology satisfying a given property is the greatest lower bound of the subset of topologies satisfying that property.
I know that a binary join (least upper bound) is a category theoretic binary coproduct, and that a binary meet (greatest lower bound) is a category theoretic binary product. And category theoretic products/coproducts are defined via a universal mapping property, thus they are a type of generalization of terminal/initial objects, respectively.
In fact, when the condition the topologies need to satisfy is the continuity of a family of functions $f$, then the strongest topology satisfying that property is called the final topology, which would correspond nicely to it being a join/coproduct/terminal object.
Likewise, weak topologies are also called initial topologies.

Question: So are strongest/weakest topologies just a special case of initial/terminal objects, respectively?

I believe this is a generalization of the following questions:
Request for gentle explanation of defining a topology with its universal property
Proof: Categorical Product = Topological Product
Is my understanding of product sigma algebra (or topology) correct?
 A: Yes, the final topology is the final object in the category of topologies on the given set making the given functions continuous, which is the poset whose objects are such topologies and whose maps are inclusions of sets, and dually for the initial topology. These words weren't necessarily invented to agree with each other, though. Arbitrary limits and colimits, a very special case of which is your lower and upper bounds in posets, are actually special cases of initial and final objects.
A: The answer to your question is kind of: strongest and weakest topologies are certainly terminal and initial objects in certain categories, but they are required to satisfy an additional property. 
What I'm writing is partially taken from the Joy of Cats which is freely available online, and partially taken from the nlab.
Explicitly, the category of topological spaces $\mathbf{Top}$ has a "forgetful" functor $\mathbf{Set}\xleftarrow{U}\mathbf{Top}$. 
It is a property of the functor $\mathbf{Set}\xleftarrow{U}\mathbf{Top}$ that initial topologies exist. This is formalized as follows.
A source in a category is simply an object $X$ together with a (possibly empty, possibly as large as a proper class) collection of morphisms $X\xrightarrow{\phi_i} X_i$. Given a functor $\mathcal C\xleftarrow{U}\mathcal D$, a source $X\xrightarrow{\phi_i} X_i$ in $\mathcal D$ is called $U$-initial if for every other source $Y\xrightarrow{\psi_i} X_i$, each morphism $UY\xrightarrow{f}UX$ factoring simultaneously $UY\xrightarrow{U\psi_i}UX_i$ as $UY\xrightarrow{f}UX\xrightarrow{U\phi_i}UX_i$ is the image under $U$ of a unique morphism $Y\xrightarrow{\phi}X$.
In the case where the functor is $\mathbf{Set}\xleftarrow{U}\mathbf{Top}$, a $U$-initial source is a collection of continuous maps $X\xrightarrow{\phi_i} X_i$ so that if the underlying set-functions of some other collection of continuous maps $Y\xrightarrow{\psi_i} Y_i$ factor as $Y\xrightarrow{f}X\xrightarrow{\phi_i}X_i$ for a set-function $Y\xrightarrow{f}X$, then $Y\xrightarrow{f}X$ is actually a continuous map $Y\xrightarrow{\phi}X$. In other words, a set-function $Y\xrightarrow{f}X$ is continuous if and only if the composites $Y\xrightarrow{f}X\xrightarrow{\phi_i}X_i$ are continuous, which says exactly that $X$ has the weakest topology for the collection of continuous maps $X\xrightarrow{\phi_i} X_i$.
An important special case are $U$-initial empty sources, i.e. when $X$ in $\mathcal D$ is a $U$-initial source by itself. This means that for any object $Y$ in $\mathcal D$, a morphism $UY\xrightarrow{f}UX$ is the image of a unique morphism $Y\xrightarrow{\phi}X$. In the case of $\mathbf{Set}\xleftarrow{U}\mathbf{Top}$, a topological space $X$ is a $U$-initial empty source if and only if it has the indiscrete topology.

Can we describe a $U$-initial source as an initial object in some category? Yes, but a) it will be more naturally described as a terminal object, and b) we have to remember that the reason we care about initial topologies is that we can put an initial topology on a set $X$ that did not have it before! and c) only terminal objects satisfying an additional condition will be $U$-initial sources.
To see that it is better to consider a $U$-initial source as a terminal object, consider again a $U$-initial empty source $X$ in the case $\mathbf{Set}\xleftarrow{U}\mathbf{Top}$, which is necessarily the indiscrete topology. It can alternatively be defined by the requirement that every set-function to $X$ is continuous; the fact that every set has an indiscrete topology, which is a terminal object, is related to the fact that taking the indiscrete objects gives a right adjoint functor to the forgetful $\mathbf{Set}\xleftarrow{U}\mathbf{Top}$, as the values of right adjoints in general are given by terminal objects.
Let's actually see what category the $U$-initial source is a terminal object of.
A $U$-source for a functor $\mathcal C\xleftarrow{U}\mathcal D$ is an object $A$ of $\mathcal D$ and collection of morphisms $A\xrightarrow{f_i}UX_i$ in $\mathcal C$. A lift of a $U$-source is a source $Y\xrightarrow{\psi_i}X_i$ and a morphism $UY\xrightarrow{f}A$ through which $UY\xrightarrow{U\psi_i}UX_i$ jointly factor as $UY\xrightarrow{f}A\xrightarrow{f_i}UX_i$. A morphism of lifts from $Y_1\xrightarrow{\phi_i}X_i$ with $UY_1\xrightarrow{f}A$ to $Y_1\xrightarrow{\psi_i}X_i$ with $UY_2\xrightarrow{g}A$ is a morphism $Y_1\xrightarrow{\chi}Y_2$ in $\mathcal D$ with $\phi_i=\psi_i\circ\chi$ and $f=U\chi\circ g$.
Going back to the case where the functor is $\mathbf{Set}\xleftarrow{U}\mathbf{Top}$, we see that a $U$-source is a set $X$ with set-functions $X\xrightarrow{f_i}X_i$ to topological spaces $(X_i,\tau_i)$. A lift in turn consists of continuous functions $(Y,\tau)\xrightarrow{\phi_i}X_i$ whose underlying set-functions jointly factor through as $Y\xrightarrow{f}X\xrightarrow{f_i}X_i$. Each such lift is an approximation of the topological space of $X$ equipped with the weakest topology, and the definition of morphisms of lifts ensures that the terminal object among such lifts are the ``closest'' approximation to putting a topology on $X$ so that the $X\xrightarrow{f_i}X_i$ will be continuous. The fact that weakest topologies exist says that the best approximation is, as a set, in bijection with $X$; this is a property of the best approximation in addition to the requirement that it be a terminal object. 
I now claim that $X\xrightarrow{\phi_i}X_i$ is a $U$-initial source if and only if the $U$-source $UX\xrightarrow{U\phi_i}UX_i$ has a lift $Y\xrightarrow{\psi_i}X_i$ with $UY\xrightarrow{f}UX$ that is a terminal object among lifts, and is such that $UY\xrightarrow{f}UX$ is an isomorphism.
To see this, note first that a lift of the $U$-source $UY\xrightarrow{U\phi_i}UX_i$ is a source $Y\xrightarrow{\psi_i}X_i$ equipped with a morphism $UY\xrightarrow{f}UX$ so that $U\psi_i=U\phi_i\circ f$. Then observe that the defining property of $X\xrightarrow{\phi_i}X_i$ being a $U$-initial source says that any lift $Y\xrightarrow{\psi_i}X_i$ with $UY\xrightarrow{f}UX$ has a unique morphism of lifts $Y\xrightarrow{\phi}X$ to the lift $X\xrightarrow{\phi_i}X_i$ with $UX\xrightarrow{\mathrm{id}_{UX}}UX$, i.e. that $X\xrightarrow{\phi_i}X_i$ with $UX\xrightarrow{\mathrm{id}_{UX}}UX$ is a terminal objects among lifts of $UX\xrightarrow{U\phi_i}UX$.
Since terminal objects are unique up to isomorphism, we see that a lift $X\xrightarrow{\phi_i}X_i$ with $UX\xrightarrow{f}A$ that is a terminal object of the $U$-source $A\xrightarrow{f_i}UX_i$ is a $U$-initial source if and only if $UX\xrightarrow{f}A$ is an isomorphism; this is the special condition I mentioned earlier.

A functor $\mathcal D\xleftarrow{U}\mathcal C$ is called topological if every $U$-source (including empty ones, including ones indexed by classes) has a $U$-initial lift. Amazingly, one can prove that a) topological functors are faithful, i.e. are injective on morphisms between a fixed pair of objects, and b) are self-dual in that every $U$-sink has a $U$-final lift. The Joy of Cats defines a topological category to satisfy the extra assumption of uniqueness of the lifts, but the proofs don't actually use that assumption for either a) or b).
